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Tag Archives: counting
do go no to
dodo do go on do no harm do to others go do likewise gogo music go no further goto considered harmful no can do it’s a no go a big nono say no to drugs what to do here or … Continue reading
A new counting system
0 = t__ough 1 = t_rough 2 = th_ough 3 = through So, for example, trough through tough though though. English is so strange.
PIE: proof by counting
Recall the setup: we have a universal set and a collection of subsets , , , and so on, up to . PIE claims that we can compute the number of elements of that are in none of the (that … Continue reading
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
PIE day
[This is part six in an ongoing series; previous posts can be found here: Differences of powers of consecutive integers, Differences of powers of consecutive integers, part II, Combinatorial proofs, Making our equation count, How to explain the principle of … Continue reading
Penn Alexander: subset counting and Gray codes
I’m volunteering again this year with the middle school math club at Penn Alexander. I’m going to try to be better this year about posting what we do each week, for posterity’s sake and in case it inspires anyone else! … Continue reading
More cookies
I recently received the following interesting problem from Shadowcat, which is a generalization of the cookie problem I’ve written about previously. We again want to count the ways to distribute identical cookies to nonidentical students, with the twist that we … Continue reading
Idempotent endofunctions
Via Topological Musings comes another neat little counting problem. A function is idempotent if applying it twice gives the same result as applying it once: that is, for any input x. Endofunction is just a fancy way of talking about … Continue reading
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
Chessboard counting: solutions and further challenges
And now for some solutions to the chessboard counting challenges. The first challenge was to count the number of squares of any size on an 8×8 chessboard. The key here (as with many counting problems) is to break the problem … Continue reading
Posted in challenges, counting, solutions
Tagged chess, chess board, counting, rectangles, squares
2 Comments