Consider the integer powers of 7: a(n) = 7^n:

7, 49, 343, 2401, 16807,... 117649, ...

Prove that the second from right digit in all those numbers (excluding the 1st) is either 0 or 4.

Three concentric circles each have radii 8, 15 and 17.

Find the largest possible area of the equilateral triangle with one vertex on each circle.

Begin with a finite sequence of blocks in a row, each in one of 3 colors: red, blue, yellow.

Below each pair of neighboring blocks place a new block with the color rule: If the blocks are the same color use that color but if they are different use the third color.

Example:

r b y y b
y r y r
b b b
b b
b

How can the color of the last block be easily predicted from the top row?

Note: I don't know the full answer but can solve special cases.