The big picture
A trick that works is interesting. A trick you can explain is one you understand.
| Idea | Remember |
|---|---|
| Digit sums | A number and its digit sum leave the same amount over on 9 and on 3. |
| Why it works | Every column is a run of nines, plus a leftover. Nines divide by 3 and by 9. |
| Odd times even | Even if either number is even. Odd only if both numbers are odd. |
| Reverse and subtract | The bigger of the two numbers minus the smaller = 9 × the gap between the two digits. |
| Many clues | Apply every clue. Count what survives: none, one, or more than one. |
| When it stops | One failed case kills a claim. Nine good cases prove nothing. |
| Saying why | Each sentence uses only what came before it. An example checks one case. It is not a step. |
| Odd and even, added | Add some numbers. Count how many are odd. If that count is even, the sum is even. If that count is odd, the sum is odd. |
| Magic squares | A magic square has equal sums in every row, column and diagonal. Use nine numbers that follow each other, like 1 to 9. The centre cell holds the middle one. The magic sum is 3 times it. |
| Counting rhythms | A rhythm of n beats starts with a 1 or a 2. After a 1, any rhythm of n − 1 beats can follow. After a 2, any rhythm of n − 2 beats can follow. Add the two counts. |
| Letters for digits | In these puzzles each letter is one digit. Different letters are different digits. A number does not start with 0. Start at the units column. |
| Even or odd expressions | The nth even number is 2n. The nth odd number is 2n − 1. With an even number in front of n, the plain number decides. With an odd one, the value can be even or odd. |