Unsolved problem

[simont]

Here's a mathematical sort of question that occurred to me last year. I've been pondering it off and on ever since, but not reached any useful conclusions; and I just came across my file of notes on it, and thought perhaps I should post it somewhere for other people to ponder as well.

Let m and n be positive integers. Suppose you have m sticks of length n, and you want to turn them into n sticks of length m, by cutting them into pieces and gluing the pieces back together. And the tricky bit is that you want to do this in a way that maximises the length of the shortest fragment involved in the dissection.

One obvious approach is to glue all your sticks end-to-end into one long stick of length mn, then cut that into n equal pieces. If you do that, the shortest fragment will have a length equal to gcd(m,n). So that's a lower bound on the solution in general, and in some cases (e.g. m=3, n=2) it really will be the best you can do.

But gcd(m,n) is not always the best you can do. For example, consider m=6 and n=5. The obvious gcd solution gives a shortest fragment of 1, but actually this case has a solution with shortest fragment 2: cut each of the six 5-sticks into pieces of length 2 and 3, then reassemble as three lots of 3+3 and two lots of 2+2+2. (In fact this was the case that first brought the problem to my attention.)

I also know that the best solution will not necessarily involve fragments of integer length. For example, consider m=5 and n=7. If you constrain yourself to only use integer-length fragments, then you can't make the shortest fragment any longer than 1. (If you try to make it 2, then you have no option but to cut up each 7-stick as 2+2+3, but then you have ten 2s and five 3s, out of which you can't make the seven 5s you need without bisecting a 2.) But with fractional fragment lengths you can improve on a shortest fragment of 1, by cutting up three 7-sticks as (8/3 + 8/3 + 5/3) and the other two as (7/3 + 7/3 + 7/3), then reassembling as six lots of (8/3 + 7/3) and one (5/3 + 5/3 + 5/3). So now your shortest fragment is one and two-thirds units long.

So, does anyone have thoughts? Other than the above examples, I haven't managed very many myself. I haven't even come up with a sensible computer search algorithm to reliably determine the best answer for a given m,n pair – so, in particular, I don't even know that 5/3 is the best answer for the example above, only that it's the best answer I know of. I have a vague idea that the best answer generally seems to involve at least one stick (on either the source or the destination side) being cut into equal-length fragments, but I wouldn't be at all surprised to find that wasn't true in all cases.

Comments

[simont]

Indeed, you certainly can't get a longer smallest stick by small perturbations of that answer, because one of the 5-sticks is entirely made up of smallest-fragments and so if you lengthen them all by any ε > 0 then that stick overflows. What I was hoping to show was that the converse is also true – that the only way in which a dissection can be locally optimal is if some stick is made up entirely of smallest-fragments, by showing that in all other cases you can find a system of ε-adjustments that lengthens every smallest-fragment. No luck yet, though!

(In fact, writinghawk has proved over on LJ that my dissection for 5-into-7 is globally as well as locally optimal, which is more than I had previously known.)

I've written a thing that looks for integer solutions

Ooh, I'd like to see whatever data it's generated.

Also it can't handle irrationals and so far can do fractions only where all stick-fragments share a denominator

In any rational dissection there must be some denominator shared by all fragments (just take the lcm of all denominators), so the latter isn't a problem. And I'm still convinced that irrationals can't appear in any solution unless there's an equally good or better one without them (I have a half-thought-out proof idea involving treating R as a vector space over Q), so I'm not worried about the former either.

[naath]

So far it is tediously slow and hasn't got anything harder than 5,7...

[naath]

Apparently I'm not exactly doing this the optimal way... because 7 x 10 in 1/3 steps is, er, ENOMEMORY. Fantastic.

5 x 8 sticks in half integer steps is best with 4 sticks cut into 2/3 and 4 uncut assembled into 4 lots of 3/5 and 1 of 2/2/2/2 which gives more than 3 stick-parts.

With m and n <= 10 and m<n in half integer steps 12 combinations have a smallest-stick-fragment larger than gcd(m,n); 4 of which are non-integer (5 x7 is the smallest). I have no idea how to prove whether any given answer could be bested by taking smaller steps other than by trying progressively smaller steps and seeing.

[simont]

5 x 8 sticks in half integer steps is best with 4 sticks cut into 2/3 and 4 uncut assembled into 4 lots of 3/5 and 1 of 2/2/2/2 which gives more than 3 stick-parts.

Borrowing writinghawk's proof technique: in a better solution, each of the five 8-sticks would have to be cut into at most three pieces (if you cut one in four then a piece must be <=2), which gives 15 pieces overall, and so at least one of the eight 5-sticks must end up as a single piece (15 isn't enough pieces for two each). The 8-stick including that whole 5-stick then has 3 units of length left over; if you divide that in two then you have a piece <=1.5 (no good) and OTOH if you leave it whole then that leaves 2 units on the 5-stick you cut it off.

So this dissection for 5 into 8 cannot be beaten even if you were to increase the denominator, and hence gerald_duck's question is indeed answered.

(I do wonder how far that proof technique can be automated. It might give rise to a better search algorithm!)