60-second revision
- Vertical mirror: left and right swap; top and bottom stay. Water image: top and bottom swap; left and right stay.
- Letters unchanged in a vertical mirror: A H I M O T U V W X Y.
- Letters unchanged in a water image: B C D E H I K O X.
- Mirror image of a clock: 11:60 − time (use 23:60 − time if the given time is between 12:00 and 12:59).
- Standard die: opposite faces add to 7. Otherwise, a face adjacent to four numbers is opposite the fifth.
- Painted n×n×n cube: 3 faces = 8, 2 faces = 12(n − 2), 1 face = 6(n − 2)², none = (n − 2)³.
- Turning anticlockwise by 90 degrees: North→West→South→East→North.
Mirror image versus water image
A mirror placed beside an object (a vertical mirror, the usual exam assumption) swaps left and right but keeps top and bottom. A water image, as in a lake below the object, swaps top and bottom but keeps left and right. Questions in text form ask which letters, words or numbers are unchanged, or what a clock appears to show.
| Symmetry | Unchanged in | Capital letters |
|---|---|---|
| About a vertical line | Mirror image | A, H, I, M, O, T, U, V, W, X, Y |
| About a horizontal line | Water image | B, C, D, E, H, I, K, O, X |
| Both lines | Both images | H, I, O, X |
A whole word looks the same in a vertical mirror only when two conditions hold: every letter is in the first row of the table, and the word reads the same backwards (WOW, MOM, TOOT). A word looks the same in a water image when every letter is in the second row, because left-right order does not change (BOX, HIDE, COOK).
Mirror image of a clock
For a vertical mirror, subtract the actual time from 11:60 (which is the same as 12:00). The rule works in both directions: the actual time is 11:60 minus the image time.
| Actual time | Working | Mirror shows |
|---|---|---|
| 2:25 | 11:60 − 2:25 | 9:35 |
| 10:10 | 11:60 − 10:10 | 1:50 |
| 6:00 | 11:60 − 6:00 | 6:00 |
| 12:20 | 23:60 − 12:20 | 11:40 |
Dice
A die has six faces; each face touches four others and is opposite one. For a standard die, opposite faces sum to 7 (1–6, 2–5, 3–4). For a non-standard die you are given views of three faces meeting at a corner. List the neighbours of a number from all views; the number that never appears with it is opposite.
Worked example: views show (2, 3, 6) and (2, 4, 5). The neighbours of 2 are 3, 4, 5 and 6, so 1 is opposite 2. For another face, if two views share two numbers, the remaining numbers in those views are opposite each other — e.g. views (1, 3, 5) and (1, 3, 6) mean 5 and 6 are opposite.
Painted cubes
A cube painted on all faces and cut into n × n × n small cubes gives four kinds of pieces. Corner pieces have three painted faces; edge pieces (not corners) have two; face-centre pieces have one; inner pieces have none.
| Painted faces | Formula | n = 5 |
|---|---|---|
| 3 | 8 (corners) | 8 |
| 2 | 12 × (n − 2) | 36 |
| 1 | 6 × (n − 2)² | 54 |
| 0 | (n − 2)³ | 27 |
| Total | n³ | 125 |
Rotation and orientation
For objects turned on a table (a clock, a compass card, a map), track one marker. Clockwise quarter turns move North → East → South → West; anticlockwise quarter turns move North → West → South → East. Every other marker moves through the same angle in the same direction.
Example: a clock lies face up with 12 pointing East. Turned 180 degrees, 12 points West and 9, which first pointed North, now points South.
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