Using an 8-way splitter, extract 1/12-th of something

Question

I have a pile of dirt. And I have an 8 way splitter.

I can split into 8 and recombine splits as many times as I wish.

What is the easiest run of splitting/combining I can do to pull out exactly 1/12th of the dirt?

Answer 1

(Not entirely serious, but too long for a comment.)

There is no solution in the strict sense, as noted in the comments already, since any legal split or recombine leaves piles which are fractions having denominators that are powers of $2\,$. Therefore, it is not possible to isolate a pile of size $1/12$ in finitely many steps.

However, assuming one has enough time and patience to work through infinitely many steps, consider that division by $4$ is available as a split-by-$8$ followed by a recombine-$2\,$. Then, first split the whole pile by $4\,$, and set $1/4$ aside. Further split one of the remaining $1/4$ piles by $4$ and add it to the $1/4$ pile previously set aside. Repeat infinitely many times (or until the atoms of dirt no longer split ;-)), then the pile which collected all the intermediate splits will have a size of:

$$ \frac{1}{4}+\left(\frac{1}{4}\right)^2+\left(\frac{1}{4}\right)^3 + \cdots \;=\; \frac{1}{4} \cdot \frac{1}{1-\cfrac{1}{4}} \;=\; \frac{1}{3} $$

Finally, divide that pile once more by $4$ to get $\;\displaystyle \frac{1}{4} \cdot \frac{1}{3} = \frac{1}{12}\,$.

Answer 2

As noted in the comments, this is not possible in a finite number of steps. However, you can approximate $1/12$ with arbitrary accuracy using $1/8$ divisions. To see how, let's write the fraction $1/12$ in its base-$8$ expansion:

$$\left(\frac1{12}\right)_8=0.0525252...$$

Thus, you can divide your original quantity into $\frac18$'s, divide one of those $\frac18$'s into $\frac1{8^2}$'s, and start with $5$ of those. To get closer, divide a remaining $\frac1{8^2}$ into $\frac1{8^3}$'s, and take $2$ of those. Continue until you're close enough!

Answer 3

Not sure if this is the quickest, but was the most obvious to me.

1. Pound the splitter down: you now have 8 piles of 1/8.

2. Combine three piles, then pound the splitter down. (this gives you 24s, seems like we're getting somewhere!) Repeat this with another 3 piles. You now have 16 piles of 1/24 and 2 piles of 1/8.

3. Combine the two piles of 1/8, with 3 piles of 1/24. Pound the splitter down on that. You now have 24 piles of 1/24.

4. Combine two piles of 1/24 and you have your 1/12th.

5. Eat the dirt, for extra credit. ;-)

Answer 4

A classic move used in Factorio: take the output of two of the split belts and feed it back into the input, until the splitter runs "clean" (this is a thing that handles dirt though, so, uh, good luck). At this point you're as close to $1/6$ in each pile as you can get. Move the splitter, feed one of the little piles in, and four of the outputs together gives 1/12.