# Which three positive whole numbers give the same result when added together as when multiplied together?

Source: https://silverapuzzles.com/riddles/which-three-positive-whole-numbers-give-the-same-result-when-added

A small number-theory puzzle you can solve by trying tiny numbers. Adding and multiplying usually give very different results; here they have to agree.

**Answer:** 1, 2 and 3.

1 + 2 + 3 = 6 and 1 × 2 × 3 = 6.

Hint: Try small numbers.

## How to think about it

1. Multiplying tends to grow faster than adding, so the numbers must be small. Start with 1s, which keep the product low.
2. Try 1, 1, 1: sum 3, product 1. Try 1, 1, 2: sum 4, product 2. Not yet.
3. Try 1, 2, 3: sum 6, product 6. They match.
4. Check that nothing else works: 1, 2, 4 gives 7 and 8, and anything larger pulls the product further ahead.

## Why “1, 2 and 3” is the answer

The numbers 1, 2 and 3 add up to 6 and multiply to 6, so the two operations agree. The 1 keeps the product from growing while still adding to the sum, which is why it has to be part of the answer. For three positive whole numbers this is the only solution; the same idea for two numbers gives 2 and 2.

## Common wrong answers

- **2, 2, 2.** Sum 6, product 8. Close, but not equal.
- **1, 1, 1.** Sum 3, product 1. The product is too small.