In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

What do you notice?

Can you convince someone what the answer would be to (a million sixes followed by a 7) ^{2} ?