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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.

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Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Mean Sequence

Stage: 3 Short Challenge Level: Challenge Level:1
Suppose the first two terms of the sequence are $x$ and $y$. The third term is $\frac{1}{2}(x+y)$, and the fourth term is $\frac{1}{4}(x+3y)$ so the fifth term is $\frac{1}{8}(3x+5y)$.

Putting $x=\frac{2}{3}$ and $y=\frac{4}{5}$ we obtain $\frac{1}{8}(2+4)=\frac{3}{4}$.

This problem is taken from the UKMT Mathematical Challenges.