If you haven’t already, take a look at my birthday puzzle. Be sure not to look at the comments if you want to solve it. Likewise the rest of this post.
Okay, now that it’s just you and me, consider this figure:
I’ve marked each of the pairs of vertical lines that are just one unit apart with the numbers 1 through 4. I’ve marked each of the pairs of horizontal lines that are one unit apart with the letters A through C.
Now, anywhere that the marked horizontal lines intersect the marked vertical lines, a unit square is the result. So since there are 4 sets of vertical lines and 3 sets of horizontal lines, they should produce exactly 4 * 3 or 12 unit squares. A quick count shows that’s correct.
Rather than counting all the squares by hand, it’s much simpler to count the gaps between the horizontal and vertical lines, and work from there.
There are 3 vertical gaps that are 2 units wide, and 4 horizontal gaps that are 2 units tall. So for 2×2 squares you know that there are 3 * 4 = 12 squares.
There are 3 vertical gaps that are 3 units wide, and 2 horizontal gaps that are 3 units tall. So there are 3 * 2 = 6 squares that are three units on a side.
This continues up to the 2 * 2 = 4 four unit squares, the 2 * 3 = 6 five unit squares, the 3 * 1 = 3 six unit squares, and the 3 * 1 = 3 seven unit squares.
Therefore the total is 12 + 12 + 6 + 4 + 6 + 3 + 3 = 46.
This method applies to the Window Pain puzzle at the U.S. Puzzle championship as well (on which this puzzle was based).