A Mathematical Chess (Dissection) Paradox

I thought I would share with our chess-loving readers the following interesting (and somewhat well-known) mathematical chess paradox , apparently proving that , and the accompanying explanation offered by Prof. Christian Hesse, University of Stuttgart (Germany).  It shows a curious connection between the well-known Cassini’s identity (related to Fibonacci numbers) and the chessboard ( being a Fibonacci number!). The connection can be exploited further to come up with similar paradoxes wherein any -square can always be “rerranged” to form a -rectangle such that the difference between their areas is either or . Of course, for the curious reader there are plenty of such dissection problems listed in Prof David Eppstein’s Dissection page.

Another elementary number theory problem

This one, by Dr. Titu Andreescu (of USAMO fame), is elementary in the sense that the solution to the problem doesn’t require anything more than arguments involving parity and congruences. I have the solution with me but I won’t post it on my blog until Jan 19, 2008, which is when the deadline for submission is. By the way, the problem (in the senior section) is from the issue of Mathematical Reflections, 2007.

Problem: Find the least odd positive integer such that for each prime is divisible by at least four (distinct) primes.