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Happy Numbers

Stage: 3 Challenge Level: Challenge Level:1

In this problem we were asked to take any number (less than $1000$), add the squares of its digits, and then go on repeating this until a pattern emerges.

You were asked to start at $145$, and you should have got the sequence $$145 \mapsto42 \mapsto20 \mapsto4 \mapsto16 \mapsto 37 \mapsto58 \mapsto89 \mapsto145$$ and, of course, the sequence then starts repeating itself.

After trying other starting points you should have been able to guess that, whatever the starting point, eventually you will always reach either $1$, when the number is called 'happy' (and then you will stay at $1$), or the `cycle' given above, when the number is called 'sad'.

You were also asked to show that whichever number we start with (less than $1000$) we always get a number less than $1000$. Any number less than $1000$ has only one, two or three digits, and each of the digits is, of course, at most $9$. This means that when we add the squares of the digits, the largest number we can get is $3 \times(9 \times9)$ which is $243$, and this is less than $1000$.