significant_solutions

Below is a (more or less) chronological listing of some of the most important Go First Dice sets that have been discovered and brief explanations about their significance.

`acbbacbccaabacbbac`

Discovered by Robert Ford (on 10 August 2010). May not have been the first such set; Eric Harshbarger remembers hearsay that a three player set had been sold previously – though he never saw it.

`abcddcbadbaccabdcbaddabccbaddabcdbaccabdabcddcba`

Discovered by Robert Ford (on 4 September 2010). Not only “go first fair” but, in fact, “permutation fair”. This is the set that, to this day is sold to the public. Originally sets were engraved and inked by Eric Harshbarger; later they were licensed to be sold at The Dice Shop in the U.S. and Maths Gear in the U.K.

`aaaaaaabcdedccccdedcbbbcdedccccdedcbbbcdedccccdedcbaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabcdedccccdedcbbbcdedccccdedcbbbcdedccccdedcbaaaaaaaaaaaabcdedccccdedcbbbcdedccccdedcbbbcdedccccdedcbaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabcdedccccdedcbbbcdedccccdedcbbbcdedccccdedcbaaaaaaa`

Discovered by Paul Vanetti. Though never physically manufactured (and the d90 is not really suitable for such production), this was the first set that demonstrated permutation fairness for 5 players. It was also the first valid *inhomogeneous* set. This solution was first announced on Byron Knoll's blog.

`edcabaccccabacddddcabaccccabacdeeeeedcabaccccabacddddcabaccccabacdedcabaccccabacddddcabaccccabacdeeeeeedcabaccccabacddddcabaccccabacdedcabaccccabacddddcabaccccabacdeeeeedcabaccccabacddddcabaccccabacde`

Discovered by Eric Harshbarger (who used the fewest-sides-known 4-player set and “induced” a 5-player set from it using Paul Vanetti's method). The d72 in this set is *nearly* suitable for a physical die.

`eabcddbacecabdeedabcdcbaeecbadcabdeebacddcbaeecbadcbadeedbacedabcdabceedabcdcabeecabddbacecabdeebacddcabeedbacdbacedabceeabcdeabcdecabdcbadeebacddcabeecabddcbaeecbaddabcecbadecbadeecabdebacddcbaeedbacdabceebacdedbacdabceecbadcabdeedabcdcabedbaceecabddbaceeabcddcabeecbadcabdeedabcdcbaeedbaceabcdcbade`

Discovered by James Grime and Brian Pollock (on 29 November 2015). This set of five 60-sided dice is “place fair” but not “permutation fair” (though it is tantalizingly close to exhibiting that stronger fairness). It was constructed using an original technique that Grime and Pollock call “binary construction”.

`dbeaceacbdcabdedcbeaacdbeebcadedbacabcedacedbbdecacebdadbaecedacbdcaebbadcedecbabcaedacebdeadbccdebabadceaedbcbcaedcbedaedabcaecdbdcbeabdceacadbeeabdc`

Discovered by Brandon Enright (2012). This set of five 30-sided dice is “place fair”. This shows that we cannot prove that a set of 5d30 permutation-fair dice is impossible via Go-First-fairness or place-fairness.

`eacdbbddcbccdbaadaacdbbcbddccdbaeeeeeacdbbddcbccdbaadaacdbbcbddccdbaeacdbbddcbccdbaadaacdbbcbddccdbaeeeeeeacdbbddcbccdbaadaacdbbcbddccdbaeacdbbddcbccdbaadaacdbbcbddccdbaeeeeeacdbbddcbccdbaadaacdbbcbddccdbae`

Discovered by Eric Harshbarger (on 4 January 2018). With the largest die in the set having only 54 sides, this was the first 5-player set that actually “worked” (all the dice roll reasonably well and it could be physically produced/used).

`eeacbddbcabcdaadcbdcabbacdeeeeeeeedcabbacdbcdaadcbacbddbcaeeeedcbaabcdbcaddacbacdbbdcaeeeeeeeeacdbbdcabcaddacbdcbaabcdeeaaecbddbcebcdeedcbdcebbecdaaaaaaaadcebbecdbcdeedcbecbddbceaaaadcbeebcdbceddecbecdbbdceaaaaaaaaecdbbdcebceddecbdcbeebcdaaddecabbaceacbeebcabceaaecbddddddddbceaaecbacbeebcaecabbaceddddbcaeeacbacebbecaecbaabceddddddddecbaabceacebbecabcaeeacbddbbecaddaceacdeedcadceaaecdbbbbbbbbdceaaecdacdeedcaecaddacebbbbdcaeeacdaceddecaecdaadcebbbbbbbbecdaadceaceddecadcaeeacdbbccaedbbdeadebaabedbeaddaebccccccccbeaddaebdebaabedaedbbdeaccccbedaadebdeabbaedaebddbeaccccccccaebddbeadeabbaedbedaadebcc`

Discovered by Mike Purcell (on 12 December 2022). This set lowered the size of a 5-player homogeneous set to 120 sides. It can be physically produced as there is an isohedral 120-sided geometric solid. This set was produced by Flying Horseduck in a very limited run.

`ecbdaadbcabcddcbadbaccabdeeeeedbaccabdcbdaadbcabcddcbaeabcddcbadbaccabdcbdaadbceeeeeeabcddcbadbaccabdcbdaadbcedbaccabdcbdaadbcabcddcbaeeeeecbdaadbcabcddcbadbaccabde`

Discovered by Bram Cohen (on 26 December 2022). This set currently holds the records for the 'fewest total sides“ and the “smallest largest die” in the set.

`abcdeedcbadcbeebcdaaaaecbddbceecbddbceaaaadcbeebcdabcdeedcbaaadcbeebcdabcdeedcbaaecdbbdceaecdbbdceaabcdeedcbadcbeebcdaaaaaaecbddbceaabceddecbdcebbecdaadcebbecdbceddecbaaecbddbceaaaaaadcebbecdaecdbbdceaabcdeedcbabcdeedcbaaecdbbdceadcebbecdaaaecbddbceabceddecbaaaadcebbecddcebbecdaaaabceddecbaecbddbcea`

Discovered by Paul Meyer (on 31 July 2023). This set improves upon Michael Purcell's 5d120 set by reducing the number of sides in a five player homogeneous set. Image rendered by Carl Hoff.

significant_solutions.txt · Last modified: 2024/02/20 11:13 by jamesgrime