Track quiz
Twelve ideas, twelve checks. Look back at the table whenever you need.
| Trick | How far it goes |
|---|---|
| Digit sum | Same leftover on 9 and on 3, for a number and its digit sum. |
| Why it works | Every column is a run of nines, plus a leftover. The leftovers decide. |
| Odd times even | Even if either number is even. Odd only if both are odd. |
| Turn it round | The bigger of the two numbers minus the smaller = 9 × the gap between the digits. |
| Several clues | Any order gives the same survivors. Count them: none, one, or more. |
| The breaking case | One failed case ends a claim. A run of good cases proves nothing. |
| Saying why | Each sentence uses only what came before it. An example checks one case. It is not a step of the argument. |
| Add odd and even | 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 square | 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. |
| Count 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. |
| Letter puzzle | 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 expression | 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. |
The digits of 6284 add up to 20. What does that tell you about dividing 6284 by 3?
Why can adding up the digits decide whether 3 divides a number?
You work out 6 × 7. What decides whether the answer pairs up completely?
A two-digit number is built from the digits 8 and 3. Take the smaller of it and its reverse away from the bigger. What do you get?
Some clues leave 48 and 84 standing. What should you write down as the result?
A claim about every number fits the first six cases. It fails on the seventh. What is it worth?
Why is 52 − 25 = 27 not a step in the argument?
You add 9 odd numbers and 4 even numbers. Is the sum odd or even?
A 3 by 3 magic square uses the numbers 11 to 19. What is its magic sum?
4 beats give 5 rhythms. 5 beats give 8 rhythms. How many rhythms have 6 beats?
In K + K + K + L = KK, what is L?
Which expression is always odd, for every whole number n?
Number Play is Chapter 6 of Part I. You arrived with a trick you could not explain. Now you can prove it, or break it.
Next: A Tale of Three Intersecting Lines. Three lines meet to make a triangle. Then we study its angles.