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Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

Please Explain

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Take a look at the two multiplications below. What do you notice?

$32 \times 46 = 1472$
$23 \times 64 = 1472$

The digits in this multiplication have been reversed, and the answer has stayed the same!

Is this surprising? Can you find other examples where this happens?

What do you notice about the pairs of two digit numbers that produce this special result?