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