Sunday, 8 November 2009

Get 55, or any other number, with three 4s.

SPOILER ALERT.   This post contains the solution to this week's  Car Talk puzzler.

The puzzle is construct the number 55 using three 4s and any mathematical symbols you like.  I remembered that somewhere I had come across a general solution to this, maybe from Halmos's book "Problems for Mathematicians Young and Old".

I half remembered it involved logs and square roots, and I managed to reconstruct it.    First of all I did it with 2's.   The first image shows you how to do it (constructed using a lovely tool called LaTeXiT.)   The number of square root symbols in a row is n, if you want to get the number n.   So for car talk it would be 55 square roots.

I was puzzling over how to do it with 4's, and my wife couldn't understand my problem.    Because she didn't realise I didn't realise that every 2 could be replaced with the square root of 4.   So the second image shows that, and so this time there are n+1 square roots, in the car talk puzzle there would be 56.

So my wife is just slightly more intelligent than me, which as we all know is the best kind of wife.     I love her.