The Times Tables That Actually Matter
Everybody is made to learn the times tables to twelve. Almost nobody is told which parts of that grid do the real work, and it is a much smaller set than it looks.
The grid is smaller than it looks
Twelve by twelve is 144 facts, which sounds enormous. It is not:
- Order does not matter. 7 × 8 and 8 × 7 are the same fact. That halves it immediately.
- The 1s, 2s, 5s and 10s are rules, not facts. Nobody memorises 7 × 10.
- The 9s have a rule too — see below.
- The 11s up to 9 are just the digit twice.
Strip those out and you are left with roughly fifteen genuinely awkward facts, clustered in the same place every time: the 6s, 7s and 8s multiplied by each other.
| The hard fifteen | ||
|---|---|---|
| 6 × 7 = 42 | 6 × 8 = 48 | 6 × 9 = 54 |
| 7 × 7 = 49 | 7 × 8 = 56 | 7 × 9 = 63 |
| 8 × 8 = 64 | 8 × 9 = 72 | 9 × 9 = 81 |
| 6 × 12 = 72 | 7 × 12 = 84 | 8 × 12 = 96 |
| 6 × 6 = 36 | 4 × 7 = 28 | 4 × 8 = 32 |
Fifteen facts is an afternoon, not a term. Everything else in the grid you can derive faster than you could recall it.
The rules worth having instead
Times 9: one less, then make nine
The tens digit is one less than the number you multiplied; the two digits add to nine.
7 × 9: one less than 7 is 6, and 6 + 3 = 9, so 63. For 8 × 9: one less is 7, plus 2 makes nine, so 72.
The other route is often quicker still: ten of them, minus one of them. 8 × 9 = 80 − 8 = 72.
Times 5: half of ten
14 × 5 → 140 ÷ 2 = 70. Always faster than five hops, and it does not degrade with bigger numbers: 68 × 5 → 680 ÷ 2 = 340.
Times 4 and times 8: double repeatedly
17 × 4 → 34 → 68. And 17 × 8 → 34 → 68 → 136. Doubling is the one operation almost everybody can do without thinking, so lean on it.
Times 6: times 3, doubled
14 × 6 → 14 × 3 = 42 → 84.
Times 11: split and add
For a two-digit number, separate the digits and drop their sum in the middle. 36 × 11 → 3, then 3+6 = 9, then 6 → 396. If the middle goes past nine, carry it: 87 × 11 → 8, 8+7 = 15, 7 → carry the 1 → 957.
Squares are worth knowing outright
Squares turn up constantly, and knowing them lets you reach neighbouring products in one step.
| n | n² | n | n² |
|---|---|---|---|
| 11 | 121 | 16 | 256 |
| 12 | 144 | 17 | 289 |
| 13 | 169 | 18 | 324 |
| 14 | 196 | 19 | 361 |
| 15 | 225 | 25 | 625 |
From a square you know, the next one up is the square plus both numbers: 17² from 16² is 256 + 16 + 17 = 289. And a product either side of a square is the square minus one: 15 × 17 = 16² − 1 = 255.
Why recall beats calculation
A fact you know costs nothing. A fact you work out costs a second and, worse, occupies the part of your mind that was holding the rest of the problem. That is why the hard fifteen are worth brute force: they are the ones that appear inside larger sums, where you cannot spare the attention.
Everything else in this article is a rule, and rules are fine — they only become expensive when they are all you have.
Carry the One‘s Difficult mode is built out of exactly these products. Multiplying larger numbers assumes them, which is a good way to find out which ones you do not actually know.
More like this: all the mental arithmetic guides.