N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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.

Knapsack

Stage: 4 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?