The easy subproblem involves a sorting routine that takes an n-element array and applies the following steps:
- Compute some a, b, c that are guaranteed to be in [0, n-1].
- Make a recursive call to sort the first a elements.
- Make a recursive call to sort the last b elements.
- Make a recursive call to sort the first c elements.

A special case makes the solution work for n ≤ 2. To solve this task without fully understanding the algorithm, we can use inductive reasoning. We start by taking an educated guess that the worst possible input will be an array sorted in reverse order. However, the algorithm is inefficient and would take a long time to terminate for larger values of n.

To optimize, we can use a standard O(n log n)-time sort instead of making recursive calls. This change makes the routine run in O(n log n), and the search for the smallest counterexample will run in O(n^2 log n). The steps are:
- Take the given code and change the recursive calls to calls to sort().
- Feed a reversed array into the modified function for each n = 1, 2, … .
- Stop once the smallest n where the modified function fails is found.

The algorithm is a generalization of the StoogeSort algorithm, where a = b = c = ⌈2n/3⌉. To understand why the algorithm works, we can draw a picture and use it to reason. The algorithm correctly sorts the array if the overlap between the first and second passes is at least as large as the total number of large elements. This condition is represented by the equation a + b + c ≥ 2n.

For the smallest n such that a + b + c < 2n, taking a reversed array as a counterexample is sufficient. The program for this subproblem generates an awful lot of swaps and conditional swaps, which are the key ingredient in sorting networks. Evaluating whether a sorting network sorts is a hard problem, but the 0-1 principle can be used to test it.

Instead of trying all possible permutations of the input, it is enough to try all possible vectors of zeros and ones. This brings the time complexity of the test from O(n!) down to O(2^n). By examining the sorting network and modifying the program to print all the swaps it performs, we can draw the comparators and see something that closely resembles a bitonic sorter.

We can divide the wires into the top half and the bottom half and determine the possible outputs of the network before the big round. We then get a restricted set of possibilities for the input of the big round. By checking two things - if there is a 1 in the top half and a 0 in the bottom half, and if not, simulating the half that contains both zeros and ones - we can find that the smallest n for which the network fails is n = 47.

An alternate solution involves making an educated guess that a bad input for a network of mostly-random comparators could be a reversed array or something similar. We can precompute the permutation done by the first part of the algorithm and compute its inverse. This tells us the order in which to provide inputs to get a reversed array just before the swaps turn into comparators. By generating lots of inputs that resemble the reversed array and feeding them into the implementation, we can find the smallest n for which the network fails.

The provided Python 3 programs can be used to check submissions for the given problems. The programs can be used to test the solutions and verify their correctness.

One possible correct output for K1 is the smallest n for which the modified function fails.
One possible correct output for K2 is the smallest n for which the network fails.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter.

The time complexity of the algorithm is O(n log n), and the search for the smallest counterexample has a time complexity of O(n^2 log n). The space complexity is O(n).

The algorithm and the programs can be used to solve the given problems and find the correct outputs. The solutions involve using inductive reasoning, optimizing the algorithm, and examining the sorting network. The programs can be used to test the solutions and verify their correctness.

The key concepts involved in the solution are the StoogeSort algorithm, the 0-1 principle, and the bitonic sorter. The solution involves using these concepts to understand why the algorithm works and to find the smallest n for which the network fails.

The programs provided can be used to check submissions for the given problems and to test the solutions. The solutions involve using inductive reasoning, optimizing the algorithm,


## Lesson Objectives
- Learn how to determine if three points are collinear using determinants.

## How to Determine if Three Points are Collinear Using Determinants
In the last lesson, we learned how to determine the area of a triangle using determinants. Now, we will see another application of this formula. If our formula yields a result of zero, the three points given are collinear, meaning they lie on the same line.

## Example #1: Determining Collinearity
Given points: $$(4, 0)$$, $$(9, 3)$$, $$(-1, -3)$$.
Let's label and plug into the formula:
$$\text{Point 1}: (4, 0)$$,
$$\text{Point 2}: (9, 3)$$,
$$\text{Point 3}: (-1, -3)$$.
The determinant is:
$$\left| \begin{array}{ccc}4&0&1\\9&3&1\\-1&-3& 1\end{array}\right|=0$$.
Since the result is zero, these points are collinear.

## Test for Collinear Points
The formula to check if three points $$(x_{1}, y_{1})$$, $$(x_{2}, y_{2})$$, and $$(x_{3}, y_{3})$$ are collinear is:
$$\left| \begin{array}{ccc}x_{1}&y_{1}&1\\ x_{2}& y_{2}& 1\\ x_{3}& y_{3}& 1\end{array}\right|=0$$.

## Skills Check:
### Example #1
Determine if $$(-2, -2), (1, -1), (7, 5)$$ are collinear.
Please choose the best answer.
A) Yes
B) No

### Example #2
Determine if $$(3, 0), (0, -12), (1, -8)$$ are collinear.
Please choose the best answer.
A) Yes
B) No

### Example #3
Determine if $$(15, 3), (2, -1), (-3, 5)$$ are collinear.
Please choose the best answer.
A) Yes
B) No


**Dynamics and Symmetry**

This textbook introduces advanced undergraduate and beginning graduate students in mathematics, physics, and engineering to the tools and language of modern geometric mechanics. The prerequisites include linear algebra, calculus, and familiarity with Euler-Lagrange variational principles and canonical Poisson brackets in classical mechanics.

The book treats the dynamics of ray optics, resonant oscillators, and the elastic spherical pendulum from a unified geometric viewpoint, using reduction by Lie group symmetries. Key concepts include Fermat's principle, differential forms on manifolds, Lie-Poisson Hamiltonian formulations, and momentum maps. The text features explicit examples, theorems, and proofs, as well as numerous exercises and worked answers to aid student understanding.

The contents include:
- Fermat's Ray Optics
- Newton, Lagrange, and Hamilton
- Differential Forms
- Resonances and S1 Reduction
- Elastic Spherical Pendulum
- Maxwell-Bloch Equations

An appendix provides introductory examples and enhanced coursework problems with worked answers to help students develop proficiency in geometric mechanics.

**Rotating, Translating, and Rolling**

This textbook introduces advanced undergraduate and beginning graduate students in mathematics, physics, and engineering to the tools and language of modern geometric mechanics. The prerequisites include linear algebra, multivariable calculus, and familiarity with Euler-Lagrange variational principles and canonical Poisson brackets in classical mechanics.

The book treats the dynamics of rotating, spinning, and rolling rigid bodies from a geometric viewpoint, using coadjoint motions generated by Lie groups. Key concepts include variational calculus on tangent spaces of Lie groups, quaternions, rotations, adjoint and coadjoint motions, Lie-Poisson Hamiltonian formulations, momentum maps, and dynamics with nonholonomic constraints.

The text features 120 exercises and 55 worked answers to help students grasp the essential aspects of the subject and develop proficiency in geometric mechanics. All theorems are stated and proved explicitly. The contents include:
- Galileo
- Newton, Lagrange, Hamilton
- Quaternions
- Quaternionic Conjugacy
- Special Orthogonal Group
- The Special Euclidean Group
- Geometric Mechanics on SE(3)
- Heavy Top Equations
- The Euler-Poincare Theorem
- Lie-Poisson Hamiltonian Form
- Momentum Maps
- Round Rolling Rigid Bodies

**Geometric Mechanics and Symmetry**

Geometric Mechanics and Symmetry is a comprehensive introduction to the geometric approach to classical mechanics, suitable for beginning graduate students or advanced undergraduates. The text fills the gap between traditional classical mechanics texts and advanced modern mathematical treatments of the subject.

After summarizing the necessary elements of calculus on smooth manifolds and basic Lie group theory, the text considers symmetry reduction of Hamilton's principle and the derivation of the Euler-Poincare equations for dynamics on Lie groups. Additional topics include rigid and pseudo-rigid bodies, the heavy top, shallow water waves, geophysical fluid dynamics, and computational anatomy.

The text features a variety of examples, figures, and exercises, making it a valuable class text. The book provides a comprehensive presentation of ideas and methods from geometric mechanics, aimed at the graduate student level, and is also suitable for specialists looking to refresh their knowledge in this modern and elegant formulation of Lagrangian and Hamiltonian mechanics.


**Naive Bayes** is a machine learning algorithm used for classification tasks. It is based on Bayes' theorem and assumes that features are conditionally independent of each other given the class label. The algorithm calculates the probability of a data point belonging to each class and assigns it to the class with the highest probability.

Naive Bayes is known for its simplicity, efficiency, and effectiveness in handling high-dimensional data. It is commonly used in various applications, including text classification, spam detection, and sentiment analysis.

The Naive Bayes theorem is used to find the probability of a hypothesis with given evidence. The equation for Bayes theorem is:

P(A|B) = P(B|A) * P(A) / P(B)

Where:
- P(B|A) is the probability of B given that A is True.
- P(A) and P(B) are the independent probabilities of A and B.

**Learning Objectives**

- Learn the concept behind the Naive Bayes algorithm.
- See the steps involved in the naive Bayes algorithm
- Practice the step-by-step implementation of the algorithm.

The Naive Bayes classifier algorithm works by calculating the posterior probability of a data point belonging to each class. The class with the highest posterior probability is chosen as the predicted class.

To understand the concept of the Naive Bayes Theorem, let's consider an example. Suppose we have a dataset of employees in a company, and we want to create a model to predict whether a person is going to the office by driving or walking using their salary and age.

The posterior probability of walking for a new data point is calculated using the Bayes theorem. The steps involved in calculating the posterior probability are:

**Step 1**: Find all the probabilities required for the Bayes theorem.
**Step 2**: Calculate the posterior probability of driving.
**Step 3**: Compare both posterior probabilities and choose the class with the highest probability.

To implement Naive Bayes in Python, we can use the following steps:

**Step 1: Importing the libraries**
```python
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import sklearn
```

**Step 2: Importing the dataset**
```python
from sklearn.preprocessing import LabelEncoder
le = LabelEncoder()
X[:,0] = le.fit_transform(X[:,0])
```

**Step 3: Train test splitting**
```python
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.20, random_state = 0)
```

**Step 4: Feature scaling**
```python
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)
```

**Step 5: Training the Naive Bayes model**
```python
from sklearn.naive_bayes import GaussianNB
classifier = GaussianNB()
classifier.fit(X_train, y_train)
```

**Step 6: Predicting the test results**
```python
y_pred = classifier.predict(X_test)
```

The Naive Bayes algorithm makes several assumptions, including:

- The features are conditionally independent of each other.
- Each feature is equal in terms of weightage and importance.
- The features follow a normal distribution.
- There is no or almost no correlation among features.

The naive Bayes algorithm is a powerful and widely-used machine learning algorithm that is particularly useful for classification tasks. Its simplicity and efficiency make it a popular choice for many data science applications.

**Key Takeaways**

- Naive Bayes is a probabilistic classification algorithm that is based on Bayes' theorem.
- There are different variants of Naive Bayes, which can be used for different tasks and can even be used for regression problems.
- Naive Bayes can be used for a variety of applications, such as spam filtering, sentiment analysis, and recommendation systems.

Q: What is the main assumption of the Naive Bayes algorithm?
A: The main assumption is that the features are conditionally independent of each other.

Q: What is the difference between Bayes theorem and the Naive Bayes algorithm?
A: Bayes theorem provides a way to calculate the conditional probability of an event based on prior knowledge of related conditions. The Naive Bayes algorithm, on the other hand, is a machine learning algorithm that is based on Bayes' theorem, which is used for classification problems.


## Calculating Mold Tonnage in Injection Molding

To calculate mold tonnage, use the formula T = A x cf, where T is the tonnage, A is the projected area of the molded part, and cf is the clamp factor. For example, if the projected area is 150 sq/in and the clamp factor is 3, the tonnage needed is 150 x 3 = 450 tons.

## Capacity of Injection Molding Machine

The capacity of an injection molding machine is typically measured in tons per day. Common capacities range from 1 to 250 tons per day. The clamping force of the machine can range from 0-100 tons to 100-300 tons. The machine type can be hydraulic, hydraulic and mechanical composite, or other types, and can process various raw materials such as HDPE, PE, PP, and PVC.

## Largest Injection Molding Machine

The largest injection molding machine is the "Cincinnati" by Milacron, with a clamping power of 2250 tons.

## Measuring Injection Mold Size

To measure injection mold size, most molders recommend 2.5 times the surface square inches of the part, plus an additional 10% as a safety factor. For example, a part with a surface area of 120 square inches would require a press size with 300 tons of pressure.

## Tonnage in Injection Molding

Tonnage refers to the measure of force required to keep the tool closed during the injection process. This force is the main factor in determining the injection molding machine size required for producing a part.

## Calculating Mold Cost

The cost of a mold is calculated using the formula: Mould price = material cost + design + process and profit + VAT + tryout cost + packing and shipping cost. The material cost typically accounts for 15-30% of the total cost, while the process and profit account for 30-50%.

## Barrel Capacity

Barrel capacity refers to a volume of 42 US gallons, which is equivalent to 158.987294928 liters.

## Types of Injection Molds

There are several types of injection molds, including:
- Cube molding
- Die casting
- Gas-assisted injection molding
- Liquid silicone rubber injection molding
- Metal injection molding
- Micro injection molding
- Reaction injection molding
- Thin-wall injection molding

## Calculating Molding

To calculate molding, divide the total number of inches by 12 to convert from inches to linear feet. For example, if a wall measures 40 inches in length and molding is being installed on the top and bottom, the total linear feet would be 40 x 2 = 80 inches, divided by 12, which equals 6.66 linear feet.

## Tonnage Calculation

Tonnage can be calculated using different formulas, depending on the context. In shipping, gross tonnage is calculated using the formula GT = K1V, where V is the volume of a ship's enclosed spaces in cubic meters and K1 is a constant. In injection molding, tonnage is calculated using the formula T = A x cf.

## Tonnage Factor

The tonnage factor provides a mechanism for converting volume to weight. In the English system, it is expressed as cubic feet per ton of ore, while in the metric system, it is the specific gravity of the ore.

## Calculating Injection Molding Cost

The cost of injection molding is calculated using the formula: price = material cost + manufacturing cost + overhead cost + profit + tax + technical value. The mold cost is a significant component of the total cost and is calculated using the formula: mold cost = material cost + manufacturing cost + overhead cost.


# Slope

In mathematics, the **slope** or **gradient** of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or the ratio of the altitude change to the horizontal distance between any two points on the line.

## Definition

The slope of a line is generally represented by the letter *m*, and is defined as the change in the *y* coordinate divided by the corresponding change in the *x* coordinate, between two distinct points on the line. Given two points (*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}), the slope *m* of the line is calculated as:

*m* = (*y*_{2} − *y*_{1}) / (*x*_{2} − *x*_{1})

## Examples

Suppose a line runs through two points: *P*(1,2) and *Q*(13,8). The slope of the line is calculated as:

*m* = (8 - 2) / (13 - 1) = 6 / 12 = 0.5

As another example, consider a line which runs through the points (4, 15) and (3, 21). The slope of the line is calculated as:

*m* = (21 - 15) / (3 - 4) = 6 / -1 = -6

## Geometry

The greater the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. A vertical line's slope is undefined.

Two lines are parallel if and only if their slopes are equal and they are not coincident or if they both are vertical and therefore have undefined slopes. Two lines are perpendicular if the product of their slopes is -1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line).

## Slope of a Road or Railway

The slope of a road or railway can be described by the angle in degrees, the slope in a percentage, or by giving one unit of rise in a certain number of horizontal units. The formulae for converting a slope as a percentage into an angle in degrees and vice versa are:

angle = arctan(slope/100)
slope = 100 * tan(angle)

## Algebra

If *y* is a linear function of *x*, then the coefficient of *x* is the slope of the line created by plotting the function. The equation of the line can be written in the slope-intercept form:

*y* = *m*x* + *b*

where *m* is the slope and *b* is the y-intercept.

## Calculus

The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point.

## Terrain Slope Analysis in GIS

In GIS mapping, slope can be calculated from a Digital Elevation Model (DEM) and is important for suitability analysis, predictive modeling, and predicting potential hazards. Slope is calculated by comparing a certain point within a raster to that point's neighbors.

## GIS&T Body of Knowledge Concept

This topic is part of the 2006 GIS&T Body of Knowledge, specifically AM6-1 (Calculating surface derivatives), AM6-4 (Intervisibility), CV4-4 (Representing terrain), and CV6-4 (Map analysis).

## Notes

References:
- ESRI. "ArcGIS Desktop Help 9.2 - Using the ArcGIS Spatial Analyst Toolbar to Calculate Slope."
- University of Colorado Geography. "GIS Modeling, Class 6: Terrain Analysis"
- Dunn M, Hickey R. "The effect of slope algorithms on slope estimates within a GIS"
- Duke University Geography Department. "Spatial Analysis; Slope and Aspect"
- DiBiase D, DeMers M, Johnson A, Kemp K, Luck T, Plewe B, Wents E. "Geographic Information Science & Technology Body of Knowledge"

## See also

- The gradient is a generalization of the concept of slope for functions of more than one variable.
- Slope definitions

## External links

- Interactive applet demonstrates how to calculate slope of a line