Square, Circle, X
Read the puzzle
A square has both of its diagonals drawn, crossing at the centre in an X. A circle is drawn inside the square, centred on the same point, just touching the middle of each of the four sides. The diagonals pass through the circle and run on into the four corners. How many closed areas can you count in this figure?
How many closed areas can you count?
Each corner outside the circle is split by a diagonal.
12
4 slices inside the circle, and each corner is cut in two: 4 + 8 = 12.
How to solve it
Count the figure zone by zone so nothing is counted twice.
Zone one, inside the circle. The two diagonals cross at the centre and cut the circle into 4 equal slices. That is 4 areas.
Zone two, outside the circle but inside the square. The circle touches each side at its midpoint, so it leaves 4 corner pieces, each shaped by two half-sides and a curved edge. Each diagonal runs straight through a corner piece and splits it in two. That gives 4 × 2 = 8 areas.
Add the zones: 4 + 8 = 12 closed areas.
The usual wrong answer is 8. It comes from counting the 4 slices and the 4 corners but missing that the diagonal cuts every corner in half. Another slip is to add larger shapes made of several pieces; the question asks for the smallest closed spaces only.
Quick check: 4 slices in the middle plus 4 corners each cut in two gives 4 + 4 × 2 = 12. Takeaway: split the drawing into rings or zones and count each one on its own.
