These problems have been a major success story for automated reasoning and constraint satisfaction. Various open problems of interest to mathematicians have been solved using a variety of different computer programs including Slaney’s constraint solver FINDER, Zhang’s Davis-Putnam procedure SATO, Stickel’s LDPP procedure, and the MGTP model generation program. Open problems solved include:
QG1 existence of idempotent QG1.12 QG2 existence of idempotent QG2.12, QG2.14-15, non-existence of idempotent QG2.10 QG3 existence of idempotent QG3.12 QG4 existence of idempotent QG4.12 QG5 non-existence of QG5.10, QG5.14, and of idempotent QG5.9-10 and QG5.12-16 QG6 existence of QG6.9 and QG6.17, non-existence of QG6.7, QG6.10-11, QG6.14-15 QG7 non-existence of QG7.7-8, QG7.10-12 and QG7.14-16
Problems that remain open include:
QG5 QG5.18, QG5.26, QG5.30, QG5.38, QG5.42, QG5.158, and many of the idempotent cases starting at QG5.18 QG6 QG6.20-21, QG6.24, QG6.41, QG6.44, QG6.48, QG6.53, QG6.60, QG6.69, QG6.77, QG6.93, QG6.96, QG6.101, QG6.161, QG6.164, QG6.173 QG7 QG7.33
For idempotent problems, the following table summarizes some of these results where e=exists, n=no such quasigroup, and ?=open (some entries left blank as status not known to problem proposer).
order 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 QG1 e n e e e e e e e e e e e e e e QG2 e n e e e n e e e e e e e e e e QG3 n n n e e n n e e n n e e n n e QG4 e n n n e n n e e n n e e n n e QG5 e n e e n n e n n n n n e ? e ? QG6 n n n e e n n n e n n e e n n ? QG7 e n n n e n n n e n n