Coffee time mathematics — any number via three twos


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.


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:


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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