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Happy Numbers

Stage: 3 Challenge Level: Challenge Level:1

Take any whole number between $1$ and $999$ inclusive and add the squares of the digits to get a new number. For example, starting from $138$ we get $1 + 3^2 + 8^2 =74.$

We can repeat this process using the squares of $7$ and $4$ to get $65$ and then continue the process indefinitely.

Start with $145$ and see what happens.

Now we come to the most important step! Show that, whatever number we start with less than $1000$, we always get another number which is less than $1000$.

Choose some numbers less than $1000$ and, each time, repeat the process until you notice a pattern.

Make some conjectures about what happens in general.