A question I got from an 8 years old kid:

Question

I got this one from an 8 old kid who got this at school

Can you get the number in the right by using the numbers on the left with the rules:

1. Each number should be used 1 time exactly,
2. The allowed operations are: $(,),+,-,\cdot,/$

A solved exercise as an example:

$12$ $11$ $20$ $2$ $= 9$ solution: $(20/2)-(12-11)$

Any ideas regarding the solutions of these three?

$8$ $2$ $1$ $32$ $= 31$

$11$ $2$ $1$ $9$ $= 111$

For example, to me it seems like the first one has no solution. I can explain it intuitively to myself but I can't prove it formally. Am I right? If yes, is there a way to formally proof it?

Thanks!

Edit exponentials and roots are not allowed (i.e. $8^2$ is not allowed). All the numbers must be used. No concatenation is allowed.

Answer 1

I wrote some code to brute-force this instead of doing any hard thinky thinking. Neither {8 2 1 32} = 31 nor {11 2 1 9} = 111 have solutions. If 111 was a typo for 11, $11 + (9+2) \cdot 1$ works.

Answer 2

For the third we have: $$5-(2+9):11=4.$$

Answer 3

A formal way to prove it could be done by exhaustion trying for all the possible allowed combinations of numbers and operations, similarly to what they did for the first proof of the famous "Four colour theorem", using a calculator.