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# Tag Archives: combinatorics

## Fermat’s Little Theorem: proof by group theory

It’s time for our third and final proof of Fermat’s Little Theorem, this time using some group theory. This proof is probably the shortest—explaining this proof to a professional mathematician would probably take only a single sentence—but requires you to … Continue reading

Posted in group theory, number theory, primes, proof
Tagged combinatorics, group, order, proof, theory

## Fermat’s Little Theorem: proof by necklaces

It’s time for our second proof of Fermat’s Little Theorem, this time using a proof by necklaces. As you know, proof by necklaces is a very standard technique for… wait, what do you mean, you’ve never heard of proof by … Continue reading

Posted in combinatorics, number theory, primes, proof
Tagged combinatorics, counting, Fermat, little, necklace, proof, theorem
4 Comments

## Making our equation count

[This is post #4 in a series; previous posts can be found here: Differences of powers of consecutive integers, Differences of powers of consecutive integers, part II, Combinatorial proofs.] We’re still trying to find a proof of the equation which … Continue reading

Posted in combinatorics, pictures
Tagged binomial coefficients, combinatorics, functions, matching, permutation
3 Comments

## Distributing cookies: solutions

And now for some solutions to the cookie distribution problem. I’m actually going to describe four different methods of solution, and thereby (re)discover some nice combinatorial identities along the way. This is what I love about combinatorics—you discover all this … Continue reading

Posted in counting, proof, solutions
Tagged binomial coefficients, combinatorial identities, combinatorics, cookies, counting, distribution
4 Comments

## Distributing cookies

Here’s a neat problem I saw in a recent post by Steven Miller on the Williams College math department blog. The problem comes from an old Putnam competition, one of the most prestigious college mathematics competitions. (It’s also one of … Continue reading