proposed by | Toby Walsh tw@cs.york.ac.uk |
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 | e | ||||