# Squaring is Daring (1)

**Number Theory**Level 4

The following sequences go by the rule:

\(a_{n+1}\) = The sum of the squares of the individual digits of \(a_{n}\)

For example, 56 becomes 61, because \(5^{2}\) + \(6^{2}\) = \(25 + 36\) = 61.

The following is an example of a bad sequence, because it reaches the number 1, and stays there.

{28, 68, 100, 1, 1, 1…}

The following is an example of a good sequence, because it repeats and is non-terminating:

{30, 3, 9, 81, 65, 61, **37**, 58, 89, 145, 42, 20, 2, 4, 16, **37**…}

If I create all sequences from 51 to 60 (inclusive), how many will be "good"?

Super Challenge: Can you find the general set of starting numbers that will make a sequence "good"/"bad"?

Ultimate Challenge: Prove It!

**Your answer seems reasonable.**Find out if you're right!

**That seems reasonable.**Find out if you're right!

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