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Author Archives: Brent
Combinatorial proofs
Continuing from a previous post, we found that if we begin with th powers of consecutive integers and then repeatedly take successive differences, it seems like we always end up with the factorial of , that is, . We then … Continue reading
Posted in combinatorics, pictures, proof
7 Comments
Differences of powers of consecutive integers, part II
If you spent some time playing around with the procedure from Differences of powers of consecutive integers (namely, raise consecutive integers to the th power, and repeatedly take pairwise differences until reaching a single number) you probably noticed the curious … Continue reading
Posted in arithmetic, iteration, pascal's triangle
Tagged binomial coefficients, consecutive, difference, integers, powers
1 Comment
17×17 4-coloring with no monochromatic rectangles
Quick, what’s special about the following picture? As just announced by Bill Gasarch, this is a grid which has been four-colored (that is, each point in the grid has been assigned one of four colors) in such a way that … Continue reading
Posted in open problems, pattern, people, pictures
Tagged 17x17, four-coloring, graph, grid, monochromatic, rectangles
4 Comments
Book review: Nine Algorithms that Changed the Future
Nine Algorithms that Changed the Future: the Ingenious Ideas that Drive Today’s Computers, by John MacCormick. Princeton University Press, 2012. I’m often wary of books written for general audiences on technical topics. It’s quite difficult to write in a way … Continue reading
Computing with decadic numbers
[This is the ninth, and, I think, final in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does "close to" mean?, The decadic metric, Infinite decadic numbers, More … Continue reading
Differences of powers of consecutive integers
Patrick Vennebush of Math Jokes 4 Mathy Folks recently wrote about the following procedure that yields surprising results. Choose some positive integer . Now, starting with consecutive integers, raise each integer to the th power. Then take pairwise differences by … Continue reading
Posted in arithmetic, pattern
Tagged consecutive, difference, integers, powers, surprising
12 Comments
u-tube
[This is the eighth in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does "close to" mean?, The decadic metric, Infinite decadic numbers, More fun with infinite decadic … Continue reading
Posted in computation, convergence, infinity, iteration, modular arithmetic, number theory, programming
Tagged decadic, Haskell, idempotent, streaming, u
1 Comment
Herbert Wilf: 13 June 1931 – 7 January 2012
I was sad to learn that Herbert Wilf died yesterday. Long-time readers of this blog may remember him as one of the discoverers of the Calkin-Wilf tree, which I wrote about in a ten-part series of posts (1, 2, 3, … Continue reading
A self-square number
[This is the seventh in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does "close to" mean?, The decadic metric, Infinite decadic numbers, More fun with infinite decadic … Continue reading
Posted in arithmetic, infinity, iteration, modular arithmetic, proof
Tagged decadic, idempotent, self, square
12 Comments
