# What is the next result?

The pound sign / number symbol / octotroph / hashtag represents a certain function. Given the examples below, can you figure out what that function is and what number should be where the ? is?

   5 # 8 = 14
11 # 6 = 18
8 # 9 = 68
12 # 8 =  ?

• question mark or pound sign? – Marius May 17 '16 at 12:34
• sorry, I am new. where is the pound sign ? You must find what does the question mark represent. (what is the result of 12 # 8) – B. Mert May 17 '16 at 12:44
• aa...sorry. I misunderstood – Marius May 17 '16 at 12:51
• +1 for using # instead of redefining + like so many annoying puzzles on Facebook seem to do. – BenM May 18 '16 at 1:07

? could be

120

Explanation

let $d(x)$ be the highest divisor of $x$ smaller than $x$
let $isprime(x)$ be $1$ if $x$ is prime or $0$ if $x$ is not prime
then $x$ # $y = (x + y)*d(x)+isprime(x)$
$5$ # $8 = (5+8)*1+1 = 14$
$11$ # $6 =(11+6)*1+1 = 18$
$8$ # $9 = (8+9)*4 +0 = 68$
$12$ # $8= (12+8)*6+0= 120$

• Probably there is more than one answer but your answer is correct. Well done! – B. Mert May 17 '16 at 14:05
• A really good answer :) – ABcDexter May 17 '16 at 14:07

assuming operator # is symmetric, ? could be equal to

59

Consider what would happen if we were to sort the operands:

5 #  8 = 14
6 # 11 = 18
8 #  9 = 68
8 # 12 =  ?


Assuming simple formula

$$f(a, b) = (a-x)\cdot(b-y) + z$$

and solving it we get the following nice numbers: $x=11$, $y=-11$ and $z=128$ which implies the following definition of $\#$

\begin{align}a\operatorname{\#}b &= \Big(\min(a,b)-11\Big)\Big(\max(a,b)+11\Big) + 128\\&=a\cdot b-11\cdot|a-b|+7\end{align}

$\ddot\smile$

• I fixed a couple of errors in the formulae for you – Jonathan Allan May 17 '16 at 22:04

There are so many different possible solutions, due to only three examples, but I got

? = 21

The rule I used is