A large triangle is cut into 9 identical small triangles by lines drawn parallel to its three sides, so that the rows hold 1, 3 and 5 small triangles reading from the apex down. Counting triangles of every size and both ways up, how many triangles are there in the drawing?

Sort them into families. Point-up triangles: 6 of the smallest size, 3 of double size, and 1 of triple size (the whole figure), giving 10. Point-down triangles: only the smallest size occurs, and there are 3 of those — one in the middle row and two in the bottom row. The total is 10 + 3 = 13. The traps: 9 counts only the nine little pieces the figure was cut into; 10 counts every point-up triangle but overlooks the upside-down ones; 14 slips in a point-down triangle of double size, which simply does not fit inside a figure only three rows deep.