http://www.ericharshbarger.org/wiki/ 2026-09-26T23:34:47+00:00 http://www.ericharshbarger.org/wiki/ http://www.ericharshbarger.org/wiki/lib/tpl/dokuwiki/images/favicon.ico text/html 2026-09-26T17:13:07+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=d7200_d8400_d9072_d25200_5d64800&rev=1790442787&do=diff This page demonstrates a permutation-fair d7200 + d8400 + d9072 + d25200 + 5d64800 configuration of Go First Dice, found by Paul Meyer on Sept 26, 2026. With an LCM of 453600, this solution can be scaled to a homogeneous 9d453600. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiihhhhh… text/html 2026-09-26T16:49:05+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=records&rev=1790441345&do=diff Current Records Here is a chart summarizing the best/smallest sets known for various player counts. Entries are known to be the best (lowest) possible values achievable (generally through exhaustive computer search). Players Smallest LCM Fewest Total Sides text/html 2026-09-26T16:46:21+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=d840_d2160_d3024_d5040_4d15120&rev=1790441181&do=diff This page demonstrates a permutation-fair d840 + d1080 + d1680 + 5d15120 configuration of Go First Dice, found by Paul Meyer on Sept 26, 2026. With an LCM of 15120, this solution can be scaled to a homogeneous 8d15120. hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhfffdabccbadcabddbacbacddcabfffefffbacddcabcabddbacdabccbadffffeffdabccbadcabddbacbacddcabefffffdabccbadcabddbacbacddcabffffbacddcabcabddbacdabccbadfffffffffffffbacddcabcabddbacdabccbadffefbacddcabcabddbacdabccbadfffffebacddcabcabddbacdabcc… text/html 2026-09-26T16:35:50+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=d360_d504_d840_4d2520&rev=1790440550&do=diff This page demonstrates a permutation-fair d360 + d504 + d840 + 4d2520 configuration of Go First Dice, found by Paul Meyer on Sept 26, 2026. With an LCM of 2520, this solution can be scaled to a homogeneous 7d2520. gggggggggggggggggggggggggggggggggggggggggfeffbacddcabcabddbacbacddcabdabccbadcabddbacdabccbaddabccbadeedabccbadcabddbacbacddcabcabddbacdabccbadbacddcabdabccbadfeffecabddbacbacddcabcabddbacbacddcabbacddcabcabddbacdabccbadefefcabddbacbacddcabbacddcabbacddcabdabccbadcabddbaccabddbacfffef… text/html 2026-09-26T14:00:23+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=d7200_d8400_d9072_d25200_4d25920_d43200&rev=1790431223&do=diff This page demonstrates a permutation-fair d7200 + d8400 + d9072 + d25200 + 4d25920 + d43200 configuration of Go First Dice, found by Paul Meyer on Sept 26, 2026. With an LCM of 907200, this solution can be scaled to a homogeneous 9d907200. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii… text/html 2026-09-26T13:45:40+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=d840_d1080_d1680_4d6048_d10080&rev=1790430340&do=diff This page demonstrates a permutation-fair d840 + d1080 + d1680 + 4d6048 + d10080 configuration of Go First Dice, found by Paul Meyer on Sept 26, 2026. With an LCM of 30240, this solution can be scaled to a homogeneous 8d30240. hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhfeabcddcbaeeeeecbaddabcedbaccabdeeeeeedbaccabdecbaddabceeeeeabcddcbaefffffecbaddabceeeeedbaccabdeabcddcbaeeeeeeabcddcbaedbaccabdeeeeecbaddabcefedbaccabdeeeeeabcddcbaecbaddabceeeeeecbaddabceabcddcbaeeeeedbaccabdeffffffedbaccabdeeeee… text/html 2026-09-26T13:41:40+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=d360_d720_d840_4d5040&rev=1790430100&do=diff This page demonstrates a permutation-fair d360 + d720 + d840 + 4d5040 configuration of Go First Dice, found by Paul Meyer on Sept 26, 2026. With an LCM of 5040, this solution can be scaled to a homogeneous 7d5040. gggggggggggggggggggggggggggggggggggggggggfeffbacddcabcabddbacbacddcabdabccbadcabddbacdabccbaddabccbadeedabccbadcabddbacbacddcabcabddbacdabccbadbacddcabdabccbadfeffecabddbacbacddcabcabddbacbacddcabbacddcabcabddbacdabccbadefefcabddbacbacddcabbacddcabbacddcabdabccbadcabddbaccabddbacfffe… text/html 2026-09-26T03:17:03+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=d20_5d180&rev=1790392623&do=diff This page demonstrates a permutation-fair d20 + 5d180 configuration of Go First Dice, found by Paul Meyer on Sept 25, 2026. With an LCM of 180, this solution can be scaled to a homogeneous 6d180. fffdabccbadcabddbacbacddcabfffefffbacddcabcabddbacdabccbadffffeffdabccbadcabddbacbacddcabefffffdabccbadcabddbacbacddcabffffbacddcabcabddbacdabccbadfffffffffffffbacddcabcabddbacdabccbadffefbacddcabcabddbacdabccbadfffffebacddcabcabddbacdabccbadfffffdabccbadcabddbacbacddcabffeffebacddcabcabddbacdabccbadf… text/html 2026-09-26T00:50:39+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=d80_d120_4d240&rev=1790383839&do=diff This page demonstrates a permutation-fair d80 + d120 + 4d240 configuration of Go First Dice, found by Paul Meyer on Sept 25, 2026. eeffabccbaddddabccbaddbccbaaaabccbddfffffddbccbaaaabccbddabccbaddddabccbaeeffeeeeabccbaddddabccbaddbccbaaaabccbddffefeeabccbaddddabccbaabccbaddddabccbaffffffffabccbaddddabccbaddbccbaaaabccbddeeabccbaddddabccbaabccbaddddabccbaeefffffefffeabccbaddddabccbaabccbaddddabccbaeeabccbaddddabccbaabccbaddddabccbaeefeffffffabccbaddddabccbaddbccbaaaabccbddfffffeeeeabccbaddddabcc… text/html 2026-09-19T19:52:33+00:00 harshec (harshec@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=inhomogeneous_solutions&rev=1789847553&do=diff Solution Counts For Inhomogeneous Sets These are configuration counts for various inhomogeneous dice sets (as found via exhaustive computer search), either with mirror-symmetry or no such restriction (note that column-grouped and column-paired symmetry can only happen with homogeneous sets). These counts avoid equivalent sets where the dice are simply relabeled. In other words, these counts assume that, if a and b are equal sized dice, then in the string representation of a solution the first text/html 2026-09-19T05:27:59+00:00 bmenrigh (bmenrigh@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=homogeneous_solutions&rev=1789795679&do=diff Solution Counts For Homogeneous Sets These counts avoid equivalent sets where the dice are simply relabeled. In other words, these counts assume that in the string representation of a solution the first “a” appears before the first “b” which appears before the first “c”, and so on. text/html 2026-09-19T05:23:19+00:00 bmenrigh (bmenrigh@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=open_questions&rev=1789795399&do=diff Open Questions 5d30? The most obvious question is whether or not a permutation fair 5d30 set exists. If such a set exists, it would make a beautiful successor to the 4d12 set that is commercially available. If such a set does not exist, it would be nice to have a proof of that (whether a mathematical proof or via exhaustive computer search). We already know (via exhaustive computer search) that a column-grouped 5d30 set can not be all-subset-place-fair, much less perm-fair. This also rules out… text/html 2026-09-19T05:20:15+00:00 bmenrigh (bmenrigh@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=fairness&rev=1789795215&do=diff Concepts of Fairness Consider the following three “levels of fairness”, starting with the “weakest” (each stronger level of fairness actually implies the weaker levels): * Go-first-fair. This is as originally proposed by James Ernest to Eric Harshbarger. Each player has an equal chance of rolling the highest number, and that person would go first. text/html 2026-09-19T05:15:25+00:00 bmenrigh (bmenrigh@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=significant_solutions&rev=1789794925&do=diff 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. 3d6 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 text/html 2026-09-17T22:46:01+00:00 harshec (harshec@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=more_info&rev=1789685161&do=diff More Information Where You Can Buy Go First Dice * U.S.: The Dice Shop at MathArtFun * U.K.: MathsGear Publications * 2012 article in The Guardian * GAMES MAGAZINE, December 2012. Much like the newspaper article above, this one is very much not math-focused; it is meant to introduce the concept of Go First Dice to laypeople. * text/html 2026-09-14T22:24:59+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=induction_chart&rev=1789424699&do=diff Induction Chart m copies of the original n-die set 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 24 30 n0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 text/html 2026-09-14T21:16:49+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=induction_and_cotesian_numbers&rev=1789420609&do=diff Note: The following explanation was largely AI-generated (via Gemini), and could likely benefit from additional manual verification by humans. Why the Induction Chart Diagonal Matches the Cotesian Numbers The diagonal numbers on the Go First Dice induction chart match the Cotesian numbers because constructing a fair ( text/html 2026-09-14T15:19:31+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=m30n8&rev=1789399171&do=diff 6+9+3+37+0+2+1+30+7+22+1+8+12+11+10+18+10+11+12+8+1+22+7+30+1+2+0+37+3+9+6 == 336 text/html 2026-09-14T15:19:22+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=m24n8&rev=1789399162&do=diff 3+100+2+3+103+36+33+92+4+10+53+15+142+15+53+10+4+92+33+36+103+3+2+100+3 == 1050 text/html 2026-09-14T15:19:14+00:00 paulmeyer (paulmeyer@undisclosed.example.com) http://www.ericharshbarger.org/wiki/doku.php?id=m20n8&rev=1789399154&do=diff 27+209+1+73+265+158+18+35+2+259+345+20+0+0+20+487+53+14+0+281+1 == 2268