There are many problems in math that no one has been able to answer. A surprising number of these problems are easy to state, and even easy to play around with.Thinking about these problems can give a student more of an idea of what math really is.
I'd like to collect problems here that would be accessible to high school and community college students. James Tanton works with students on open problems, and they've gotten results. I think that requires finding little known problems. The problems I'm listing below have been worked on long enough that any new results from us amateurs are unlikely. But we can still have fun playing.
The word conjecture means something that people believe is likely to be true, but which has not been proved.
1. Goldbach's Conjecture: Every even number greater than 2 can be expressed as the sum of two primes.
2. Twin Prime Conjecture: There are an infinite number of twin primes (n and n+2).
3. Collatz Conjecture: Play this game: Start with a positive whole number, n1. If n1 is even divide by 2, if n1 is odd multiply by 3 and add 1. You now have n2. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1, which gives you a loop of 1, 4, 2, 1, ...
4. The Integer Brick Problem, to find a brick size in which the 3 lengths along the sides, the 3 diagonals along the faces, and the interior diagonal are all integers, or to prove that this cannot be done.
5. Is every prime larger than 3 the average of two primes? (from James Tanton)
6. A perfect number is a number that equals the sum of all its factors (including 1, but not including itself). Are there any odd perfect numbers? (from Wikipedia's list of open problems in mathematics)
7. The Lonely runner conjecture: if k + 1 runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be more than a distance 1 / (k + 1) from each other runner) at some time? (same source as #6)
8. Towers of Hanoi with 4 pegs: There is a stack of n disks on a peg, with the disks decreasing in size as you move up the stack. If you are allowed to move one disk at a time from one peg to another, and cannot put a bigger disk over a smaller one, what is the least number of moves it takes to move the stack. (seen twice recently, once here)
9. The Goormaghtigh conjecture: The only numbers that can be represented as 11...1 in more than one base are 31 and 8191. 31 can be represented in base 2 as 11111 and in base 5 as 111. 8191 can be represented in base 2 as 1111111111111 and in base 90 as 111. (We don't look 11 (two digits) in this problem, as any number n will be represented as 11 in the base n-1, making it easy to find numbers with two different representations that are all 1's. For example, 13 is 11 in base 12 and is 111 in base 3.) (thanks to Mary O'Keeffe)
10. Legendre's conjecture: There is a prime number between n2 and (n + 1)2 for every positive integer n. (mentioned in Bob and Ellen Kaplan's book, Hidden Harmonies)
Here's a good question: Which two problems on this list go together? If one were solved the other would be too. Are the two equivalent, or could one of them be solved and still leave the other open? (thanks to Christopher Sears)
What other open problems (simple to state and to play around with) can you add to this list?
[Edit date: 1/15]