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How is prisoner's dilemma different from chicken? |
A probability problem I love.
Take a shuffled deck of cards. Deal off the cards one by one until you reach any Ace. Turn over the next card, and note what it is.
**The question**: which card has a higher probability of being turned over, the Ace of Spades or the Two of Hearts? |
Okay, so this question was bound to come up sooner or later- the hope was to ask it well before someone asked it badly...
###We all love a good puzzle
To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculati... |
Can someone help me with a math/logic problem my friend and I are struggling with? Dealing with poisoned wines and servants |
I covered homotopy theory in a recent maths course. However I was never presented with any reasons as to why (or even if) it is useful.
Is there any good examples of its use outside academia? |
I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented:
>The King has $1000$ wines, $1$ of which is poisoned. He needs to identify the poisoned wine as soon as possible, and with the least resources, so he hires the protagonist, a Mathematician.... |
Can someone help me with a math/logic problem my friend and I are struggling with? Dealing with identifying poisoned wines with minimal servants. |
I've learned that multiplying two generating functions f(x) and g(x) gives you the result $\sum_{k=0}^\infty\left(\sum_{j=0}^k a_j\,b_{k-j}\right)x^k$.
I've used the result, but it was presented in my class without proof and I'm having some trouble tracking one down. Weak google-foo today, I suppose. Can anyone gi... |
One thing I have been wondering about lately is how to write an equation that describes a pattern of numbers. What I mean is:
x 0 1 2
y 1 5 9
If I have this, I can tell that an equation that describes this would be y=4x+1. In fact, I don't even need the 3rd pair of numbers. It's very eas... |
How can I write an equation that matches any sequence? |
EDIT: For some reason, the jsMath plugin isn't displaying plus signs. How odd.
If you know your relationship is going to be a **polynomial**, then there are some pretty (conceptually) simple ways you can do this.
If you know what degree your polynomial is (line, parabola, cubic, etc.) then your job will be much ... |
###Motivatation
I am presently looking at a structure that I am trying to pin down- my strategy being to pull the thing up into the greatest possible generality (based on the bits I'm sure about) and narrow it down from there.
The situation I have is somewhat similar to the dual space structure in vector spaces,... |
I believe the right generalization of dual spaces is that of a [dual object][1] in a [tensor category][2], which I will assume symmetric for convenience.
Recall what makes a dual space of a vector space work: We have a map $V \times V^* \to k$ (for $k$ the ground field). The problem is, this isn't a homomorphism in ... |
If you know your relationship is going to be a **polynomial**, then there are some pretty (conceptually) simple ways you can do this.
If you know what degree your polynomial is (line, parabola, cubic, etc.) then your job will be much easier. But if not, then you simply need to look at the number of points you have.... |
If you know your relationship is going to be a **polynomial**, then there are some pretty (conceptually) simple ways you can do this.
If you know what degree your polynomial is (line, parabola, cubic, etc.) then your job will be much easier. But if not, then you simply need to look at the number of points you have.... |
I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented:
>The King has $1000$ wines, $1$ of which is poisoned. He needs to identify the poisoned wine as soon as possible, and with the least resources, so he hires the protagonist, a Mathematician.... |
If you know your relationship is going to be a **polynomial**, then there are some pretty (conceptually) simple ways you can do this.
If you know what degree your polynomial is (line, parabola, cubic, etc.) then your job will be much easier. But if not, then you simply need to look at the number of points you have.... |
I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented:
>The King has $1000$ wines, $1$ of which is poisoned. He needs to identify the poisoned wine as soon as possible, and with the least resources, so he hires the protagonist, a Mathematician.... |
I just came back from my Introduction to Rotational Kinematics class, and one of the important concepts they described was *Rotational Inertia*, or *Moment of Inertia*.
It's basically the equivalent of mass in Netwon's $F = m a$ in linear motion. The equivalent rotational equation is $\tau = I \alpha$, where $\tau$... |
Finding Moment of Inertia (rotional inertia?) $I$ using integration? |
I have heard that one can think of stacks on a site as taking sheaves but instead of the restrictions being equal, we just loosen it to isomorphic, and treat the sheaf conditions with the "obvious" coherence relations.
How seriously can one take this analogy? Note that my background is stacks is feeble at best.
I... |
In my linear algebra class we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a 2 by 2 matrix by the formula. Our teacher showed us how to compute the d... |
What's an intuitive way to think about the determinant? |
Your trouble with determinants is pretty common. They’re a hard thing to teach well, too, for two main reasons that I can see: the formulas you learn for computing them are messy and complicated, and there’s no “natural” way to interpret the value of the determinant, the way it’s easy to interpret the derivatives you d... |
In my AP chem class, I often have to balance chemical equations like the following:
Al + O<sub>2</sub> ---> Al<sub>2</sub>O<sub>3</sub>
The goal is to make both side of the arrow have the same amount of atoms by adding compounds in the equation to each side.
An solution
4Al + 3O<sub>2</sub> ---> 2Al<sub>2</... |
I just came back from a class on Probability in Game Theory, and was musing over something in my head.
Assuming, for the sake of the question:
- Playing cards in their current state have been around for approximately eight centuries
- A deck of playing cards is shuffled to a random configuration one billion time... |
When you randomly shuffle a deck of cards, what is the probability that it is a unique permutation never before configured? |
I just came back from a class on Probability in Game Theory, and was musing over something in my head.
Assuming, for the sake of the question:
- Playing cards in their current state have been around for approximately eight centuries
- A deck of playing cards is shuffled to a random configuration one billion time... |
Randomly break a stick (or a piece of dry spaghetti, etc.) in two places, forming three pieces. The probability that these three pieces can form a triangle is 1/4 (coordinatize the stick form 0 to 1, call the breaking points x and y, consider the unit square of the coordinate plane, shade the areas that satisfy the tr... |
About two years ago, I started thinking about the following problem: You're given an N and an S, positive integers. You start with an N-gon that has positive integer labels at each vertex, such that the labels sum to S. A "move" is replacing the label at some vertex with the positive difference of the labels at its two... |
How do Lagrange multipliers work to find the lowest value of a function subject to a constraint? |
Your trouble with determinants is pretty common. They’re a hard thing to teach well, too, for two main reasons that I can see: the formulas you learn for computing them are messy and complicated, and there’s no “natural” way to interpret the value of the determinant, the way it’s easy to interpret the derivatives you d... |
Two mathematicians each come into a coffee shop at a random time between 8:00 a.m. and 9:00 a.m. each day. Each orders a cup of coffee then sits at a table, reading a newspaper for 20 minutes before leaving to go to work.
On any day, what is the probability that both mathematicians are at the coffee shop at the same... |
Randomly break a stick (or a piece of dry spaghetti, etc.) in two places, forming three pieces. The probability that these three pieces can form a triangle is 1/4 (coordinatize the stick form 0 to 1, call the breaking points x and y, consider the unit square of the coordinate plane, shade the areas that satisfy the tr... |
What property of certain regular polygons allows them to be faces of the Platonic Solids? |
How can I randomly generate trees? |
Knuth says to look at it as generating all nested parentheses in lexicographic order.
Look here for the details
http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBIQFjAA&url=http%3A%2F%2Fwww-cs-faculty.stanford.edu%2F~uno%2Ffasc4a.ps.gz&ei=l_NLTOagLIHCsAPS4oHkCw&usg=AFQjCNGFPnsHAYgcsoaKZ2ieiwJ-pn6hiw&sig2=-dVe... |
Suppose that I am buying cakes for a party. There are k different types and I intend to buy a total of n cakes. How many different combinations of cakes could I possibly bring to the party? |
Suppose we model traffic flow between two points with a directed graph. Each route has a maximum capacity of how much traffic it can hold and how long it takes to travel that route. We assume that each driver wishes to minimise their own travel time and we assume that the drivers form a Nash equilibria. Can removing a ... |
Complex numbers are the final step in a sequence of increasingly "unreal" extensions to the number system that humans have found it necessary to add over the centuries in order to express significant numerical concepts.
The first such "unreal number" was zero, back in the mists of time. It seems obvious to us now, b... |
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain **to a non-mathematician** that complex numbers are necessary and m... |
Do complex numbers really exist? |
I need n cakes for a party. I go to the cake shop and there are k different kinds of cake. For variety, I'd like to get at least one of every cake. How many ways can I do this? |
In least-squares approximations the <a href="">normal equations</a> act to project a vector existing in N-dimensional space onto a lower dimensional space, where our problem actually lies, thus providing the "best" solution we can hope for (the orthogonal projection of the N-vector onto our solution space). The "best"... |
Are there variations on least-squares approximations? |
Recently I discussed an experiment with a friend. Assume we start a random experiment. At first there is an array with size 100 000, all set to 0. We calculate at each round a random number modulo 2 and select one random position in that array. If the number in the array is 1, nothing is changed and otherwise the pre-c... |
The calculus of relations is an algebra of operations over sets of pairs of individuals, where for any relation R, we can express the relation in the usual infix manner: x R y iff (x,y) ∈ R. This allows all of the properties of relations to be expressed in equational form.
There are three fundamental relations... |
The calculus of relations is an algebra of operations over sets of pairs of individuals, where for any relation R, we can express the relation in the usual infix manner: x R y iff (x,y) ∈ R. This allows all of the properties of relations to be expressed in equational form.
There are four fundamental relations ... |
I've just came back from my Mathematics of Packing and Shipping lecture, and I've run into a problem I've been trying to figure out.
Let's say I have a rectangle of length $l$ and width $w$.
Is there a simple equation that can be used to show me how many circles of radius $r$ can be packed into the rectangle? So... |
How many circles of a given can be packed into a given rectangular box? |
I've just came back from my Mathematics of Packing and Shipping lecture, and I've run into a problem I've been trying to figure out.
Let's say I have a rectangle of length $l$ and width $w$.
Is there a simple equation that can be used to show me how many circles of radius $r$ can be packed into the rectangle? So... |
How many circles of a given radius can be packed into a given rectangular box? |
Couldn't all transformation which send each point (x,y) to another point (x',y') which can be computed from the first one by performing only the four operations and extraction of square root? |
I've just came back from my Mathematics of Packing and Shipping lecture, and I've run into a problem I've been trying to figure out.
Let's say I have a rectangle of length $l$ and width $w$.
Is there a simple equation that can be used to show me how many circles of radius $r$ can be packed into the rectangle, in ... |
Is there symbolism (or even a name) for groups of statements in which any fixed-number of them imply all the rest?
For example, in linear algebra, a basis is sometimes defined as a set of n-dimensional vectors which:
* Is linearly independent
* Spans the vector space
* Contains exactly n elements
However,... |
Given
1. A straight line of arbitrary length
2. The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude yields its own magnitude.
Is there a way to geometrically construct (using only a compass and straightedge) th... |
Compass-and-straightedge construction of the square root of a given line? |
Say there are three jars, $j_1, j_2, j_3$ filled with different binary sequences of length two.
The distribution of the binary sequences in each of the jars is given by the $p_i^k(1-p_i)^{n-k}$, where
$p_i = \frac{i}{m}$ where m is the number of jars, i is the jar index, k is number of 1's and n is the length of... |
Given
1. A straight line of arbitrary length
2. The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude yields its own magnitude.
Is there a way to geometrically construct (using only a compass and straightedge) th... |
I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form
y = a x<sup>2</sup> + b x + c
My teacher gave me the formula:
$x = -\frac{b}{2a}$
as the x coordinate of the vertex.
I asked her why, and she told me not to tell her how to do her job.
My sm... |
I read [this question][1] the other day and it got me thinking: the area of a circle is $\pi r^2$, which differentiates to $2 \pi r$, which is just the perimeter of the circle.
>Why doesn't the same thing happen for squares?
If we start with the area formula for squares, $l^2$, this differentiates to $2l$ which... |
Why does area differentiate to perimeter for circles and not for squares? |
The notion (rank-2) "tensor" appears in many different parts of physics, e.g. stress tensor, moment of inertia tensor, etc.
I know mathematically a tensor can be represented by a 3x3 matrix. But I can't grasp its geometrical picture — unlike scalar (a number) and vector (an arrow with direction and magnitude) which ... |
How to visualize a rank-2 tensor? |
I know that the harmonic series 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n + ... diverges, but what about the alternating harmonic series 1/1 - 1/2 + 1/3 - 1/4 + 1/5 - ... + (-1)^n/n + ...? Does it converge? If so, what is its sum? |
I have two WGS-84 coordinates on the earths surface with times $t$ and $t+1$ and would like to calculate a heading vector for those. How do I accomplish that? |
How to calculate a heading vector on the earths surface? |
I have two WGS-84 coordinates on the earths surface with times $t$ and $t+1$ and would like to calculate a heading vector for those using the "True North" as reference. How do I accomplish that? |
Is there an algorithm for working out the best way (i.e. fewest multiplications) of calculating A^n in a structure where multiplication is associative?
For example, suppose A is a square matrix. Matrix multiplication is associative, and I can compute A^9 with 4 multiplications:
A^2 = A*A
A^3 = A^2*A
... |
There seems to be no efficient algorithm for this. Quoting Wikipedia on [*Addition-chain exponentiation*](http://en.wikipedia.org/wiki/Addition-chain_exponentiation):
> … the addition-chain method is much more complicated, since the determination of a shortest addition chain seems quite difficult: **no efficient opt... |
I had an answer before, but I looked into it a bit more and my answer was incorrect so I removed it. This link may be of interest: [Circle Packing in a Square (wikipedia)][1]
It was suggested by KennyTM that there may not be an optimal solution yet to this problem in general. Further digging into this has shown me t... |
I keep hearing questions about whether something is NP-complete, but they never really mention what it is. Why do people care so much about NP-complete problems? |
What are NP-complete problems and why are they so important? |
To expand on Mau's answer, you should care about NP-complete problems because there is an [entire family][1] of them that spans a large number of seemingly basic algorithms across a wide range of disciplines. These aren't obscure problems, but extremely important and highly practical questions. For examples, consider t... |
To expand on Casebash's answer:
Any problem for which a solution (once found) can be *verified* as a solution quickly is said to be "in **NP**" (Here, "quickly" means in polynomial-time). Any problem for which a solution can be *found* quickly is said to be "in **P**." **P** is a subset of **NP** - that is, any p... |
To expand on Casebash's answer:
Any problem for which a solution (once found) can be *verified* as a solution quickly is said to be "in **NP**" (Here, "quickly" means in polynomial-time). Any problem for which a solution can be *found* quickly is said to be "in **P**." **P** is a subset of **NP** - that is, any p... |
Is there a way of taking a number known to limited precision (e.g. 1.644934) and finding out an "interesting" real number (e.g. $\pi^2/6$) that's close to it?
I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers.
The intended use would be: write a program to calc... |
Can someone point me to a proof that the set of irrational numbers is uncountable. I know how to show that the set **Q** is countable but how would you then show that the irrationals are uncountable? |
Are there any real online mathematics (applied math, statistics, ...) degree programs out there?
I'm full-time employed, thus not having the flexibility of attending an on campus programs. I also already have a MSc in Computer Science. My motivation for a math degree is that I like learning, and am interested in the... |
I was coming back from my Driver's Education class, and something mathsy really stuck out to me.
One of the essential properties of a car is its current speed. Or speed at a current time. For example, at a given point in time in my drive, I could be traveling 40 mph. But what does that *mean*?
From my basic al... |
What does it mean to be going 40 mph (or 64 kph, etc.) at a given moment? |
Are there any real online mathematics (applied math, statistics, ...) degree programs out there?
I'm full-time employed, thus not having the flexibility of attending an on campus program. I also already have a MSc in Computer Science. My motivation for a math degree is that I like learning and am interested in the s... |
I was coming back from my Driver's Education class, and something mathsy really stuck out to me.
One of the essential properties of a car is its current speed. Or speed at a current time. For example, at a given point in time in my drive, I could be traveling 40 mph. But what does that *mean*?
From my basic al... |
To expand on Mau's answer, you should care about NP-complete problems because there is an [entire family][1] of them that spans a large number of seemingly basic algorithms across a wide range of disciplines. These aren't obscure problems, but extremely important and highly practical questions. For examples, consider t... |
What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples?
For instance, if f'(a) = b, then f(a) exists, f is continuous at a, f is differentiable at a, but f' need not be continuous at a. A function for which this is tru... |
I asked this question on Stack Overflow but it was closed as "not programming related". So I think this is probably the best place for it...
---
I read over the wikipedia [article][1], but it seems to be beyond my comprehension. It says it's for optimization, but how is it different than any other method for opti... |
What is linear programming? |
> You have sheets of 42-cent stamps and
> 29-cent stamps, but you need at least
> $3.20 to mail a package. What is the
> least amount you can make with the 42-
> and 29-cent stamps that is sufficient
> to mail the package?
A contest problem such as this is probably most easily solved by tabulating the possible... |
Any problem for which a solution (once found) can be quickly *verified* as a solution is said to be "in **NP**" (Here, "quickly" means in polynomial-time). Any problem for which a solution can be *found* quickly is said to be "in **P**." **P** is a subset of **NP** - that is, any problem for which a solution can be ... |
The [standard form][1] (and [example][2]) sections pretty well describe what it is.
> How is it different than any other method for optimizing things?
It's, well, just another method. However, it is somewhat special in that many other optimization algorithms either [use linear programming][3] as part of their so... |
The [standard form][1] (and [example][2]) sections pretty well describe what it is.
> How is it different than any other method for optimizing things?
It's, well, just another method. However, it is somewhat special in that many other optimization algorithms either [use linear programming][3] as part of their so... |
The absolute value of a 2 by 2 matrix determinant is the area of a corresponding parallelogram with the 2 row vectors as sides.
The absolute value of a 3 by 3 matrix determinant is the volume of a corresponding parallelepiped with the 3 row vectors as sides.
Can it be generalized to n-D? The absolute value of a n... |
Say there are three jars, $j_1, j_2, j_3$ filled with different binary sequences of length two.
The distribution of the binary sequences in each of the jars is given by the $p_i^k(1-p_i)^{n-k}$, where
$p_i = \frac{i}{m + 1}$ where m is the number of jars, i is the jar index, k is number of 1's and n is the lengt... |
Here's one that popped into my mind when I was thinking about binary search.
I'm thinking of an integer between 1 and *n*. You have to guess my number. You win as soon as you guess the correct number. If your guess is not correct, I'll give you a hint by saying "too high" or "too low". What's your best strategy?
... |
I have a 3-D sphere of radius R, centered at the origin.
(x<sub>1</sub>,y<sub>1</sub>,z<sub>1</sub>) and
(x<sub>2</sub>,y<sub>2</sub>,z<sub>2</sub>) are two points on the sphere. The Euclidean distance is easy to calculate, but what if I were to restrict myself to traveling on the surface of the sphere?
Two appr... |
How do I measure distance on a globe? |
I've come back from a Skydiving Training lecture, and I've run into a conceptual problem that I can't seem to grasp.
If I'm given a certain, non-varying acceleration, I can graph velocity over time easily.
In fact, if the acceleration varies over depending on a known function (constant, polynomial, exponential, t... |
Wind resistance -- upwards acceleration, typically varies either linearly or quadratically by the current velocity.
There is a constant downward acceleration due to gravity.
How can we model the velocity over time of a falling object, subject only to wind resistance and downwards gravity?
I don't have much exp... |
RHS is a well known test for determining the [congruency of triangles][1]. It is easy enough to prove it works, simply use Pythagorus' theorem to reduce to SSS. I thought that it seems strange that this only works for an angle being 90 degrees - or does it? What if I tried changed the given angle to 89 degrees or 91 de... |
RHS Congruency test - What makes 90 degrees different? |
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