A very interesting riddle for those of you with some basic background in Set Theory.

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A very easy riddle. Four frogs are sitting on the corners of the unit square (i.e. they have coordinates (0,0), (0,1), (1,1) and (1,0) ). Each turn, a frog can jump over any other frog, thereby transferring itself to the symmetrical point on the other side of the static frog. For example, if the frog at (0,0) jumps over the frog at (1,1) it will land on (2,2).

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 This is a cute puzzle. Consider an infinite checkerboard divided in two with an infinite line lying along the x-axis, as depicted below:

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In this post I will give a basic example of the necessity (or uselessness – its up to you to choose – pun intended) of the axiom of choice. I will also present a less known proof of the Cantor–Bernstein–Schroeder theorem.

The Cantor–Bernstein–Schroeder Theorem

Let A and B be sets. Let f:A->B and g:B->A be injective functions. Then there exists a bijection h:A->B.

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Dany Valevsky gave me this very cool riddle.

You are given a vector of size N, the elements of which are numbers in the range 1,…,N-1. I.e. there is at least one repeating element. Give an algorithm that finds a repeating element (it does not matter which one, in case there are several) with O(N) time complexity and O(1) memory complexity.

NOTE - the time and memory complexities are calculated in integers. I.e. the input is of size N, not N*logN.

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A ship in the plane has integer coordinates. It also has integer velocity (again in ZxZ). Each turn the ship advances according to its velocity. Here is an example of a ship with velocity (3, 1).

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 Galois Theory is an algebraic theory providing a powerful connection between fields and groups. Many complicated problems involving fields can be converted to (possibly) simpler problems involving groups.

Galois Theory, albeit being extremely beautiful (it answers some very elementary questions, which were all open problems until its arrival), is far from being wildly known (i.e. by non-mathematicians). One of the reasons for this, in my opinion, is that is it a vast subject and most books on the topic are filled with too many details and so are inadequate for a quick read.

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 I recently watched this interesting video by Shai Avidan and Ariel Shamir from MERL. They developed an extremely cool image resizing technique called seam carving. They explain all about it in this article.

One of the coolest things about their idea is that is it easy to implement. I implemented a semi-optimized version of their algorithm in a couple of hours of coding. All the images in this post were generated by my code.

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When considering the extra-credit of the 23 and 2000 riddle, I thought of an additional interesting problem.

Let N be a positive integer. A partition of N into m parts (an m-partition of N) is a multiset of m positive integers, such that their sum equals N. A multiset is a set which may contain repeated elements.

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You are given 23 whole numbers, not necessarily distinct, in a row.

You cannot change the order of the numbers.

Prove that there exists an arrangement of the symbols ’+’, ‘×’, ‘(‘ and ‘)’ in-between the 23 numbers, such that the final result is a valid formula, whose evaluated value equals 0 mod 2000.

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