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# Reversals

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Age 11 to 14

Challenge Level

*Reversals printable worksheet*

*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.*

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?

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.

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.