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Tag Archives: proof
The chromatic number of the plane, part 3: a new lower bound
In my previous post I explained how we know that the chromatic number of the plane is at least 4. If we can construct a unit distance graph (a graph whose edges all have length ) which needs at least … Continue reading
The chromatic number of the plane, part 2: lower bounds
In a previous post I explained the HadwigerNelson 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
Comments Off on Fermat’s Little Theorem: proof by group 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, twoplayer, zerosum
Comments Off on Computing optimal play for the greedy coins game, part 4