The square root of 12345678987654321

Question

What is the square root of 12,345,678,987,654,321? The number rises from 1 to 9 and falls back to 1. You are not allowed a calculator, and you should be able to explain why the answer works rather than just verify it.

Accepted Answer

111,111,111 (nine ones). The pattern. 11 squared = 121. 111 squared = 12,321. 1,111 squared = 1,234,321. Each time you add a 1, the square counts up one further and back down. With nine 1s the square counts up to 9 and back: 12,345,678,987,654,321. Why it works. Multiply 111,111,111 by itself in long multiplication. You are adding nine copies of 111,111,111, each shifted one place further left. In the units column only one copy contributes a 1; in the tens column two copies overlap, giving 2; and so on up to the middle column, where all nine copies overlap, giving 9. Then the overlaps decreas…

The square root of 12345678987654321

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