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.*