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# Think of Two Numbers

##### Age 11 to 14Challenge Level

Although we had lots of answers to this problem, very few of you explained your solutions carefully. Alistair from Histon and Impington Infant School sent us this solution:

I started by working out a few answers, and spotted that (if A is the first number and B is the second), A goes up and the number of the answer goes up by 10, and B goes up and the answer goes up by 1. My chart:

 1 2 3 4 5 1 20 30 40 50 60 2 21 31 41 51 3 22 32 42 52 4 23 5 24

So then I made a new chart, showing what numbers you would get at different steps of the operations:

 A 1 1 1 2 3 4 2 B 1 2 3 1 1 1 2 A+1 2 2 2 3 4 5 3 A+1 x5 10 10 10 15 20 25 15 5A+5 +1 11 11 11 16 21 26 16 5A+6 x2 22 22 22 32 42 52 32 10A+12 -1 21 21 21 31 41 51 31 10A+11 +B 22 23 24 32 42 52 33 10A+11+B +2 24 25 26 34 44 54 35 10A+13+B x2 48 50 52 68 88 108 70 20A+26+2B -8 40 42 44 60 80 100 62 20A+18+2B /2 20 21 22 30 40 50 31 10A+9+B

So the answer is always 10A + B + 9. If you are told the answer, you must take away nine, then the digits of the answer are the numbers that your friend thought of.

Chris, Isobel, Wui Shen and Alex from Maadi British International School in Cairo noticed something else:

We thought about the problem and worked it out together first. When we had our answer we tried to work out how to get our first numbers.

Wui Shen said we could take 1 away from the tens column and + 1 to the units column.

Isobel said we could + the two numbers and find two numbers that equal the same number.

We then each did our own sum and discovered Wui Shen's worked every time but Isobel's didn't because there was more than one answer to the sum.

Jordan from London explained why Wui Shen's method works:

10A + B + 9 gives the final answer.
If you subtract 9 from this you are left with 10A + B.

A and B are whole numbers less than 10; this means that multiplying A by 10 would give a result with 0 in the units column and A in the tens column.
After that the addition of B to 10A would put B in place of the 0, so A would be the tens digit and B would be the units digit.

When 9 is taken away from a number it is the same as subtracting 10 (the tens column decreases by 1), and then adding 1 (the units column increases by 1).

Therefore decreasing the tens column value by 1 and doing the opposite to the units column value has the same effect as subtracting 9. Both leave you with A and B as the two digits.

This is what Wui Shen found.