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# Tag Archives: proof

## The chromatic number of the plane, part 2: lower bounds

In a previous post I explained the Hadwiger-Nelson problem—to determine the chromatic number of the plane—and I claimed that we now know the answer is either 5, 6, or 7. In the following few posts I want to explain how … Continue reading

## Iterating squared digit sum

Another fun fact I learned from John Cook. Let be the function which takes a positive integer and outputs the sum of the squares of its digits. For example, . Since the output is itself another positive integer, we can … Continue reading

Posted in arithmetic, computation, proof
Tagged digit, loop, Porges, proof, squared, sum
12 Comments

## The chromatic number of the plane, part 1

About a week ago, Aubrey de Grey published a paper titled “The chromatic number of the plane is at least 5”, which is a really cool result. It’s been widely reported already, so I’m actually a bit late to the … Continue reading

## Every positive integer is a sum of three palindromes

I recently learned from John Cook about a new paper by Javier Cilleruelo, Florian Luca, and Lewis Baxter proving that every positive integer can be written as a sum of three palindromes. A palindrome is a number that is the … Continue reading

Posted in arithmetic, computation, links
Tagged algorithm, constructive, news, palindromes, proof, sum
14 Comments

## Properties of orthogons I

First things first: from now on, when talking about polygons with only right angles, instead of calling them “orthogonal polygons” I’m going to start calling them “orthogons”, which sounds cool, is much less clunky than “orthogonal polygons”, and doesn’t seem … Continue reading

Posted in combinatorics, geometry, proof
Tagged concave, convex, orthogonal, orthogons, polygons, proof, properties, vertices
10 Comments

## 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

## Computing optimal play for the greedy coins game, part 4

Last time I explained a method for computing best play for instances of the greedy coins game, which is feasible even for large games. This general approach is known as dynamic programming and is applicable whenever we have some recursively … Continue reading

Posted in computation, games, recursion
Tagged coins, dynamic, game, optimal, play, programming, proof, recurrence, strategy, tree, two-player, zero-sum

## Computing optimal play for the greedy coins game, part 3

In a previous post we saw how we can organize play sequences in the greedy coins game into a tree. Then in the last post, we saw how to work our way from the bottom of the tree upward and … Continue reading

Posted in computation, games, recursion
Tagged coins, game, optimal, play, proof, recurrence, strategy, tree, two-player, zero-sum
2 Comments