The cube of a whole number ends in the digit 7. In which digit must the number itself end?

Cube each of the ten digits and read off the endings: 0 → 0, 1 → 1, 2 → 8, 3 → 7, 4 → 4, 5 → 5, 6 → 6, 7 → 3, 8 → 2, 9 → 9. Only 3 gives a cube ending in 7, as $3^3=27,13^3=2197$ and $23^3=12167$ all show. The trap 7 swaps the pair round: 7 is the digit whose CUBE ends in 3, not the digit whose cube ends in 7. Notice how different cubes are from squares — every ending comes from exactly one digit, whereas a square ending in 6 can come from a number ending in either 4 or 6.