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One of my friend brought this simple-looking math puzzle to me, but it was harder than it seems. I tried several ideas including simple arithmetic operations, and even considering base-n.

5#2=18 
8#2=11 
7#4=45
3#5=33 
6#3=?

The ?(answer) must be an integer.

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    $\begingroup$ @msh210 You're a 10k-er, why would you comment something like that? $\endgroup$
    – Bass
    Commented Jun 14, 2023 at 7:54
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    $\begingroup$ @msh210 Let's recap: you made up a way to add arbitrary stuff to the problem, and then proceeded to solve it for a result that is completely dependent on said arbitrary stuff, and posted a link to that result. One of the few things that's known for certain about that result is that it is never going to be the one unique solution to the question as posted by OP. This, then, brings us to my original question. $\endgroup$
    – Bass
    Commented Jun 14, 2023 at 8:31
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    $\begingroup$ @Moti You are completely missing the point. If you get four data points, you can find an infinite number of different functions that match exactly those values at exactly those four points. msh210 chose one of those functions, without any justification why that function should be chosen rather than any other, and then calculated the value of that function at the fifth point. That is not finding a solution, it is saying "Look ma, I drew a line through all the numbers in this colour-by-numbers puzzle". $\endgroup$
    – Bass
    Commented Jun 16, 2023 at 21:01
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    $\begingroup$ @Moti Well, since you asked nicely, I'll humour you. Try plugging in the left-hand side values into $$\frac{1040}{9}x + \frac{89}{6}y - \frac{7391}{378}xy - \frac{29780}{189}\frac{x}{y}$$ and see what happens. Then try $$ -\frac{223}{78}x^2 + \frac{209}{6}x + \frac{506}{39}y - \frac{4315}{39}$$These both use 4 coefficients, both fit the given data exactly, and both will of course give wildly different values for x=6,y=3 (70-something and 34-something, respectively). $\endgroup$
    – Bass
    Commented Jun 17, 2023 at 21:51
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    $\begingroup$ @SinonOW Would knowing the subject be relevant to the problem? Ex. we might be looking for a number theory answer vs a calculus answer. $\endgroup$ Commented Jun 18, 2023 at 5:01

1 Answer 1

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I feel like this problem is underspecified, even if constrained by an obligation to find something terse (a point being made in the comments, I think).

For example, suppose we had a code phrase of "A CRY OF THE CRYPTOGRAPHER" (or in fact any phrase starting with "A CRY OF TH", and n#m meant to find the $n$th and $m$th letter (in "ACRYOFTHECRYPTOGRAPHER"), then translate them to numbers with $a=1, b=2, \ldots, z=26$, and finally add them up. That takes a while to write out, but is still pretty occam if I can shamelessly verb that (ugh, amiright?)

Anyway, that gives:

\begin{eqnarray} 5\#2&\rightarrow O,C \rightarrow 15+3 =& 18\\ 8\#2&\rightarrow H,C \rightarrow 8+3 =& 11\\ 7\#4&\rightarrow T,Y \rightarrow 20+25 =& 45\\ 3\#5&\rightarrow R,O \rightarrow 18+15 =& 33 \end{eqnarray}

So the answer is (okay, I'll put in a spoiler, but I can't believe that's what this is):

$6\#3\rightarrow F,R \rightarrow 6+18 = 24$

A little more on generating this solution (such as it is).

We have 2 small digits on the left, both less than 8. And we have 11 to 45 on the right. This range possibly suggests the sum of two letters, where letters are actually numbers from 1 to 26.

If this premise is correct (and I'm not saying it is, but it's at least conceivable), then we'd be looking for 8 letters to index against. Equations 1, 2, and 4, give us 10 possibilities:

  • -ap-q--j,
  • -bq-p--i,
  • -cr-o--h,
  • -ds-n--g,
  • -et-m--f,
  • -fu-l--e,
  • -gv-k--d,
  • -hw-j--c,
  • -ix-i--b,
  • -jy-h--a,

Equation 3 gives us 8 more possibilities:

  • ---s--z-
  • ---t--y-
  • ---u--x-
  • ---v--w-
  • ---w--v-
  • ---x--u-
  • ---y--t-
  • ---z--s-

for a total of 80 possible templates, such as -cryo-th, which mixes the third of the first group (-cr-o--h) with the seventh of the second group (---y--t-). However, any of the 80 combinations would yield a valid solution. There are no one- or two-word solutions that fit (which would have been better, occam-wise I think), but there are many others that are a bit meh, but still work.

A couple of others:

  • "A CRY OUT H..."
  • "OF US LAZED"
  • "FIX XII UBIquitously" or XIV or XIX.

The last of these translates as:

\begin{eqnarray} 5\#2&\rightarrow I,I \rightarrow 9+9 =& 18\\ 8\#2&\rightarrow B,I \rightarrow 2+9 =& 11\\ 7\#4&\rightarrow U,X \rightarrow 21+24 =& 45\\ 3\#5&\rightarrow X,I \rightarrow 24+9 =& 33\\ 6\#3&\rightarrow I,X \rightarrow 9+24 =& 33\\ \end{eqnarray}

Coincidentally, the same answer. "FIX XIX UBI" would have given us 48. Should I find one that gives an answer of 42?

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