
Recent Posts
Archives
 May 2020
 April 2020
 March 2020
 June 2018
 July 2016
 June 2016
 May 2016
 March 2015
 February 2015
 January 2015
 December 2014
 December 2013
 November 2013
 July 2013
 June 2013
 May 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
 April 2012
 March 2012
 February 2012
 January 2012
 December 2011
 November 2011
 October 2011
Categories
Meta
Pages
Category Archives: Uncategorized
Free Groups and Tiling
Introduction Consider the following simple problem. Prove that the shape on the left cannot be completely tiled by 20 polygons of the types shown on the right. The solution is rather simple: colour the shape in the following manner. This … Continue reading
Posted in Uncategorized
Tagged combinatorics, free groups, group theory, groups, polyominoes, tiling, words
Leave a comment
Solving PermutationBased Puzzles
Introduction In the previous article, we described the SchreierSims algorithm. Given a small subset which generates the permutation group G, the algorithm constructs a sequence such that for: we have a small generating set for each Specifically, via the Sims … Continue reading
Posted in Uncategorized
Tagged group actions, group theory, permutations, rubik's cube, schreiersims, symmetric group
Leave a comment
SchreierSims Algorithm
Introduction Throughout this article, we let G be a subgroup of generated by a subset We wish to consider the following questions. Given A, how do we compute the order of G? How do we determine if an element lies in G? Assuming , how … Continue reading
Posted in Uncategorized
Tagged group actions, group theory, permutations, programming, rubik's cube, schreiersims, symmetries
Leave a comment
Primality Tests III
SolovayStrassen Test This is an enhancement of the Euler test. Be forewarned that it is in fact weaker than the RabinMiller test so it may not be of much practical interest. Nevertheless, it’s included here for completeness. Recall that to … Continue reading
Posted in Uncategorized
Tagged cryptography, elementary, jacobi symbol, legendre symbol, number theory, primality tests, primes, programming
Leave a comment
Primality Tests II
In this article, we discuss some ways of improving the basic Fermat test. Recall that for Fermat test, to test if n is prime, one picks a base a < n and checks if We also saw that this method would utterly fail … Continue reading
Posted in Uncategorized
Tagged carmichael numbers, cryptography, elementary, number theory, primality tests, primes, pseudoprimes
Leave a comment
Primality Tests I
Description of Problem The main problem we wish to discuss is as follows. Question. Given n, how do we determine if it is prime? Prime numbers have opened up huge avenues in theoretical research – the renowned Riemann Hypothesis, for … Continue reading
Posted in Uncategorized
Tagged carmichael numbers, cryptography, elementary, number theory, primality tests, primes, pseudoprimes
Leave a comment
Polynomials and Representations XXXIX
Some Invariant Theory We continue the previous discussion. Recall that for we have a equivariant map which induces an isomorphism between the unique copies of in both spaces. The kernel Q of this map is spanned by for various fillings T with shape and entries … Continue reading