Abstract thinking follows shapes through turns and flips, works out directions and clock angles, and counts squares, rectangles and painted cubes.
| Clock angle at h:m | hour hand 30 × h + 0.5 × m, minute hand 6 × m; the smaller angle is at most 180° |
| Rectangles in an m × n grid | (m + 1)m ÷ 2 × (n + 1)n ÷ 2 |
| Example, 2 × 3 grid | 3 × 6 = 18 rectangles |
| Squares in a 2 × 3 grid | 6 + 2 = 8 |
| Painted cube, n along each edge | 8 corners; 12(n − 2) edges; 6(n − 2)2 faces; (n − 2)3 inside |
| Fold k times | 2ᵏ layers |
(a) How many rectangles (squares included) are there in a grid of 5 rows and 5 columns of small squares?
(b) The corners of a regular dodecagon are numbered 1 to 12 clockwise, with corner 1 at the top. It is turned clockwise about its centre through 180°. Which corner is now at the top?
Working: (a) choose 2 of the 6 horizontal lines (15 ways) and 2 of the 6 vertical lines (15 ways); (b) each 30° moves every corner one place clockwise; 180° = 6 places, so the corner 6 places anticlockwise of corner 1 comes to the top
Answer: (a) 225; (b) 7