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Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

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Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Three Frogs

Age 14 to 16 Challenge Level:

3 Funny coloured frogs

Three frogs came in through the kitchen door

Said one to the others "let's explore"

The frogs hopped in and onto the table.

They sat in a line and admired the view - the red frog on the left, the green in the middle, and the blue frog on the right.

In their excitement they hopped over each other - one frog hopping randomly over any adjacent frog.

After 999 hops they stopped!

Why I do not know - perhaps they were exhausted.

Is it possible, after their 999 hops, for them to have ended up in the position in which they started?