## 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.

1. if you use n (or n+1) sqrt signs, isn't that kind of cheating since it is effectively embedding unary n in the answer?
Consider for example, I could get any n with zero 4's if I did |{x,....,x}| with n x's

Joe C.

2. Oops, in the previous {,} should have been <> (vector not set).
Or something like the set of nested sets on empty e.g. |{ {}, {{}}, ... {{{{...}}}}} }| if you object to x's.

3. Actually, after thinking a few more minutes, of course the number has to be embedded. Anyway, a cool problem and analysis.
Joe C.

4. Very cool idea, thanks for writing it up.

(Graeme)

5. Joe, as you say the number has to be embedded somehow, but then it would be supercool to embed it in binary or something!

Graeme, just to be fair remember I got the idea from somewhere else, which I can't remember now.

And I like the zero 4s solution.

6. I thought that you couldn't use the square root int he puzzler because with a square root, it was impled that there was a 2 there.

7. You must have been able to use a square root because the official answer used one!