### Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

### Big Powers

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

# More Mathematical Mysteries

##### Age 11 to 14 Challenge Level:

Meg offers the following solution:

If you take any 3 digit number, and then rearrange the digits of the number and find the difference between the two numbers, the number will always be a multiple of 9.

Think about the digits as being a,b and c.
These can be rearranged and subtracted in the following ways:
abc-acb= (10b+c)-(10c+b)=9(b-c)
abc-bac=(100a+10b+c)-(100b+10a+c)=90a-90b=9(10a-10b)
abc-bca=(100a+10b+c)-(100b+10c+a)= 99a-90b-9c=9(11a-10b-c)
abc-cab=(100a+10b+c)-(100c+10a+b)=90a+9b-99c=9(10a+b-11c)
abc-cba=(100a+10b+c)-(100c+10b+a)=99a-99c=9(11a-11c)

The digits of any multiple of 9 can always be reduced to give 9. eg
18: 1+8= 9
27: 2+7 = 9
99:9+9 = 18; 1+8 = 9
981:9+8+1=18;1+8=9

So whichever numbers you use, you will always end up with 9.
9-5=4
4 translates to D using the code
The only countries beginning with D are Denmark, Djibati, Dominica and the Dominican Republic. Most people would choose Denmark.
You'd be hard-pushed to think of another jungle animal beginning with E other than elephant.
Animals beginning with K are not many in number, so Koala may come to mind.
This leads to a fruit beginning with A, which is most obviously apple.