Numbers ending in 5

Square any number ending in 5 in your head, with no working, using one multiplication you already know.

Square any number ending in 5 without writing anything down.

The trick

Take the digits before the 5. Multiply them by the next number up. Write 25 after the result.

Why it works

Write the number as 10n + 5. Squaring gives 100n² + 100n + 25, which is 100 × n(n+1) + 25. The n(n+1) lands in the hundreds and above; the 25 always fills the last two places.

That is the whole proof, and it explains the shape of the rule: the 25 is not a coincidence or a memorised ending, it is what 5² contributes once the cross terms have been folded into the hundreds.

It goes further

The same idea works for any two numbers with the same tens digit whose units add to 10.

Squaring a number ending in 5 is just the case where the two numbers are equal.