Arithmetic clues
How to Use Digit Sums and Remainders in Number Puzzles
Some clues test the candidate directly. Others ask you to add its digits first. Writing down which number you are dividing prevents a small arithmetic slip from crossing out the right answer.
Add the digits, not the place values
For 286, the digit sum is 2 + 8 + 6 = 16. It is not 2 + 80 + 600, and it is not the candidate number 286. A digit-sum comparison such as 'greater than 15' would keep 286 because 16 is greater than 15.
- Write each digit separately.
- Recount a repeated digit every time it appears.
- Zero adds nothing to the sum.
Calculate a remainder from the right starting value
For the clue 'remainder 1 when divided by 3', the number 286 leaves 1 because 3 × 95 = 285. For a digit-sum remainder clue, use 16 instead: 16 divided by 3 also leaves 1. These happen to agree here, but that is not guaranteed for every divisor.
- Find the nearest multiple no larger than the tested value.
- Subtract that multiple to get the remainder.
- Check whether the clue names the number or its digit sum.
Use a short table if needed
With only a few candidates left, jot their digit sums and remainders beside them. For 42 and 48, the digit sums are 6 and 12. If the clue says the digit sum is greater than 10, only 48 remains. Checking the arithmetic once is faster than repeatedly rereading the same two clues.
- 42 → 4 + 2 = 6.
- 48 → 4 + 8 = 12.
- Keep the result attached to its original candidate.