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Think of Two Numbers Poster

Age 11 to 14
Challenge 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.