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Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

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Attractive Tablecloths

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Collatz-ish

Age 14 to 16 Short Challenge Level:

The terms of the sequence are$$6, 3, 14, 7, 34, 17, 84, 42, 21, 104, 52, 26, 13, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, \dots$$As can be seen, there will now be no other terms in the sequence other than $4$, $2$, and $1$. It can also be seen that the only values of $n$ for which the $n$th term equals $n$ are $13$ and $16$.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.