## 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:

.

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

.

and .

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

,

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

More later,

Nalin Pithwa

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