Diamond Window
Read the puzzle
A square window has both diagonals drawn, crossing at the centre. A diamond joins the midpoints of the four sides. A small circle sits at the centre, wholly inside the diamond and not touching it; the diagonals pass through the circle. How many closed areas (panes) can you count?
How many closed areas can you count?
Work outward: circle slices, then the diamond, then the corners.
16
8 corner pieces + 4 pieces round the circle + 4 slices inside it: 16.
How to solve it
Work from the middle outward, one layer at a time.
Layer one, the circle. The two diagonals cross at its centre and cut it into 4 slices: 4 areas.
Layer two, the band between the circle and the diamond. The circle does not touch the diamond, so a ring of space surrounds it. The two diagonals cut that ring into 4 pieces: 4 areas.
Layer three, the corners outside the diamond. The diamond joins the side midpoints, so it leaves 4 corner triangles. Each diagonal passes through a corner and splits its triangle in two: 4 × 2 = 8 areas.
Total: 4 + 4 + 8 = 16 closed areas.
The usual wrong answer is 12. It forgets layer two, the 4 pieces around the circle inside the diamond. Another route to 12 is treating the corner triangles as whole and forgetting the diagonals cut them: 4 + 4 + 4.
Quick check: the drawing has 3 layers (circle, band, corners) and every layer is cut by the same two diagonals, so count each layer separately and add.
