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Knapsack
Age 14 to 16
Challenge Level 
There is only one way to make a length of 1cm from your knapsack so the first letter is easy.
Your problem with the superincreasing series is that there can be more than one way to make each of the totals from your knapsack and you need to look at all the possibilities to work out the message
Why can you just subtract the largest length in the case of superincreasing series in order to decode?
The superincreasing series 1, 2, 4, 8, 16, ... allows you to make all numbers but the one given in the question does not. Why doesn't this matter?
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Novemberish
a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.