Coffee time mathematics — any number via three twos

Problem:

Here’s a witty algebraic brain teaser that had amused participants of a congress of physicists in the erstwhile USSR. The problem is to represent any number that must be positive and whole (any positive integer) using three twos and mathematical symbols.

Solution:

Let us take a particular case, and think “inductively”. Suppose we are given the number 3. Then, the problem is solved thus:

$3=-\log_{2} \log_{2} \sqrt{\sqrt{\sqrt{2}}}$ .

It is easy to see that the equation is true. Indeed,

$\sqrt{\sqrt{\sqrt{2}}}= ((2^{1/2})^{1/2})^{1/2}= 2^{\frac{1}{2^{3}}}=2^{{2}^{-3}}$ .

$\log_{2}2^{2^{-3}}=2^{-3}$ and $-\log_{2}2^{-3}=3$ .

If we were given the number 5, we would proceed in the same manner:

$5=-\log_{2}\log_{2}\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{2}}}}}$ .

It will be seen that we have made use of the fact that the index 2 is dropped when writing the square root.

The general solution looks like this. if the given number is N, then

$N=-\log_{2}\log_{2}\underbrace{\sqrt{\sqrt{\ldots \sqrt{\sqrt{2}}}}}_{N times}$ ,

the number of radical signs equalling the number of units in the given number.

More later,

Nalin Pithwa

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