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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.