The solution to this puzzle is one I came up with when working on this puzzle: What is the answer of this math puzzle?

For two natural numbers a,b, can you figure out how a#b is defined, going from the following examples?

2#0 = 10
2#2 = 2
2#16 = 8
3#3 = 3
4#9 = 3
5#5 = 5
5#11 = 7
5#17 = 9
6#13 = 7
9#12 = 12
9#16 = 16
10#3 = 7

Following this rule, work out 3#19.

This is a mathematical puzzle, i.e., all numbers actually represent numbers and not letters of the alphabet. Also for the input the numerical value of the inputs and their representation (i.e., the base in which they are writen in) are relevant, not the number of holes, its English name or anything like that. Programming and similar technical help can help, but is not required.

If it takes a while until it is solved, I would like to make this puzzle a bit more interactive. Instead of (or in addition to) giving hints, I will provide more examples on some of your suggested inputs that you think will help you find the pattern.

Hint #1:



Hint #2:


Hint #3:



Hint #4:



Hint #5:

a#b is always valid. However, there may be cases where one could argue it is a bit ambiguous. I haven't seen such cases yet, but also couldn't rule out that they exist.

Hint #6:


Hint #7:

While it should be possible to write a#b as a single function, it is much easier and more intuitive to think of it as a procedure that takes the inputs and performs some algorithmic steps.

Hint #8:




Hint #9:



Hint #10:

There are no explicit "magic constants" involved (e.g., divide by 3). Someone mathematically interested might however argue that there is one implicit constant involved.

Hint #11:


0xF#0xF=3, however 15#15=5 (which should clarify the last hint)

Hint #12:

Hint #5, that a#b is always valid, was meant for base 10 when I added it. Now with people exploring other bases I should say that there is one base for which a#b may not be valid. In fact (if a certain conjecture holds) there are infinitely many inputs for which a#b is undefined in that base. As far as I can tell (no guarantees on this) in other bases a#b is always valid.

Hint #13:

The operator utilizes concatenation.

Hint #14:


  • 2
    $\begingroup$ This would be a hint, so your call when or whether to tell us: Is it always true that a#b=b#a? $\endgroup$
    – quarague
    Jun 19, 2023 at 7:04
  • 4
    $\begingroup$ the # is the function f(x, y) = -0.2679x² + 0.0536y² + 0.6429x - 0.4286y + 11.2857 and the value of 3#19 is approximately 2.7. (note, this should not be taken seriously) $\endgroup$
    – Marius
    Jun 19, 2023 at 8:49
  • 2
    $\begingroup$ What happens if $a=0$? $\endgroup$
    – Dr Xorile
    Jun 20, 2023 at 21:50
  • 2
    $\begingroup$ Is $a\#b$ valid for all integers $a$ and $b$? $\endgroup$
    – Dr Xorile
    Jun 20, 2023 at 21:58
  • 4
    $\begingroup$ the only striking thing for me (so far) is rot13(vs o vf bqq (be rira) gur erfhyg vf nf jryy.). But maybe there will be counter examples in the future? $\endgroup$
    – theozh
    Jun 21, 2023 at 13:27


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