In February of 2023 Eric Harshbarger started using resources at the Alabama Supercomputer Authority to search certain spaces for possible permutation-fair sets. The hope is that such improved speeds will help fill in entries farther down the list on the page of Inhomogeneous Solution Counts. Previous reliance on single computers to conduct such searchs allowed Eric, Landon Kryger, and Brandon Enright to make quite a bit of progress in eliminating configurations by exhaustive search. However, eventually the search spaces became too large; things started becoming too big around 18-18-18-24-30 or so.
With the discovery of a valid 20-36-36-36-36 set by Bram Cohen in December 2022, Eric was encouraged that the use of a high performance cluster (HPC, aka, supercomputer) might help search this remaining window of configurations. An HPC will likely only speed things up by a few orders of magnitude, and while that would not help in spaces as vast as the 5d30 space (1098 in size with no symmetry restrictions), a factor of 1000x or so might help discover an inhomogeneous set between 18-18-18-24-30 and 20-36-36-36-36 (or, definitively rule out the existence of such sets).
To take advantage of the HPC, Eric had to modify the search code in use. The original code had been written by Landon Kryger and was extraordinarily fast; but, again, it had pretty much reached its limit when running a single search on a single core processor. The java code was modified by Eric to do a couple of things.
First, an option to write out all found solutions was added. The original program actually had the ability to start a search not always from the n=1 stage, but it could, given an n-player configuration, try to find an n+1-configuratuion from that seed. So, being able to write out all previous solutions of a smaller set would generate a file of many, many seeds that could be used. And, importantly, that data file could be easily partitioned up and searched by parallel processes.
So, secondly, Eric then wrote in the ability for the search program to not only read in seeds of small configuratuions, but to read in only certain sections of that data file.
All that was left to do was to write a few bash scripts on the supercomputer login node so that multiple searches could be run across multiple cores, each search using a different sector of the seed file.
The first successful implementation of all of this was used to search the 18-18-18-24-30-m space (that is asking: “Is there a 5-player, permutation-fair set that uses 3d18+d24+d30 and has mirror symmetry?”)
First Eric generated all valid 3-player, 18-18-18-m configurations. There are 767975 of these, and it took a couple of seconds to generate them all on a laptop. Then, using the HPC, he ran eight searches in parallel (looking to see how many ways a d24 could be added to create 18-18-18-24-m solutions), partitioning those 767975 seeds into eight groups of 100000 each (the last group not having a full count). Those eight searches were completed in less than ten minutes; and each wrote out its own output file. Those output files were concatenated together into one large file, and the total number of 18-18-18-24-m configurations was confirmed to be 3159772 (this was known from a previous search done long ago). That data file of 3159772 4-player configurations was then partitioned up into 32 sections (with 100,000 configurations in each, save the final), and the HPC let loose with 32 individual, parallel searches trying to add in a d30 as a fifth die.
An interesting thing about those parallel searches: the amount of time to complete a particular partition can vary drastically depending upon how “fertile” the section is (that is, how well an n-configuration lends itself to possible n+1-configurations). The “quickest” sector of 100000 was searched in under ten minutes, while the “slowest” one took nearly five days. In fact, when that final sector still had about 33,000 seeds to search through, Eric simply broke that section of 33,000 seeds into 100 groups of 330 each, and let the HPC finish up the job in a couple of minutes.
The end result of all of this is rather anticlimactic: there are no 18-18-18-24-30-m permutation-fair sets.
But, at least that “0” can be added way down the chart of Inhomogeneous Solution Counts. And, there is no doubt that more results (whether negative or positive) can be added, results that a single-core search would take months or years to accomplish (next on the list: 18-18-18-30-36-m).
It is at this point Eric should formally that the Alabama Supercomputer Authority by including their recommended blurb: “This work was made possible in part by a grant of high performance computing resources and technical support from the Alabama Supercomputer Authority.” Any results garnered from ASA resources will include the initial “ASA” next to them.