Hi! We're E and Pi, here to introduce you to Jon Bettencourt's web page on unsolved math problems. It's his final project for COSMOS, the math and science program he's in. Is he going to be surprised when he sees what we've done!

Our favorite problem is the one about e

Anyway, I have been gathering information on unsolved problems from the Internet and other sources. Here are a few of the problems I have found and some information on some of them.

Some of these problems seem very simple, but end up with no simple proof or solution to them. As an example, Fermat's Last Theorem was very simple, but it took hundreds of years and some parts from very advanced branches of mathematics to solve it. So if one seems simple, don't try to solve it unless you know a lot of math.

More information about unsolved problems can be found with a search for unsolved problems on Google.com. Some of these are a little advanced, though; I only put the simpler ones up here.

The Collatz Conjecture

Coloring the Plane

Cube Triangulation

The Degree/Diameter Problem

e

Folding a Napkin

A Formula for Any Digit of Pi

Goldbach's Conjecture

Maximum Matching

Odd Perfect Numbers

Perfect, Amicable, and Sociable Numbers

Perfect Rational Triangles

Poncelet's Theorem

Radical Differences

Repetition-Resistant Sequence

Smallest Possible Area

Tiling a Plane with Pentagons

Triangles and Tetrahedra

Twin Primes

My Problems

Sources Used

Given a centrally symmetric convex compact body K in 3-space and a point p on its surface, the point on the surface farthest from p in the surface metric need not be the antipode of p. (Cf. the Knuth-Kotani puzzle, which asks one to find the point on the surface of a 1-by-1-by-2 box that is farthest from one of the corners; it is not the opposite corner.)Back to Contents

However, one might still conjecture that the maximum distance between two points in the surface metric, asbothof them vary over the surface, is achieved by a pair of antipodal points. (This is believed to be true in the case of an a-by-b-by-c box for all a, b, and c, though to my knowledge no proof is known even for this very special case.)

Can anyone prove or disprove the conjecture? (Is the convexity condition necessary?)

If we start with 13, we will get the sequence:

13 40 20 10 5 16 8 4 2 1

So far, no counterexamples have been found. It has been verified that if you start with a number less than 700 billion, you will eventually end up with 1.

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- The graph can be 7-colored (try a hexagonal tessellation)
- There is a configuration that requires 4 colors

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This raises an interesting open question for higher dimensional hypercubes: what is the minimum number of simplices into which they can be cut? The long-diagonal construction generalizes to d! simplices in d dimensions, which can be reduced to O(d!/kI have here the original thread from which this problem comes from:^{d}) for some k by using Cartesian products of the five-tetrahedron solution. There is also a lower bound of ( (d!) k^{d}) for another constant k coming from volume arguments (What is the largest simplex you can fit into a hypercube? This is equivalent to asking what's the largest determinant you can get with a 0-1 matrix, and is answered by the theory of Hadamard matrices.) Both bounds can be tightened slightly but there still remains a big gap between them. I'm not sure if it makes a difference whether you allow additional vertices or just stick with the original set of 2^{d}vertices.

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Here are the properties the graph needs:

- It is planar. Its edges can be drawn in a flat plane without crossing each other.
- The diameter is 3.
- It is 3-regular.

Some of the values of Exp(Pi*(n)) are very close to integers. A prize will be awarded to anyone who can either convincingly argue that this is coincidence, or who can explain why this is so in terms intelligible to an intelligent college senior. Something else that might help to lift the veil on this mystery would be a predictor: for a given value of n, is Exp(Pi*(n)) close to an integer?

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It is easy to show that one folding decreases the perimeter, but more folds may increase it again--can the perimeter ever be larger than it was from the start?

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So far, the only ways of calculating a specific digit of Pi require calculations of all the previous digits.

However, a formula has been found that gives any

This is no longer an unsolved problem, but I put it up here because it is interesting that nobody even thought a digit-extraction formula for pi was possible, and now here it is.

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Given 6 arbitrary points in the euclidean plane, find a maximum matching for these 6 points. Let the matching be ((aHere's the original thread:_{1}, b_{1}), (a_{2}, b_{2}), (a_{3}, b_{3})). Prove that there must exist a point x such that for all i, 1 <= i <= 3, dist(a_{i}, x) + dist(x, b_{i}) < 2/(3) * dist(a_{i}, b_{i}) where dist(x, y) is the Euclidean distance between x and y.

One interesting consequence is that if such a point exists for any 6 points in the plane, then it also must exist foranyeven number of points in the plane.

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So far, no odd perfect number has been found. However, it has been found that if there

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Perfect, Amicable, and Sociable Numbers

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Here's the original thread:

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Show that if the path returns to the original point after n points, i.e. A

If n is even, say n = 2k, show that A

If n is odd, say n = 2k + 1, let T

There is a proof of the first result using elliptic integrals - is there an easier proof?

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What is the minimum nonzero difference between two sums of square roots of integers? In particular, find a lower bound on r(n,k), the minimum positive value of:

(awhere a_{i}) - (b_{i})

Examples:

r(20,2) = (10)+(11)-(5)-(18) = approx .0002Back to Contents

r(20,3) = (5)+(6)+(18)-(4)-(12)-(12) = approx .000005

For each N, let L(N) be the longest length of the repeated string in the first N terms of R; e.g., L(19)=3, since within the first 19 terms, a string of length 3 (namely 121) repeats, and no longer string repeats.

You can write out many terms of R, using this rule: after the initial 123, at each stage, ask: when I append the next number (1, 2, or 3), will L(N + 1) increase, no matter which of the three I choose? If so, choose 1; otherwise, choose the least of the three for which L(N + 1) does not increase.

A finite string of 1's, 2's, and 3's is called a word. Here's the big question: does every word occur in R?

If so, then it is easy to see that every word repeats infinitely many times in R, which is notable, since the rule for generating R tries to resist repetition.

This problem has a reward of $50.

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(A

Take a tetrahedron ABCD. A sphere touches the three faces containing A and touches internally the circumsphere of ABCD at A

(A

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Given any integer n, does n appear in an infinite decimal expansion of pi?

Given any integer n, does n appear in an infinite decimal expansion of e?

Given any integer n, does n appear in an infinite decimal expansion of 2?

Given any integer n, does n appear in an infinite decimal expansion of (1+ 5)/2 (the golden ratio)?

Given any integer n, does n appear in an infinite decimal expansion of sin()?

More generally, does any integer n appear in any irrational number?

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The Bailey-Borwein-Plouffe Pi Algorithm

The Exp(Pi*Sqrt(n)) Page

Games on Graphs

MathSoft Unsolved Problems

Open Problems for Undergraduates

Perfect, Amicable, and Sociable Numbers

sci.math (a discussion forum for mathematics)

Unsolved Problems

Unsolved Problems and Rewards

Unsolved Problems with Elementary Formulations

As mentioned before, more information about unsolved problems can be found with a search for unsolved problems on Google.com.

E and Pi (and Bromine) are of my own creation.

Some of these problems were brought up during my classes here at COSMOS.

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