# Prove that it is impossible to make 18

Today, I was shown the following puzzle:

Find a way to make the numbers $7$ through $18$ using any three adjacent numbers in the grid below: \begin{array} {|c|c|} \hline 10 & 10 & 9 \\ \hline 5 & 7 & 3 \\ \hline 8 & 11 & 11 \\ \hline \hline \end{array}

Operations allowed: $+$ $-$ $\div$ $( )$. Parenthesis could be used for multiplication.

Example: $15 = 11 + 11 - 7$ (using numbers in an "L" shape from the bottom right corner)

Because of the allowed operators, you can't do something like $12 = 5 + 7 + \frac{d}{dx} 3$

I was able to make all the numbers but 18. After half an hour of trying each combination, I was told that it was impossible to make 18.

Am I being told the truth? Or am I missing something?

• Are you allowed to use the brackets for multiplication, like $(10)(10)(9)=900$? – David Jun 6 '14 at 0:30
• @David Yes, parenthesis can be used for multiplication. I'll update the question with adjacency rules – Joe the Person Jun 6 '14 at 3:53
• Since you didn't specify that we could only use each of those three numbers once, I'm going to say the answer is clearly $8+10\div 5 + 10\div 5 + 10\div 5 + 10\div 5 + 10\div 5$. ;b – user137731 May 2 '15 at 22:52

I really don't know how I would approch such a problem "mathematically" without running into a ton of cases. Anyways, here is some code that checks in the most unoptimized fashion that there is no solution for $18$.