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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