# Express the given Fractions as Continued Fractions

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

A. $$136/11$$ =

B. $$235/19$$=

C. $$4131/334$$=

• 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? – Gareth McCaughan Jul 5 '19 at 10:18
• Typical continued fractions involve plus, minus, division..I will post an example. – Uvc Jul 5 '19 at 10:21
• 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.) – Gareth McCaughan Jul 5 '19 at 15:15
• 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? – Gareth McCaughan Jul 5 '19 at 15:18
• 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. – Rubio Jul 5 '19 at 23:40

## 1 Answer

$$\frac{136}{11}$$

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

$$\frac{235}{19}$$

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

$$\frac{4131}{334}$$

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

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