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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.

Reversals

Age 11 to 14 Challenge Level:

You may wish to try Always a Multiple? and Special Numbers before tackling this problem.

Where should you start, if you want to finish back where you started?

Alison chose a two-digit number, divided it by $2$, multiplied the answer by $9$, and then reversed the digits. 
Her answer was the same as her original number!
Can you find the number she chose?

Then she chose another two-digit number, added $1$, divided the answer by $2$, and then reversed the digits.
Again, her answer was the same as her original number!
Can you find the number she chose this time?

Charlie chose a two-digit number, subtracted $2$, divided the answer by $2$, and then reversed the digits.
His answer was the same as his original number!
What was Charlie's number?

Choose a number, subtract $10$, divide by $2$ and reverse the digits.
What number should you start with so that you finish with your original number?


Extension

Choose a 3-digit number where the last two digits sum to the first (e.g. $615$).

Rotate the digits one place, so the first digit becomes the last (so for the example, we get $156$).

Subtract the smallest number from the largest and divide by $9$ (which is always possible).

What do you notice about the result? Can you explain why?





With thanks to Don Steward, whose ideas formed the basis of this problem.