Fast checks, friendly numbers, fewer dead ends
Speed maths for number puzzles
In a target-number puzzle, speed maths is less about showing off and more about eliminating bad routes quickly. If you can spot divisibility, reshape awkward multiplication, and see the distance to a friendly number almost instantly, you spend more of the round solving and less of it checking arithmetic.
1
Use digit sums to spot multiples of 3 and 9
Add the digits. If that total is divisible by 3, the original number is divisible by 3. If the total is divisible by 9, the original number is divisible by 9.
In SumGoal this is useful before attempting a division: you can reject an impossible ÷3 or ÷9 route without calculating the quotient.
Quick check
738
Which statement is true?
2
Test divisibility by 7: double the last digit and subtract
Take the last digit, double it, and subtract that from the number formed by the remaining digits. If the result is a multiple of 7, so was the original number. Repeat if needed.
An equivalent shortcut found in Bill Handley's Speed Mathematics material is to multiply the last digit by 5 and add it to the preceding part. For 343: 34 + 15 = 49. Both tests work; the double-and-subtract version is usually easier to remember.
Divisible by 7?
343
3
Check 4 and 8 from the end of the number
For divisibility by 4, only the last two digits matter. For divisibility by 8, only the last three digits matter.
This is particularly useful when an intermediate result is large. You do not need to divide the whole number to know whether ÷4 or ÷8 will stay legal.
Which divisions are exact?
1,524
4
Multiply by 9 using 10× minus the original
The quickest general rule is often simply n × 9 = n × 10 − n. For 47 × 9, think 470 − 47 = 423.
There is also a digit-by-digit Trachtenberg-style shortcut. Working from the right: subtract the final digit from 10; for each inner digit subtract it from 9 and add the neighbour on its right; finish with the leading digit minus 1. Carry normally when a step is 10 or more.
×9 sprint
47 × 9
5
Divide by 9 by spotting friendly chunks
First use the digit-sum test to see whether exact division is possible. Then split the number into chunks that are obvious multiples of 9.
For two-digit numbers, another useful check from Handley's material is that the first digit gives the initial quotient and the digit sum gives the initial remainder; if that remainder is 9 or more, convert another 9 into one more quotient. For target puzzles, however, chunking into known multiples is usually clearer.
÷9 sprint
729 ÷ 9
6
Double one factor and halve the other
The product does not change if you double one factor and halve the other. Keep going until one side becomes a number that is effortless to multiply.
This is exceptionally useful with 25, 50 and sometimes 75. For example, 75 × 8 → 150 × 4 → 300 × 2 = 600.
Reshape the product
25 × 16
Try doubling 25 and halving 16.
7
Know complements to 10, 50, 100 and 1,000
When you overshoot or work backwards, the important arithmetic is often the gap. Train yourself to see complements instantly: 63 needs 37 to make 100; 875 needs 125 to make 1,000.
Fast complement recognition makes the anchor-and-repair strategy much quicker because you can evaluate several nearby routes without writing anything down.
Find the complement
63 → 100
A five-minute speed routine
- One minute on digit sums and divisibility.
- One minute on ×9 and ÷9.
- One minute on doubling and halving.
- One minute on complements and gaps.
- One SumGoal round where you deliberately use at least one shortcut.
For puzzle-specific route planning, continue with the SumGoal solving strategy guide. Or play a round and try to notice one of these patterns before your first move.
Sources and background
The page draws on standard divisibility rules, Bill Handley's Speed Mathematics / Speed Math for Kids material, and the Trachtenberg approach to multiplication by 9. The methods here are rewritten and selected specifically for arithmetic target puzzles.