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Double Digit

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

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.