| ## Page 1 | |
| 948 | |
| Fibonaccimal Base | |
| The well known Fibonacci sequence is obtained by starting with 0 and 1 and then adding the two last | |
| numbers to get the next one. For example the third number in the sequence is 1 (1=1+0), the forth is | |
| 2 (2=1+1), the fifth is 3 (3=2+1) and so on. | |
| i | |
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 | |
| 9 | |
| Fib(i) | |
| 0 | |
| 1 | |
| 1 | |
| 2 | |
| 3 | |
| 5 | |
| 8 | |
| 13 | |
| 21 | |
| 34 | |
| Figure 1 - The first numbers in the Fibonacci sequence | |
| The sequence appears on many things in our life, in nature, and has a great significance. Among | |
| other things, do you know that all positive integer numbers can be represented as a sum of numbers in | |
| the Fibonacci sequence? More than that, all positive integers can be represented as a sum of a set of | |
| Fibonacci numbers, that is, numbers from the sequence, without repetition. For example: 13 can be the | |
| sum of the sets {13}, {5,8} or {2,3,8} and 17 is represented by {1,3,13} or {1,3,5,8}. Since all numbers | |
| have this property (do you want to try to prove this for yourself?) this set could be a nice way to use as | |
| a "base" to represent the number. But, as we have seen, some numbers have more than one set whose | |
| sum is the number. How can we solve that? Simple! If we add the constraint that the sets cannot | |
| have two consecutive Fibonacci numbers, than we have a unique representation for each number! This | |
| restriction is because the sum of any two consecutive Fibonacci numbers is just the following Fibonacci | |
| number. | |
| Now that we know all this we can prepare a nice way to represent any positive integer. We will use | |
| a binary sequence (just zeros and ones) to do that. For example, 17 = 1 + 3 + 13 (remember that no | |
| two consecutive Fibonacci numbers can be used). Let’s write a zero for each Fibonacci number that is | |
| not used and one for each one that is used, starting at the right. Then, 17 = 100101. See figure 2 for a | |
| detailed explanation. In this representation we should not have zeros at the left, this is, we should only | |
| write starting with the first one. In order for you to understand better, note that in this scheme, not | |
| using two consecutive Fibonacci numbers means that the binary sequence will not have two consecutive | |
| ones. When we use this representation for a number we say that we are using the Fibonaccimal base, | |
| and we write it like 17 = 100101 (fib). | |
| 17 = | |
| 1 | |
| 0 | |
| 0 | |
| 1 | |
| 0 | |
| 1 | |
| 13+3+1 = | |
| 13 | |
| 8 | |
| 5 | |
| 3 | |
| 2 | |
| 1 | |
| Figure 2 - Explaining the representation of 17 in Fibonaccimal base | |
| Given a set of numbers in decimal base, your task is to write them in the Fibonaccimal base. | |
| Input | |
| The first line of input contains a single number N, representing the quantity of numbers that follow | |
| (1 ≤N ≤500). | |
| Than follow exactly N lines, each one containing a single positive integer smaller than 100 000 000. | |
| These numbers can come in any order. | |
| Output | |
| You should output a single line for each of the N integers in the input, with the format ‘DEC BASE | |
| = FIB BASE (fib)’. DEC BASE is the original number in decimal base and FIB BASE is its | |
| representation in Fibonaccimal base. See the sample output for an example. | |
| ## Page 2 | |
| Sample Input | |
| 10 | |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 | |
| 9 | |
| 10 | |
| Sample Output | |
| 1 = 1 (fib) | |
| 2 = 10 (fib) | |
| 3 = 100 (fib) | |
| 4 = 101 (fib) | |
| 5 = 1000 (fib) | |
| 6 = 1001 (fib) | |
| 7 = 1010 (fib) | |
| 8 = 10000 (fib) | |
| 9 = 10001 (fib) | |
| 10 = 10010 (fib) |