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# Think of Two Numbers Poster

##### Age 11 to 14Challenge Level

I subtract 9 from your answer, and the digits of the number that I'm left with, are your two numbers.

For example, if your final answer is 74, when I subtract 9, I get 65, so I know your numbers were 6 and 5.

How does it work?

If the two numbers are $a$ and $b$,

 Take one of your numbers $a$ Add $1$ $a+1$ Multiply by $5$ $5(a+1) = 5a+5$ Add $1$ again $5a+5+1 = 5a+6$ Double your answer $2(5a+6) = 10a+12$ Subtract $1$ $10a+12-1 = 10a+11$ Add your second number $10a + 11 + b$ Add $2$ $10a + 11 + b + 2 = 10a + b + 13$ Double again $2(10a + b + 13) = 20a + 2b + 26$ Subtract $8$ $20a + 2b + 26 - 9 = 20a + 2b + 18$ Halve this number $\frac12(20a + 2b + 18) = 10a + b + 9$

Final answer is $10a + b + 9$, so I subtract $9$ to leave $10a+b$

$10a + b$ is $a$ tens and $b$ ones

 Tens Ones $a$ $b$

Since $a$ and $b$ are both less than $10$, they fit in the tens and ones columns and so they are the digits of the number.