torquestomp

So, most people should be familiar with the fail-safe way for 2 people to divide an object:

- The first person cuts the object into 2 pieces

- The second person chooses the object they wish to take first.

Since the second person is going to choose the largest available piece, the first person has a strong incentive to make an even cut, as if it is un-even, they will get less of the final product.

So, I got around to thinking about this problem in the case of N people. Here is a procedure that I think is incentive-compatible for 3 people:

- The first person makes 2 pieces, designating one as a 'small' piece and one as a 'large' piece

- The second person gets 2 choices. They may:

- Take the 'small' piece for themselves. The large piece is then divided using the 2-person division procedure between the first and third person

- Give the 'small' piece to the first person, and split the large piece with the 3rd person - using the same procedure used for 2 people.

So, here's a bunch of somewhat open-ended questions.

- Does my procedure for 3 people work? Can you prove it, preferably somewhat elegantly? If it does not work, show how it can be broken.

- The procedure for 3 people build recursively on the 2 person procedure. If this procedure works... can we construct a general procedure for N-people that also works recursively?

What do you all think?

We say that a set S in R^n is D-bounded, D > 0, if, for all x,y ∈ S, |x-y| <= D. We say that S is maximally D-bounded if, for any z ∉ S, S ∪ {z} is not D-bounded.

It is trivial to prove that any maximally D-bounded set is necessarily closed and bounded. My challenge is this: Prove that, for any maximally D-bounded set S, there exists a subset B of S, of Lebesgue Measure zero, such that S is equal to the intersection of all circles C of radius D with center b ∈ B.

Let H be the set of all functions h for which there exists a continuously differentiable real function f such that h(x) = f(f(x)). Such functions include h(x) = x, h(x)=e^(e^x), etc. (H is clearly infinite in size).

What can we derive in general about h? Are there any non-trivial operations under which H is closed? What are we describing if we extend H to transformations in R^n?

Has H been studied at all? Or is it just a cramped, relatively pointless set of functions?

I am not looking for any specific answers here, just some discussion with luck some enlightening deductions. However, for those who fear these kinds of questions, here are some more specific questions (more food for thought than anything. These have relatively trivial proofs):

a) Prove that if h is bounded if f is bounded, but f unbounded does not force h unbounded

b) Prove that f is not unique, even when h(x) constant

c) Prove that h(x) = -x is not a member of H

For those of you who are not aware, the Putnam Collegiate Mathematics Competition was held today, from 10:00-6:00PM US Eastern Time (So these are now fair game :) )

Here is a second problem that piqued my fancy. I feel as though I got really close to a solution, but was unable to close. Here it is:

B5) Does there exist a strictly increasing function f : R --> R such that f'(x) = f(f(x)) ?

My notes & progress:

If f exists, there exists a quantity L in (-1, 0) such that:

f(x) --> L as x --> - infinity

f(L) = 0

Feel free to post explanations for my notes and/or full proofs. Again, no cheating!

For those of you who are not aware, the Putnam Collegiate Mathematics Competition was held today, from 10:00-6:00PM US Eastern Time (So these are now fair game :) )

Here is one question that piqued my fancy, as I was unable to solve it in the time I had, given that it was stubbornly resilient to my class of usual approaches. Anyone else?

No cheating by looking at the website.

A6) Let f : R --> R be strictly decreasing with limit f(x) --> 0 as x --> infinity. Prove that:

∫_0^∞ (f(x)-f(x+1))/f(x) dx

diverges

More specifically, given a non-trivial (i.e. self-referential, Godelian) statement in ZFC that is true, but unprovable, can one in general prove that the statement cannot be proven in ZFC? Alternatively, can one extend Godel's Incompleteness Theorem to show that not only are there unprovable truths, but there are truths whose inherent unprovability is also unprovable?

I have only had brief experience with Godel's work, and a very lacking one at that. I encourage verbosely justified arguments.

Given two positive integers a,b > 1, is there a formula to obtain the most significant decimal digit (or for any base m, for that matter) of a^b, without computing a^b? The number of digits in a^b? Please provide proof.

...and its not the one between males and females.

http://www.cs.northwestern.edu/~riesbeck/mathphyse...

I found these hilarious. Anyone else?

I found this Flash game online called "The Codex of Alchemical Engineering"

http://www.kongregate.com/games/krispykrem/the-cod...

A surprisingly intellectual game: a rather rare find in the pool of common, simple and addictive flash games. My question is this: In the later levels of this game (Elixir of Life, Philosopher's Stone, etc.) where complex molecules are formed, is this a true excercise of the mind any longer? Or is this higher complexity merely a disguise for repetition and quantity, as opposed to quality and novelty?

In general really, when does some intellectual task, such as studying an Academic field, learning a new practice, or studying any type of complex sustem, stop being an excercise in critical thinking and instead becomes just a test of memorization and computation?

Obviously this question has no concrete answer. I'd like to hear the thoughts of the Y!A community on this dilemma - anyone with knowledge on the psychology or neurology of this please step forward.

P.S. For the Flash game addicts, (I confess I am one of them), there's a more challenging version of this game here: http://www.kongregate.com/games/krispykrem/the-cod...

I haven't dared to attempt any of these yet - I still need to beat the last level int the first one.

This test is not for the faint of heart. I got the idea from my Logic professor.

-- Extreme Multiple Choice --

Every question has exactly one correct answer. The other three are incorrect.

1) Which of the following is true?

a) This test is easy

b) The answer to question 8 is A

c) The answer to question 2 is C

d) The answer to question 4 is neither A nor C

2) How many questions on this test have A as the correct answer?

a) 2

b) 1

c) 4

d) 3

3) What is the letter answer to question 7?

a) D

b) It’s neither C nor D

c) Same as the letter answer to question 6

d) Same as the letter answer to question 8

4) Which of the following is true?

a) The answer to question 6 is A

b) The answer to question 8 is C

c) I had pizza while writing this test

d) The answer to question 6 is not B

5) What is the answer to this question?

a) The Euler-Mascheroni constant

b) A phrase of five words: “the answer to this question”

c) The meaning of life

d) It’s incomputable

6) Which of the following is false?

a) The answer to question 9 is D

b) I was staring at a street lamp when I wrote this question

c) The answer to question 1 is B

d) The answer to question 8 is not B

7) Which of the following is true?

a) The answer to question 1 is C

b) The answer to question 5 is either B or C

c) Note of the above

d) The answer to question 10 is the same as the answer to this question

8) What number am I thinking of?

a) 1 + The number of questions on this test with the answer A

b) It’s a perfect square

c) π squared over 6

d) A triangular number!!!

9) Seriously, what number am I thinking of? I haven’t changed it, honest.

a) The number of a question with the answer B

b) 4

c) The number of questions on this test with answer C

d) 3

10) This is an ambiguous question?

a) And this is an ambiguous answer.

b) This answer is incorrect. Ambiguous paradox!

c) Yes, this is an ambiguous question.

d) Um, your point?

-----

First post with a 100% score will get best answer

I will not give hints, so don't ask. If questions seem ambiguous and undecidable, ignore the actual answers - assume that each answer has a hidden "correctness" value, which must coincide with the other questions and answers. Deductively, there will exist one unique answer that can be marked as "correct".

1) Prove or disprove that every positive rational number can be represented as the finite sum of the reciprocals of distinct positive integers. Namely:

For any number m in Q+, there exists a finite sequence of distinct integers {a1, a2, a3, ...} such that m = 1/a1 + 1/a2 + 1/a3 + ...

2) Prove or disprove that for every positive rational number there exists a sequence satisfying the above conditions that is pairwise coprime. If it does exist, prove or disprove that the sequence is unique.

3) (Fun non-exactly-number-theory Bonus)

Recall the epsilon-delta definition of continuity (the names might be switched): A function f is continuous at a point x0 if for every ϵ > 0, there exists a number δ > 0 such that for all x, x-δ < x < x+δ, f(x0)-ϵ < f(x) < f(x0)+ϵ. Prove, by example, that there exists a function f : R --> R that is continuous at all the irrational numbers but discontinuous at all the rational numbers.

Follow this link:

http://xkcd.com/356/

Please, no comments on the nature of xkcd.com. My friend told me about this site, and I was bored with an hour to kill, so I gave in. A fair portion of the comics are actually quite clean, though-provoking and entertaining.

Anyways - the Physics Problem! I took AP Physics and know much of the basic laws of resistors and the Diff EQ's that govern inductors and capacitors, but Kirchoff's rule is failing me. The current is supposed to follow the path of least resistance - if we partition the grid into two infinite sets of parallel resistors towards the top and bottom, does the effective resistance become just 1? Or would it be 0?

On the 2008 Putnam, there was a question posted about a Matrix game:

--

Consider a 2008 x 2008 matrix, initially empty. Alex and Barbara, the players in the game, take turns placing a real number of their choosing at any empty spot in the grid. Alex goes first, Barbara second.

--

In the original problem, Barbara wins at the end of the game if and only if the determinant of the completed matrix was 0. Alex wins if the determinant is non-zero. Obviously, Barbara has the winning strategy here, because she can just mirror Alex's moves in a different row, forcing two rows in the matrix to be equal, and making the determinant 0.

However, I propose the following twist - what if Alex went second? Could Alex then have a winning strategy? Would aforementioned strategy apply to any n x n matrix?

Incomplete answers welcome.

I know what transcendal means - my question is, how do you prove that an irrational number in general is transcendental? Or does someone know how to prove that pi is transcendental? Is e transcendental?

Also, I have a conjecture: Let a function f(x) be "transcendentally conservative" if for any non-transcendental number y, f(y) is also non-transcendental. Can we prove that for any transcendental number z, for all "transcendentally conservative" functions f, f(x) is also transcendental?

Best answer goes to the person who answers the most questions without reiterating * cough - copying - cough * previous answers.

How can I get the songs from RB1 to be playable on RB2? Does it cost Microsoft Points?

I have both disks, but I want to give the RB1 disk to someone else and still be able to play the songs on RB2.

Follow up to:

http://answers.yahoo.com/question/?qid=20081212103...

The original problem, as posted on the Putnam 2008 Collegiate Math Comptetion, was this:

Q) What is the radius of the largest possible circle that can be inscribed in a four dimensional unit hypercube?

The solution was proved by Putnam staff as sqrt(2)/2, using Cauchy-Schwarz Inequalities. However, the proof is somewhat lengthy, and purely analytical in nature and terminology.

What we're looking for is a solution constructed from simple geometric relationships between dimensions. For example, the maximum radius of a circle in a unit square is 1/2, the maximum in a cube is sqrt(3/8), and for a hypercube it is sqrt(2)/2, which are all equivalent to sqrt(n/8), for n = 2, 3, and 4 respectively (the dimension of the container)

So - what can the knowledgable community of Yahoo! Answers procure today? Can we prove/disprove the sqrt(n/8) hypothesis? What else can we come up with?

This was a question on the collegiate Putnam 2008 Math Competition. I was wondering if anyone had a nice, simple way to solve it.

Q: Consider a four-dimensional unit hypercube. What is the greatest possible radius a two-dimensional circle inscribed in this hypercube can have?

Two of us got the answer sqrt(11/18), but after analyzing our methods, we know that's wrong. My professor used vector calculus and optimization to get sqrt(1/2), so we think that's the right answer. Does anyone by chance have a more elegant way to do this? The professor's proof was entirely analytical, and took over two pages of work - We're looking for something fundamental.

Alright, so I was experimenting with my TI-89, and I found out that the calculator had memorized the solution to ζ(2) - nameley, when I entered the summation formula explicity as the sum from k = 1 to infinity of 1 / k^2, it gave me the exact answer π^2/6.

Further experimentation yielded ζ(4) = π^4/90, ζ(6) = π^6/945, and onwards up the trail until I got bored, unable to recognize any elegant pattern.

My question is, is there an explicit formula for ζ(n) for all positive even integers n? And will this always yield an answer in the form c*π^n, where c is a rational number? (c is in the form 1 / d for the first five even integers, but once you hit n = 12 that nice hypothesis goes out the window. ζ(12) = 691 π^12 / 638512875)

Insights, proofs, explanations - all are welcome. I'd like to see where this leads.

Let's see what clever statements you users can come up with. I will award 10 points to the responder that meets the criteria of this question.

1) All of the answers to this question are false, except for the correct one.

2) If the statements below yield a contradiction, then the most recent answer that can be removed to fix the contradiction will be ignored.

Please don't be cheap and change your answers...If I notice, your post will be disqualified.

As soon as I am able to select a Best Answer, I will post the deadline time as Additional Details. The deadline will be at least six hours after I post it.

This is a follow up to the question posted here:

http://answers.yahoo.com/question/index;_ylt=ArIze...

My main question is, can the best answer posted there be proven to be correct? As I mentioned in my comment, the statement made by the Best Answer does not prove in and of itself that all < u_i , u_j > are 0, only that a certain collection of sums are.

For the case n = 4, there are six different inner products and four equations. I can get solutions to the equations that are nonzero, but then I start hitting problems when trying to construct unique Vectors of magnitude 1 to fit these values. This is expected, seeing as the statement can be proven by other means: but is there a way to algebraically verify stephenmdalton's claim with Vector properties?

As an addition to my old post, here is the proof I came up with:

Let W_k = U_k / || U_k ||, where U_k is the cross product of all vectors u_i for i = 1 to n excluding k. W_k is a unit vector, and a member of V, so with it we obtain the equation:

|| W_k || ^ 2 = Sum from i = 1 to n of < W_k , u_i >

Because W_k is proportional to U_k, which is orthogonal to all u_i for i not equal to k, W_k is also orthogonal to these Vectors. Therefore:

|| W_k || ^ 2 = < W_k , u_k > = || W_k || * || u_k || * cos(θ_k)

This simplifies to cos(θ_k) = 1, showing that W_k = some scalar λ * u_k, where λ can equal ±1. Since u_k is proportional to W_k, u_k is also orthogonal to all u_i for i not equal to k. This goes for all k from 1 to n, so all < u_i , u_j > = 0 for i not equal to j, making this an orthonormal set.

Different proofs or insights welcome as well. Please contribute.