### 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:

This problem is based on the idea of Knapsack codes.

You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value that is given in the lookup table below.

You go off on a camping trip with 5 sticks in your knapsack. They have lengths of 1,3,5,10 and 20 centimetres, these will help you decode any message your friend sends.

A coded message arrives. It is the number 31. To decode the message you must work out which sticks you need to make a length of 31cm and convert this information into the binary code that tells you the letter. Decoding will be easy because your sticks form part of a superincreasing series (each stick is longer than the sum of the lengths of all the smaller sticks).

Taking the largest length off first leaves 31-20 = 11 so the coded letter used the 20 cm stick. 11-10 = 1 so the coded letter also involves the 10cm stick. With 1 left the 5-stick and the 3-stick are not used, just the 1-stick.

This gives you a binary code of 10011 (1x1cm+0x3cm+0x5cm+1x10cm+1x20cm).

So using the binary lookup table, the number 31 represents is 10011, which is the letter s.

Lookup Table

 Letter Binary Reference Letter Binary Reference a 00001 n 01110 b 00010 o 01111 c 00011 p 10000 d 00100 q 10001 e 00101 r 10010 f 00110 s 10011 g 00111 t 10100 h 01000 u 10101 i 01001 v 10110 j 01010 w 10111 k 01011 x 11000 l 01100 y 11001 m 01101 z 11010

Using the sticks in your knapsack decode the message: 33,18, 20, 1, 31, 20, 30, 33.

That was easy, but say your knapsack code involved a non-superincreasing series that used 1,2,3,4,5cm sticks.

Using this knapsack can you decode the message 1, 5, 14, 4, 5, 8, 10, 5, 4, 7, 9?

Can you explain why superincreasing series are so much easier to decode?