This is one I come back to when I'm bored and all I have is pen and paper. I also read in a biography of Benjamin Franklin that he used to do this when he was stuck in boring meetings. Construct a 3x3 grid of numbers, using numbers 1 through 9, and arrange the numbers in the square such that every row, column, and diagonal (diagonals through the center) adds up to 15.
Too easy? Now construct a 4x4 grid made out of numbers 1 through 16, such that every row, column, and diagonal adds up to 34. This one is killing me because I figured it out once, but can't seem to rediscover the solution. There are ways to do this for grids 5x5, 6x6, and up, though they no doubt get very difficult.
One solution to the 3x3 square is below. There are other solutions that are simply rotations of this one - can you find a solution that is not a rotation of the square below? As for the 4x4 square, I'm still trying to figure it out.