# 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? May 17, 2016 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) May 17, 2016 at 12:44
• aa...sorry. I misunderstood May 17, 2016 at 12:51
• +1 for using # instead of redefining + like so many annoying puzzles on Facebook seem to do.
– BenM
May 18, 2016 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! May 17, 2016 at 14:05
• A really good answer :) May 17, 2016 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 May 17, 2016 at 22:04

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

? = 21

The rule I used is

Add the two numbers together.
If neither number is a perfect square, add 1.
If one of the numbers is a perfect square, multiply by one more than its square root.
There may be an additional rule for when both numbers are perfect squares, but that situation doesn't arise in the given examples.

? is :

90

Rules for how I got there:

if prime--sum numbers, add 1 (smallest denominator of LHS, trivial) (5 + 8 + 1) = 14, (11 + 6 + 1) = 18 if both composite--multiply numbers, subtract largest denominator of LHS (8 * 9 = 72, 72 - 4 = 68) --Solution (12 * 8 = 96, 96 - 6 = 90)