Using only the numbers, $1$, $2$, $12$.

No concatenations allowed.

Only permitted signs are plus and division.

Brackets are not needed.

Expressions should be as concise as possible.

Typical example:

enter image description here

Fill in the right hand side for each of the three following cases:

A. $136/11$ =

B. $235/19$=

C. $4131/334$=

  • $\begingroup$ The title says "continued fractions", the question says "plus and division". The first of those is a stricter condition than the second. E.g., the trivial solutions that look like $\frac{1+1+\cdots+1}{1+1+\cdots+1}$ satisfy the second but not the first. What's the actual requirement? $\endgroup$ – Gareth McCaughan Jul 5 '19 at 10:18
  • $\begingroup$ Typical continued fractions involve plus, minus, division..I will post an example. $\endgroup$ – Uvc Jul 5 '19 at 10:21
  • $\begingroup$ I know what a continued fraction is. The question is whether, as the title suggests, you literally mean that you want a continued fraction; or whether, as the question text suggests, you are content with any expression built out of +,-,/. (If the former then I don't even understand why you need to specify +-/ only. If the latter then I don't understand why continued fractions are in the title.) $\endgroup$ – Gareth McCaughan Jul 5 '19 at 15:15
  • $\begingroup$ It looks as if the answer is that you did mean continued fractions, without even permitting "generalized continued fractions" where the numerator isn't 1. If so, then I don't see how there's the slightest element of puzzliness to the question. There is a standard, simple, obvious algorithm for computing continued fractions, and the only degree of freedom left to the solver is how to write each coefficient as a sum of 1s, 2s and 12s, and since each of those numbers divides the next there's also an obvious way to optimize that. Am I missing something? $\endgroup$ – Gareth McCaughan Jul 5 '19 at 15:18
  • 1
    $\begingroup$ I think the concern here is that a “puzzle” solved by mechanical application of a well understood algorithm/process isn’t really a puzzle. See the comments here as well as this answer to a very relevant question on our Meta, which feels exactly like this question. Compare the guidance here. $\endgroup$ – Rubio Jul 5 '19 at 23:40


$12 + \frac{1}{2 + \frac{1}{1 + \frac{1}{1 + 2}}}$


$12 + \frac{1}{2 + \frac{1}{1 + \frac{1}{2 + \frac{1}{2}}}}$


$12 + \frac{1}{2 + \frac{1}{1 + \frac{1}{2 + \frac{1}{1 + \frac{1}{1 + \frac{1}{12+2+2+1}}}}}}$

  • $\begingroup$ Yes..this is the form I am expecting for the other 2 also as shown in the example $\endgroup$ – Uvc Jul 5 '19 at 10:35
  • $\begingroup$ Check the last 2..don’t add up right $\endgroup$ – Uvc Jul 5 '19 at 10:48
  • $\begingroup$ Last 1 ok..checking 2nd one.. $\endgroup$ – Uvc Jul 5 '19 at 10:50
  • $\begingroup$ All 3 ok..got all $\endgroup$ – Uvc Jul 5 '19 at 10:52

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