How Many Triangles?
Read the puzzle
A large triangle has its apex at the top. Two lines run from the apex down to the base, dividing the base into three parts. A fourth line runs from the bottom-left corner up to a point on the right side, crossing both inner lines on the way. How many triangles of any size are in the figure?
How many triangles are in the figure?
15
Single pieces plus every combined shape: 15 in all.
How to solve it
Group the triangles by where their corners sit, and count each group.
First, triangles with a corner at the apex and their bottom side on the base. Four lines come down from the apex (the two sides and the two inner lines), and any two of them cut out a triangle: 4 x 3 / 2 = 6.
Second, triangles with a corner at the apex and their bottom side on the slanted line. The same four lines meet the slanted line at four points (the bottom-left corner, two crossings and the point on the right side), so again any two give a triangle: 6 more.
Third, triangles with a corner at the bottom-left, one side on the base and one on the slanted line. Each line that rises from the base to the right closes one: the left inner line, the right inner line and the right side. That is 3 more.
Total: 6 + 6 + 3 = 15.
The usual wrong answer is around 8 to 10, from counting the small single pieces and only a few large ones. Quick check: each triangle needs three corners joined by drawn straight lines.
Learn the method: How Many Squares Puzzle: Count Squares, Rectangles and Triangles
