I haven’t written new posts for a while now, as I have been very busy lately.

I think this is a very interesting post and I hope it will make up for the lack of updates. I also want to take this opportunity to thank all of you for posting lots of interesting comments and for sending me many ideas and riddles, thank you!

In this post (and its sequel) I will describe Hilbert’s 3rd problem and show how it is solved. I based the posts mainly on a lecture by Prof. David Gilat.

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There are 100 red dots and 100 blue dots on the plane (a lot of planar riddles lately). The dots are arranged such that no three are on the same line.

A pairing of the red dots and the blue dots is a one-to-one function that assigns one blue dot to each red dot.

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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(log(N)) memory complexity.

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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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 In this post I will define turing machines and demonstrate a simple one in action.

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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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