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Comment by jgrahamc | Hacker News Reader

It doesn't score very well on my metric. Not badly but not well.

    [[ 1  9 17 24 16  8]
     [ 5 13 21 20 12  4]
     [ 3 11 19 22 14  6]
     [ 7 15 23 18 10  2]]

    400.634917287

Compare with this one found using the 'genetic algorithm':

    [[ 9  4 18 10 15 20]
     [16 21 12  1 22  7]
     [ 2  6 24  5 17 13]
     [14  8 19 11 23  3]]

    386.229085028

View on HN

Best I've found (with a random shuffle of the numbers every 5M swaps) is (score verified with `advent.py`):

         [[11 16  8 13  2 20]
          [18 21  5 24  7 15]
          [ 3  9  1 19 22  4]
          [14 23  6 17 10 12]]

         382.46224448143795

That's a lot of swaps! I played around with a simulated annealing approach in my lunch break which consistently gets down to 383.xx with about 10000 swaps, or 379.xx in 100000 swaps. I've yet to tweak the annealing parameters to see if I can get down to (or beat) the 378 example in the article.

It's basically the same as the swap() function in advent.py except you sometimes allow swaps that result in worse scores according to an ever-decreasing probability ("temperature"). The added controlled randomness allows you to break out of local minima at the start and hone in on the local optimum towards the end.

EDIT: after a little tweaking of the temperature schedule I got 377.59 after 10K swaps.

The matrix

   8 20 15 10  5 17 
  13  4 22 12 23  1 
  18  2 24 21  3  7 
  11  6 16  9 14 19

gets a score 376.9629. This came from starting with a random matrix, trying all possible swaps and taking the best one, and iterating until a local minimum is reached.

This is a purely random approach. Currently on 382.08832 after 1.4B swaps. Might have a look at tweaking things later when I'm not at work...

Interesting approach. Can you share your code?

Sure. My best so far is:

    np.matrix('12 19 6 14 4 9; 8 17 1 24 22 16; 21 3 23 2 20 11; 15 10 5 13 18 7')

which has a score of 376.899. This was produced in 20K swaps with the line:

    best, score = anneal(n=10000)

which was then further refined with:

    best, score = anneal(start_temp=0.2, advent=best, n=10000)

where anneal() is defined here: https://pastebin.com/xBVGJfQd

The score() function is a bit slow at the moment. If I get some time later, I might see if I can speed it up a bit, which will allow for some faster experimentation.

EDIT: replaced code with pastebin link to save space in comments

Thanks. That's great.

Slight improvment to the above:

  [[12, 19,  6, 14,  4,  9],
   [ 8, 17,  1, 24, 22, 16],
   [21,  3, 23, 20,  2, 11],
   [15, 10,  5, 13,  7, 18]]

Score = 376.6144674353488

Found by increasing number of iterations to 100000

Sim.annealing followed by an exhaustive pair-swap search to find the local minimum would have found this more efficiently. Possibly there are some further refinements left -- I haven't run the exhaustive pair search on the above!

My best so far, but not using the improvements above, just small tweaks to the code from the blog post:

  [[ 9 15  4 20  6 12]
   [18 23 11  1 22 17]
   [ 7  2 16 24  3  8]
   [13 21  5 10 19 14]]
  376.364049355

My Go code, slightly tweaked from the pure random approach, got this one:

    [[17 11  6 19 14  8]
     [ 4 22 24  1  3 21]
     [ 9 15  2 23 12 16]
     [13 20  7 18  5 10]]
    375.998672775885

Ooh nice! That's the first one I've seen below 376.0

I've gotten close, but not cracked it yet. I was wondering if anyone would break the 376.0 barrier!

375.923809971073 :-)

https://gist.github.com/jffry/fab43b5b65499c3f513fea70159780...

Added to https://rjp.is/calendars/topthree.png (which is now four but nvm) to show the exceptional symmetry. I'm thinking we might be right on the lower limit here - doesn't seem much room for more symmetricalising.

I'm wondering if vertical symmetry might be involved (and a way of optimising future efforts) - plotting the journeys of the top three on this post definitely seems to imply that.

https://rjp.is/calendars/topthree.png

Compare and contrast the original set from @jgc's article:

https://rjp.is/calendars/originals.png

Thanks for keeping track. Quite fascinating how algorithms employing so much randomization can arrive at such structure and symmetry.

Finally got a < 376.0 from my own code using simulated annealing method...

375.998672775885

  [[13 20  7 18  5 10]
   [ 9 15  2 23 12 16]
   [ 4 22 24  1  3 21]
   [17 11  6 19 14  8]]

...only to realise it's the same as yours, but with the rows reversed! Rather surprised, but I now wonder if there is only a small number of very-low-scoring solutions. So perhaps this is less of a coincidence than it first appears.

I'm now using a much more aggressive temperature drop-off to find decent candidates early, followed by a tempering phase to search for nearby solutions, and a final cool-off to refine the final answer. I'm still using only random pair swaps in Python, so probably wasting a lot of cycles, but I'm still quite surprised how quickly it converges to some pretty decent scores. Beyond that I'm just going to try lots of random starting layouts.

I'm interested to see if my method can find any of the other posted solutions or (fingers crossed!) any new ones, but I may need to crunch through a lot more candidates... I will have to translate from Python into something faster to up my game!

Not entirely scientific but starting from the trivial 1-24 board, I did 100M random swaps (in Go, keeping all of them, takes about 4 minutes) and counted how many scores were in each bucket of 10 (ie 370-379.99, 380-389.99, etc.) Having done this a few times and never got anything in 37, I'm thinking the number of solutions below 380 must be relatively small, yeah.

    37	0
    38	2377
    39	1103812
    40	16535778
    41	39376324
    42	29525491
    43	10609914
    44	2394340
    45	395239
    46	51463
    47	4880
    48	363
    49	19
    50	0

The symmetry thing makes sense, but now I wonder if there's are really "interesting" or not. Does the symmetry spoil the fun of searching or not?

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