Maximum number of normalized diagonal Latin squares that can be orthogonal to the same diagonal Latin square of order n, https://oeis.org/A287695 n=1, a(1)=1 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, V. S. Titov, Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (in Russian). http://evatutin.narod.ru/evatutin_co_ls_odls_cnt_1_7.pdf Way of finding: brute force for source diagonal Latin squares + Euler-Parker method + DLX 0 n=2, a(2)=0 - n=3, a(3)=0 - n=4, a(4)=1 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, V. S. Titov, Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (in Russian). http://evatutin.narod.ru/evatutin_co_ls_odls_cnt_1_7.pdf Way of finding: brute force for source diagonal Latin squares + Euler-Parker method + DLX 0 1 2 3 2 3 0 1 3 2 1 0 1 0 3 2 n=5, a(5)=1 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, V. S. Titov, Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (in Russian). http://evatutin.narod.ru/evatutin_co_ls_odls_cnt_1_7.pdf Way of finding: brute force for source diagonal Latin squares + Euler-Parker method + DLX 0 1 2 3 4 2 3 4 0 1 4 0 1 2 3 1 2 3 4 0 3 4 0 1 2 n=6, a(6)=0 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, V. S. Titov, Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (in Russian). http://evatutin.narod.ru/evatutin_co_ls_odls_cnt_1_7.pdf Way of finding: brute force for source diagonal Latin squares + Euler-Parker method + DLX 0 1 2 3 4 5 1 2 0 5 3 4 4 3 5 0 2 1 3 5 1 4 0 2 5 4 3 2 1 0 2 0 4 1 5 3 n=7, a(7)=3 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, V. S. Titov, Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (in Russian). http://evatutin.narod.ru/evatutin_co_ls_odls_cnt_1_7.pdf Way of finding: brute force for source diagonal Latin squares + Euler-Parker method + DLX 0 1 2 3 4 5 6 2 3 1 5 6 4 0 5 6 4 0 1 2 3 4 0 6 2 3 1 5 6 2 0 1 5 3 4 1 5 3 4 0 6 2 3 4 5 6 2 0 1 n=8, a(8)=824 Announcement 1: -, M. O. Manzuk, Mar 14 2016 (a(8)>=824) Article 1: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, V. S. Titov, Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (in Russian). http://evatutin.narod.ru/evatutin_co_ls_odls_cnt_1_7.pdf (a(8)>=824) Announcement 2: addition to OEIS, E. I. Vatutin, May 30 2017 (a(8)=824) Article 2: Vatutin E.I., Manzuk M.O., Titov V.S., Kochemazov S.E., Belyshev A.D., Nikitina N.N. Orthogonality-based classification of diagonal latin squares of orders 1-8 (in Russian) // High-performance computing systems and technologies. Vol. 3. No. 1. 2019. pp. 94-100. http://evatutin.narod.ru/evatutin_ls_all_structs_n1to8_art.pdf (a(8)=824) Way of finding: row permutable search for a(8)>=824, brute force for source diagonal Latin squares + Euler-Parker method + DLX for a(8)=824 0 1 2 3 4 5 6 7 4 2 1 7 0 6 5 3 6 7 3 2 5 4 0 1 7 6 5 4 3 2 1 0 5 3 7 1 6 0 4 2 3 5 6 0 7 1 2 4 1 0 4 5 2 3 7 6 2 4 0 6 1 7 3 5 n=9, a(9)=614 Announcement 1: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=100&postid=938, Natalia Makarova, Aug 15 2020 (a(9)>=614) Announcement 2: https://vk.com/wall162891802_1485, E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, Dec 07 2020 (a(9)=614) Way of finding: brute force using X-based fillings for source diagonal Latin squares + Euler-Parker method + DLX 0 1 2 3 4 5 6 7 8 4 2 3 1 7 8 5 6 0 6 5 7 0 1 4 3 8 2 5 8 0 4 2 7 1 3 6 3 4 8 6 5 1 0 2 7 8 3 1 2 0 6 7 4 5 2 6 4 7 3 0 8 5 1 7 0 5 8 6 3 2 1 4 1 7 6 5 8 2 4 0 3 n=10, a(10)>=10 Announcement: https://boinc.progger.info/odlk/forum_thread.php?id=1&postid=1742, Natalia Makarova, Apr 25 2018 Way of finding: neighborhoods of generalized symmetries for source diagonal Latin squares + Euler-Parker method + DLX 0 1 2 3 4 5 6 7 8 9 1 2 0 4 5 3 7 9 6 8 9 7 6 2 3 8 5 1 4 0 7 5 1 8 6 9 2 0 3 4 3 9 8 6 7 4 1 2 0 5 2 4 5 9 8 1 0 3 7 6 4 8 3 0 2 6 9 5 1 7 8 6 7 5 9 0 3 4 2 1 6 3 9 1 0 7 4 8 5 2 5 0 4 7 1 2 8 6 9 3 n=11, a(11)>=32462 Announcement: https://boinc.tbrada.eu/forum_thread.php?id=3104&postid=4149, Tomas Brada, Jul 09 2020 Way of finding: ? (refound by Eduard I. Vatutin within the squares of combinatorial structure from one of cyclic diagonal Latin squares + Euler-Parker method + DLX) 0 1 2 3 4 5 6 7 8 9 10 1 2 3 5 6 9 7 10 0 4 8 7 10 8 0 3 1 5 9 6 2 4 3 5 9 4 10 6 8 0 2 7 1 4 6 7 10 1 8 2 3 9 0 5 5 9 4 6 8 7 0 1 3 10 2 10 8 0 1 5 2 9 4 7 3 6 6 7 10 8 2 0 3 5 4 1 9 8 0 1 2 9 3 4 6 10 5 7 2 3 5 9 7 4 10 8 1 6 0 9 4 6 7 0 10 1 2 5 8 3 a(12)>=3855983322 Announcement (square): https://vk.com/wall162891802_1732, Eduard I. Vatutin, Aug 18 2021 Announcement 1 (ODLS number, preliminary value value 3855938736, detected that a small part of results need additional validation): https://vk.com/wall162891802_1782, Eduard I. Vatutin, Oct 24 2021 Announcement 2 (right ODLS number for 1421 transversal (1847414), right value 3855938736-1802828+1847414=3855983322): https://boinc.ru/forum/topic/proekt-gerasimhome/?part=119#postid-4917, private communications, Eduard I. Vatutin, Oct 24-27 2021 Announcement 3 (independent confirmation): https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=3169, Mynx, Nov 11 2021 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 9 8 11 5 10 0 6 7 5 8 10 6 11 4 1 3 9 7 0 2 11 7 5 8 10 2 9 1 3 6 4 0 7 5 8 10 6 3 0 2 4 11 9 1 9 0 1 2 3 7 10 11 5 4 8 6 6 11 7 5 8 1 4 0 2 10 3 9 10 6 11 7 5 0 3 9 1 8 2 4 3 4 9 0 1 6 5 10 11 2 7 8 2 3 4 9 0 10 7 8 6 1 11 5 4 9 0 1 2 11 8 6 7 3 5 10 8 10 6 11 7 9 2 4 0 5 1 3 n=13, a(13)>=9 Announcement: https://vk.com/wall162891802_1576, Eduard I. Vatutin, Mar 14 2021 Way of finding: from known cliques 0 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12 0 1 4 5 6 7 8 9 10 11 12 0 1 2 3 6 7 8 9 10 11 12 0 1 2 3 4 5 8 9 10 11 12 0 1 2 3 4 5 6 7 10 11 12 0 1 2 3 4 5 6 7 8 9 12 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 12 0 3 4 5 6 7 8 9 10 11 12 0 1 2 5 6 7 8 9 10 11 12 0 1 2 3 4 7 8 9 10 11 12 0 1 2 3 4 5 6 9 10 11 12 0 1 2 3 4 5 6 7 8 11 12 0 1 2 3 4 5 6 7 8 9 10 a(13)>=248703? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=123&postid=1462, T. Brada, Feb 12 2021 Way of finding: cyclic square 0 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12 0 1 4 5 6 7 8 9 10 11 12 0 1 2 3 6 7 8 9 10 11 12 0 1 2 3 4 5 8 9 10 11 12 0 1 2 3 4 5 6 7 10 11 12 0 1 2 3 4 5 6 7 8 9 12 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 12 0 3 4 5 6 7 8 9 10 11 12 0 1 2 5 6 7 8 9 10 11 12 0 1 2 3 4 7 8 9 10 11 12 0 1 2 3 4 5 6 9 10 11 12 0 1 2 3 4 5 6 7 8 11 12 0 1 2 3 4 5 6 7 8 9 10 n=14, a(14)>=1 Announcement: https://vk.com/wall162891802_1576, Eduard I. Vatutin, Mar 14 2021 Way of finding: from known cliques 0 3 10 4 7 1 8 2 6 13 11 12 5 9 13 1 4 10 5 2 9 3 7 11 12 6 0 8 11 13 2 5 10 3 0 4 8 12 7 1 9 6 12 11 13 3 6 4 1 5 9 8 2 0 7 10 9 12 11 13 4 5 2 6 0 3 1 8 10 7 6 7 8 9 0 10 12 13 11 5 4 3 2 1 8 9 0 1 2 13 11 10 12 7 6 5 4 3 5 6 7 8 9 11 13 12 10 4 3 2 1 0 7 8 9 0 1 12 10 11 13 6 5 4 3 2 2 10 3 6 8 0 7 1 5 9 13 11 12 4 10 2 5 7 3 9 6 0 4 1 8 13 11 12 1 4 6 2 12 8 5 9 3 10 0 7 13 11 3 5 1 12 11 7 4 8 2 0 10 9 6 13 4 0 12 11 13 6 3 7 1 2 9 10 8 5 a(14)>=295309? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=1912, Natalia Makarova, Apr 29 2021 Way of finding: ? 0 2 4 6 12 11 7 9 13 10 8 5 3 1 12 1 3 11 8 7 5 4 9 6 13 2 0 10 10 0 2 13 6 9 4 5 7 8 11 3 1 12 7 11 8 3 5 0 12 10 1 4 2 6 13 9 9 13 6 2 4 1 10 12 0 5 3 8 11 7 6 12 1 7 13 5 3 2 4 11 9 0 10 8 4 9 11 1 2 12 6 8 10 3 0 13 7 5 1 3 5 8 10 13 9 7 11 12 6 4 2 0 2 5 7 10 0 6 13 11 8 1 12 9 4 3 13 8 10 5 7 3 1 0 2 9 4 12 6 11 3 4 9 12 1 8 11 13 6 0 10 7 5 2 5 7 13 0 3 10 8 6 12 2 1 11 9 4 8 10 0 9 11 4 2 3 5 13 7 1 12 6 11 6 12 4 9 2 0 1 3 7 5 10 8 13 a(14)>=307662? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=123&postid=1977, Natalia Makarova, May 10 2021 Way of finding: Gergely method 0 1 2 3 4 5 6 7 8 9 10 11 12 13 12 2 10 4 5 6 0 13 7 8 9 3 11 1 2 3 4 8 6 0 12 1 13 7 5 9 10 11 3 4 5 6 13 12 2 11 1 0 7 8 9 10 4 5 6 0 12 11 3 10 2 1 13 7 8 9 5 6 0 12 2 3 9 4 10 11 1 13 7 8 7 0 12 2 3 4 5 8 9 10 11 1 13 6 8 7 13 1 11 10 4 9 3 2 12 0 6 5 9 8 7 13 1 2 10 3 11 12 0 6 5 4 10 9 8 7 0 1 11 2 12 13 6 5 4 3 11 10 9 5 7 13 1 12 0 6 8 4 3 2 1 11 3 9 8 7 13 0 6 5 4 10 2 12 13 12 11 10 9 8 7 6 5 4 3 2 1 0 6 13 1 11 10 9 8 5 4 3 2 12 0 7 n=15, a(15)>=3 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=115&postid=1271, Natalia Makarova, Jan 19 2021 Way of finding: from known cliques 0 6 11 9 8 2 14 7 1 13 10 3 12 4 5 2 1 7 12 10 9 3 8 0 11 4 13 5 6 14 1 3 2 8 13 11 10 9 12 5 0 6 7 14 4 13 2 4 3 9 0 12 10 6 1 7 8 14 5 11 7 0 3 5 4 10 1 11 2 8 9 14 6 12 13 3 8 1 4 6 5 11 12 9 10 14 7 13 0 2 10 4 9 2 5 7 6 13 11 14 8 0 1 3 12 8 9 10 11 12 13 0 14 7 6 5 4 3 2 1 5 10 8 7 1 14 4 6 13 0 12 9 2 11 3 9 7 6 0 14 3 2 5 4 12 13 11 8 1 10 6 5 13 14 2 1 9 4 8 3 11 12 10 7 0 4 12 14 1 0 8 13 3 5 7 2 10 11 9 6 11 14 0 13 7 12 5 2 3 4 6 1 9 10 8 14 13 12 6 11 4 7 1 10 2 3 5 0 8 9 12 11 5 10 3 6 8 0 14 9 1 2 4 13 7 n=16, a(16)>=13 Announcement: https://vk.com/wall162891802_1576, Eduard I. Vatutin, Mar 14 2021 Way of finding: from known cliques 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 n=17, a(17)>=13 Announcement: https://vk.com/wall162891802_1576, Eduard I. Vatutin, Mar 14 2021 Way of finding: from known cliques 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 a(17)>=421? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=1917, Natalia Makarova, Apr 29 2021 Way of finding: semi-cyclic diagonal Latin squares 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 a(17)>=829? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2210, Natalia Makarova, Jun 01 2021 Way of finding: semi-cyclic squares 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 n=18, a(18)>=1 Announcement: http://www.natalimak1.narod.ru/diagon.htm, Natalia Makarova, Jun 06 2011 Way of finding: from known cliques 1 5 15 16 17 18 2 14 4 13 3 7 12 10 8 6 11 9 8 2 6 15 16 17 18 1 5 14 4 13 11 9 7 12 10 3 14 9 3 7 15 16 17 2 6 1 5 12 10 8 13 11 4 18 13 1 10 4 8 15 16 3 7 2 6 11 9 14 12 5 18 17 12 14 2 11 5 9 15 4 8 3 7 10 1 13 6 18 17 16 11 13 1 3 12 6 10 5 9 4 8 2 14 7 18 17 16 15 3 12 14 2 4 13 7 6 10 5 9 1 8 18 17 16 15 11 9 10 11 12 13 14 1 15 16 17 18 8 7 6 5 4 3 2 6 7 8 9 10 11 12 18 17 16 15 5 4 3 2 1 14 13 5 6 7 8 9 10 11 16 15 18 17 4 3 2 1 14 13 12 7 8 9 10 11 12 13 17 18 15 16 6 5 4 3 2 1 14 4 15 16 17 18 1 8 13 3 12 2 14 6 11 9 7 5 10 15 16 17 18 14 7 9 12 2 11 1 3 13 5 10 8 6 4 16 17 18 13 6 8 3 11 1 10 14 15 2 12 4 9 7 5 17 18 12 5 7 2 4 10 14 9 13 16 15 1 11 3 8 6 18 11 4 6 1 3 5 9 13 8 12 17 16 15 14 10 2 7 10 3 5 14 2 4 6 8 12 7 11 18 17 16 15 13 9 1 2 4 13 1 3 5 14 7 11 6 10 9 18 17 16 15 12 8 n=18, a(18)>=2 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2013, Natalia Makarova, May 14 2011 Way of finding: ? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 9 17 7 0 2 3 4 16 10 8 1 12 13 14 15 5 6 11 12 8 6 16 17 0 2 5 1 10 7 13 14 15 4 11 9 3 13 12 10 11 5 6 17 4 7 1 16 14 15 3 9 8 2 0 14 13 12 1 9 4 11 3 16 7 5 15 2 8 10 0 17 6 15 14 13 12 7 8 3 2 5 16 4 0 10 1 17 6 11 9 17 15 14 13 12 16 10 0 4 5 3 1 7 6 11 9 8 2 2 0 17 6 11 9 8 12 13 14 15 3 4 5 16 7 1 10 11 9 8 10 1 7 16 14 15 12 13 6 17 0 2 3 4 5 3 2 0 17 6 11 9 15 14 13 12 4 5 16 7 1 10 8 6 11 9 8 10 1 7 13 12 15 14 17 0 2 3 4 5 16 10 3 4 5 16 17 0 1 9 11 8 2 6 12 13 14 15 7 4 5 16 7 0 2 1 10 11 6 9 8 3 17 12 13 14 15 16 7 1 2 3 10 15 8 6 17 11 5 9 4 0 12 13 14 1 10 3 4 8 15 14 9 17 0 6 7 16 11 5 2 12 13 8 4 5 9 15 14 13 11 0 2 17 10 1 7 6 16 3 12 5 16 11 15 14 13 12 6 2 3 0 9 8 10 1 17 7 4 7 6 15 14 13 12 5 17 3 4 2 16 11 9 8 10 0 1 n=19, a(19)>=15 Announcement: https://vk.com/wall162891802_1576, Eduard I. Vatutin, Mar 14 2021 Way of finding: from known cliques (cyclic diagonal Latin squares) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a(19)>=471? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=132&postid=2166, Natalia Makarova, May 26 2021 Way of finding: cyclic and semi-cyclic squares 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a(19)>=775? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=139&postid=2480, Natalia Makarova, Jun 21 2021 Way of finding: cyclic and semi-cyclic squares ? a(19)>=1383? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=139&postid=2484, Natalia Makarova, Jun 22 2021 Way of finding: cyclic and semi-cyclic squares ? n=20, a(20)>=2 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2012, Natalia Makarova, May 14 2021 Way of finding: ? 0 11 9 18 17 6 5 14 3 12 1 10 8 19 16 7 4 15 2 13 10 1 19 8 7 16 15 4 13 2 11 0 18 9 6 17 14 5 12 3 4 15 2 13 11 0 19 8 7 16 5 14 3 12 10 1 18 9 6 17 14 5 12 3 1 10 9 18 17 6 15 4 13 2 0 11 8 19 16 7 8 19 6 17 4 15 13 2 1 10 9 18 7 16 5 14 12 3 0 11 18 9 16 7 14 5 3 12 11 0 19 8 17 6 15 4 2 13 10 1 2 13 10 1 8 19 6 17 15 4 3 12 11 0 9 18 7 16 14 5 12 3 0 11 18 9 16 7 5 14 13 2 1 10 19 8 17 6 4 15 16 7 4 15 12 3 10 1 8 19 17 6 5 14 13 2 11 0 9 18 6 17 14 5 2 13 0 11 18 9 7 16 15 4 3 12 1 10 19 8 11 0 18 9 6 17 14 5 12 3 10 1 19 8 7 16 15 4 13 2 1 10 8 19 16 7 4 15 2 13 0 11 9 18 17 6 5 14 3 12 15 4 13 2 0 11 8 19 16 7 14 5 12 3 1 10 9 18 17 6 5 14 3 12 10 1 18 9 6 17 4 15 2 13 11 0 19 8 7 16 19 8 17 6 15 4 2 13 10 1 18 9 16 7 14 5 3 12 11 0 9 18 7 16 5 14 12 3 0 11 8 19 6 17 4 15 13 2 1 10 13 2 1 10 19 8 17 6 4 15 12 3 0 11 18 9 16 7 5 14 3 12 11 0 9 18 7 16 14 5 2 13 10 1 8 19 6 17 15 4 7 16 15 4 3 12 1 10 19 8 6 17 14 5 2 13 0 11 18 9 17 6 5 14 13 2 11 0 9 18 16 7 4 15 12 3 10 1 8 19 a(20)>=2? Announcement: ?, Natalia Makarova, May 21 2011 Way of finding: ? ? n=21, a(21)>=1 Announcement: https://boinc.progger.info/odlk/forum_thread.php?id=162&postid=6486, Harry White, Oct 03 2020 Way of finding: SODLS 0 12 3 2 17 19 20 18 13 15 16 14 1 8 10 11 9 4 6 7 5 2 1 0 12 20 18 17 19 16 14 13 15 3 11 9 8 10 7 5 4 6 12 3 2 1 18 20 19 17 14 16 15 13 0 9 11 10 8 5 7 6 4 1 0 12 3 19 17 18 20 15 13 14 16 2 10 8 9 11 6 4 5 7 8 10 11 9 4 12 7 6 0 2 3 1 5 17 19 20 18 13 15 16 14 11 9 8 10 6 5 4 12 3 1 0 2 7 20 18 17 19 16 14 13 15 9 11 10 8 12 7 6 5 1 3 2 0 4 18 20 19 17 14 16 15 13 10 8 9 11 5 4 12 7 2 0 1 3 6 19 17 18 20 15 13 14 16 17 19 20 18 13 15 16 14 8 12 11 10 9 4 6 7 5 0 2 3 1 20 18 17 19 16 14 13 15 10 9 8 12 11 7 5 4 6 3 1 0 2 18 20 19 17 14 16 15 13 12 11 10 9 8 5 7 6 4 1 3 2 0 19 17 18 20 15 13 14 16 9 8 12 11 10 6 4 5 7 2 0 1 3 3 2 1 0 7 6 5 4 11 10 9 8 12 16 15 14 13 20 19 18 17 4 6 7 5 0 2 3 1 17 19 20 18 14 13 12 16 15 8 10 11 9 7 5 4 6 3 1 0 2 20 18 17 19 16 15 14 13 12 11 9 8 10 5 7 6 4 1 3 2 0 18 20 19 17 13 12 16 15 14 9 11 10 8 6 4 5 7 2 0 1 3 19 17 18 20 15 14 13 12 16 10 8 9 11 13 15 16 14 8 10 11 9 4 6 7 5 18 0 2 3 1 17 12 20 19 16 14 13 15 11 9 8 10 7 5 4 6 20 3 1 0 2 19 18 17 12 14 16 15 13 9 11 10 8 5 7 6 4 17 1 3 2 0 12 20 19 18 15 13 14 16 10 8 9 11 6 4 5 7 19 2 0 1 3 18 17 12 20 n=22 a(22)>=1? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=1949, Natalia Makarova, May 7 2021 Way of finding: Li Zhu method 0 21 20 19 18 17 15 14 16 8 6 5 4 3 9 2 11 12 10 7 13 1 4 1 21 20 19 18 0 16 15 9 7 6 5 10 3 12 13 11 8 14 2 17 11 5 2 21 20 19 18 1 17 10 8 7 6 4 13 14 12 9 15 3 0 16 5 12 6 3 21 20 19 18 2 11 9 8 7 14 15 13 10 16 4 1 17 0 15 6 13 7 4 21 20 19 18 12 10 9 8 16 14 11 17 5 2 0 1 3 17 16 7 14 8 5 21 20 19 13 11 10 9 15 12 0 6 3 1 2 4 18 16 0 17 8 15 9 6 21 20 14 12 11 10 13 1 7 4 2 3 5 18 19 14 17 1 0 9 16 10 7 21 15 13 12 11 2 8 5 3 4 6 18 19 20 3 15 0 2 1 10 17 11 8 16 14 13 12 9 6 4 5 7 18 19 20 21 13 14 15 16 17 0 1 2 3 19 18 21 20 12 11 10 9 8 7 6 5 4 6 7 8 9 10 11 12 13 14 20 21 18 19 5 4 3 2 1 0 17 16 15 2 3 4 5 6 7 8 9 10 18 19 20 21 1 0 17 16 15 14 13 12 11 1 2 3 4 5 6 7 8 9 21 20 19 18 0 17 16 15 14 13 12 11 10 21 20 19 18 16 14 13 15 0 7 5 4 3 17 2 8 1 10 11 9 6 12 20 19 18 15 13 12 14 17 11 6 4 3 2 21 16 1 7 0 9 10 8 5 19 18 14 12 11 13 16 10 4 5 3 2 1 20 21 15 0 6 17 8 9 7 18 13 11 10 12 15 9 3 6 4 2 1 0 19 20 21 14 17 5 16 7 8 12 10 9 11 14 8 2 5 7 3 1 0 17 18 19 20 21 13 16 4 15 6 9 8 10 13 7 1 4 6 5 2 0 17 16 11 18 19 20 21 12 15 3 14 7 9 12 6 0 3 5 4 13 1 17 16 15 8 10 18 19 20 21 11 14 2 8 11 5 17 2 4 3 12 1 0 16 15 14 6 7 9 18 19 20 21 10 13 10 4 16 1 3 2 11 0 12 17 15 14 13 7 5 6 8 18 19 20 21 9 n=23, a(23)>=19 Announcement: - (well known) Way of findins: cyclic MODLS 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 1 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 11 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20 21 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 1 2 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 11 12 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 13 14 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 11 12 15 16 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 17 18 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 19 20 21 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 21 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a(23)>=525? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=132&postid=2183, Natalia Makarova, May 27 2021 Way of finding: semi-cyclic squares ? n=24, a(24)>=2 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=1942, Natalia Makarova, May 07 2021 Way of finding: from known LS (SageMath) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 3 17 4 0 8 14 9 23 10 6 2 20 15 16 18 13 11 7 21 22 12 19 5 1 13 8 15 14 17 6 19 2 21 20 23 0 10 11 7 22 18 9 4 5 1 16 12 3 23 18 6 11 13 15 17 12 0 5 19 21 2 20 9 10 1 22 8 14 3 4 7 16 11 21 22 23 9 18 5 15 16 17 3 12 1 0 19 2 20 4 7 6 13 8 14 10 22 9 5 1 21 2 16 3 11 7 15 8 14 4 17 12 0 19 20 10 23 18 6 13 7 6 16 18 23 13 1 0 22 12 17 19 9 5 21 20 2 10 3 11 15 14 8 4 18 10 23 7 22 21 12 4 17 1 16 15 20 2 3 5 19 0 14 8 9 11 13 6 5 22 20 21 10 1 11 16 14 15 4 7 17 19 0 8 3 12 23 13 6 2 9 18 4 3 1 19 5 20 10 9 7 13 11 14 16 15 12 17 6 8 22 21 18 23 0 2 19 14 8 4 12 3 13 20 2 10 18 9 6 7 11 16 15 23 0 1 5 22 21 17 8 12 13 6 14 16 2 18 19 0 20 22 21 23 10 9 7 11 15 17 4 3 1 5 9 5 21 20 2 10 3 11 15 14 8 4 7 6 16 18 23 13 1 0 22 12 17 19 20 2 3 5 19 0 14 8 9 11 13 6 18 10 23 7 22 21 12 4 17 1 16 15 17 19 0 8 3 12 23 13 6 2 9 18 5 22 20 21 10 1 11 16 14 15 4 7 16 15 12 17 6 8 22 21 18 23 0 2 4 3 1 19 5 20 10 9 7 13 11 14 6 7 11 16 15 23 0 1 5 22 21 17 19 14 8 4 12 3 13 20 2 10 18 9 21 23 10 9 7 11 15 17 4 3 1 5 8 12 13 6 14 16 2 18 19 0 20 22 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3 4 5 6 7 8 9 10 11 15 16 18 13 11 7 21 22 12 19 5 1 3 17 4 0 8 14 9 23 10 6 2 20 10 11 7 22 18 9 4 5 1 16 12 3 13 8 15 14 17 6 19 2 21 20 23 0 2 20 9 10 1 22 8 14 3 4 7 16 23 18 6 11 13 15 17 12 0 5 19 21 1 0 19 2 20 4 7 6 13 8 14 10 11 21 22 23 9 18 5 15 16 17 3 12 14 4 17 12 0 19 20 10 23 18 6 13 22 9 5 1 21 2 16 3 11 7 15 8 a(24)>=10 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=1956, Natalia Makarova, May 7 2011 Way of finding: ? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 4 10 1 7 8 6 5 3 12 2 9 23 0 14 21 11 20 18 17 15 16 22 13 19 6 7 11 2 5 4 8 1 14 23 3 10 13 20 0 9 22 15 19 18 21 12 16 17 7 6 5 12 3 2 1 8 23 14 13 4 19 10 9 0 15 22 21 20 11 18 17 16 5 3 7 1 14 8 4 2 13 11 23 17 6 0 12 10 21 19 15 9 22 16 20 18 3 5 6 8 7 9 2 4 11 13 22 0 23 1 10 12 19 21 14 16 15 17 18 20 1 8 4 5 2 3 23 6 9 16 12 13 10 11 7 14 17 0 20 21 18 19 15 22 2 4 8 6 1 7 3 13 18 12 0 14 9 23 11 5 10 20 16 22 17 15 19 21 12 21 9 23 0 14 13 11 19 10 6 16 15 17 18 20 3 5 22 8 7 1 2 4 15 9 10 11 12 13 14 23 0 22 21 20 3 18 17 16 7 6 5 4 19 2 1 8 13 11 23 9 17 0 12 10 2 20 16 22 14 15 19 21 18 4 8 6 1 7 3 5 11 13 14 0 23 22 10 12 20 18 17 15 16 9 5 19 4 2 1 7 8 6 21 3 14 23 20 10 13 12 0 9 17 8 11 21 18 19 15 22 1 16 4 5 2 3 7 6 23 14 13 19 11 10 9 0 16 17 18 12 20 21 22 7 8 1 2 3 4 5 6 15 10 12 0 14 9 23 11 18 21 19 15 1 22 16 20 13 5 3 7 17 6 8 4 2 9 0 12 13 10 11 16 14 22 15 19 18 21 4 23 17 6 7 3 2 5 20 8 1 21 19 15 17 22 16 20 5 10 4 8 6 1 7 3 18 2 12 0 14 9 23 11 13 22 15 19 18 21 20 7 17 1 0 4 5 2 3 16 6 9 8 12 13 10 11 23 14 20 18 17 15 16 1 21 19 3 5 14 8 7 22 2 4 11 13 6 0 23 9 10 12 18 20 16 22 6 15 19 21 5 3 7 9 17 8 4 2 13 11 23 1 14 0 12 10 16 17 18 4 20 21 22 15 7 6 5 19 11 2 1 8 23 14 13 12 3 10 9 0 17 16 3 21 18 19 15 22 6 7 20 2 5 12 8 1 14 23 11 10 13 4 0 9 19 2 22 16 15 17 18 20 4 21 1 7 8 6 13 3 12 10 9 23 0 14 5 11 8 22 21 20 19 18 17 16 15 1 2 3 4 5 6 23 0 9 10 11 12 13 14 7 a(24)>=824? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2016, Natalia Makarova, May 14 2021 Way of finding: ? ? n=25, a(25)>=21 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=117&postid=1224, Natalia Makarova, Jan 17 2021 Way of finding: from known cliques (full MOLS of order n=25^2) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2 3 4 0 1 7 8 9 5 6 12 13 14 10 11 17 18 19 15 16 22 23 24 20 21 4 0 1 2 3 9 5 6 7 8 14 10 11 12 13 19 15 16 17 18 24 20 21 22 23 1 2 3 4 0 6 7 8 9 5 11 12 13 14 10 16 17 18 19 15 21 22 23 24 20 3 4 0 1 2 8 9 5 6 7 13 14 10 11 12 18 19 15 16 17 23 24 20 21 22 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 12 13 14 10 11 17 18 19 15 16 22 23 24 20 21 2 3 4 0 1 7 8 9 5 6 14 10 11 12 13 19 15 16 17 18 24 20 21 22 23 4 0 1 2 3 9 5 6 7 8 11 12 13 14 10 16 17 18 19 15 21 22 23 24 20 1 2 3 4 0 6 7 8 9 5 13 14 10 11 12 18 19 15 16 17 23 24 20 21 22 3 4 0 1 2 8 9 5 6 7 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 22 23 24 20 21 2 3 4 0 1 7 8 9 5 6 12 13 14 10 11 17 18 19 15 16 24 20 21 22 23 4 0 1 2 3 9 5 6 7 8 14 10 11 12 13 19 15 16 17 18 21 22 23 24 20 1 2 3 4 0 6 7 8 9 5 11 12 13 14 10 16 17 18 19 15 23 24 20 21 22 3 4 0 1 2 8 9 5 6 7 13 14 10 11 12 18 19 15 16 17 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 7 8 9 5 6 12 13 14 10 11 17 18 19 15 16 22 23 24 20 21 2 3 4 0 1 9 5 6 7 8 14 10 11 12 13 19 15 16 17 18 24 20 21 22 23 4 0 1 2 3 6 7 8 9 5 11 12 13 14 10 16 17 18 19 15 21 22 23 24 20 1 2 3 4 0 8 9 5 6 7 13 14 10 11 12 18 19 15 16 17 23 24 20 21 22 3 4 0 1 2 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 17 18 19 15 16 22 23 24 20 21 2 3 4 0 1 7 8 9 5 6 12 13 14 10 11 19 15 16 17 18 24 20 21 22 23 4 0 1 2 3 9 5 6 7 8 14 10 11 12 13 16 17 18 19 15 21 22 23 24 20 1 2 3 4 0 6 7 8 9 5 11 12 13 14 10 18 19 15 16 17 23 24 20 21 22 3 4 0 1 2 8 9 5 6 7 13 14 10 11 12 a(25)>=50? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=132&postid=2206, Natalia Makarova, May 29 2021 Way of finding: semi-cyclic squares 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 n=26 a(26)>=1? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=1951, Natalia Makarova, May 7 2011 Way of finding: ? 0 25 24 23 22 21 20 18 17 19 16 12 9 6 4 3 5 2 13 8 15 11 7 14 10 1 4 1 25 24 23 22 0 21 19 18 20 13 10 7 5 6 3 14 9 16 12 8 15 11 2 17 7 5 2 25 24 23 22 1 0 20 19 14 11 8 6 4 15 10 17 13 9 16 12 3 18 21 5 8 6 3 25 24 23 22 2 1 21 15 12 9 7 16 11 18 14 10 17 13 4 19 0 20 17 6 9 7 4 25 24 23 22 3 2 16 13 10 8 12 19 15 11 18 14 5 20 1 21 0 13 18 7 10 8 5 25 24 23 22 4 17 14 11 9 20 16 12 19 15 6 21 2 0 1 3 21 14 19 8 11 9 6 25 24 23 22 18 15 12 10 17 13 20 16 7 0 3 1 2 4 5 18 0 15 20 9 12 10 7 25 24 23 19 16 13 11 14 21 17 8 1 4 2 3 5 6 22 15 19 1 16 21 10 13 11 8 25 24 20 17 14 12 0 18 9 2 5 3 4 6 7 22 23 1 16 20 2 17 0 11 14 12 9 25 21 18 15 13 19 10 3 6 4 5 7 8 22 23 24 20 2 17 21 3 18 1 12 15 13 10 0 19 16 14 11 4 7 5 6 8 9 22 23 24 25 19 20 21 0 1 2 3 4 5 6 7 23 22 25 24 18 17 16 15 14 13 12 11 10 9 8 8 9 10 11 12 13 14 15 16 17 18 24 25 22 23 7 6 5 4 3 2 1 0 21 20 19 3 4 5 6 7 8 9 10 11 12 13 22 23 24 25 2 1 0 21 20 19 18 17 16 15 14 2 3 4 5 6 7 8 9 10 11 12 25 24 23 22 1 0 21 20 19 18 17 16 15 14 13 25 24 23 22 20 19 17 16 18 15 0 11 8 5 3 21 2 4 1 12 7 14 10 6 13 9 24 23 22 19 18 16 15 17 14 21 8 10 7 4 2 25 20 1 3 0 11 6 13 9 5 12 23 22 18 17 15 14 16 13 20 7 11 9 6 3 1 24 25 19 0 2 21 10 5 12 8 4 22 17 16 14 13 15 12 19 6 10 3 8 5 2 0 23 24 25 18 21 1 20 9 4 11 7 16 15 13 12 14 11 18 5 9 2 6 7 4 1 21 22 23 24 25 17 20 0 19 8 3 10 14 12 11 13 10 17 4 8 1 5 9 6 3 0 20 15 22 23 24 25 16 19 21 18 7 2 11 10 12 9 16 3 7 0 4 8 1 5 2 21 19 13 14 22 23 24 25 15 18 20 17 6 9 11 8 15 2 6 21 3 7 0 5 4 1 20 18 10 12 13 22 23 24 25 14 17 19 16 10 7 14 1 5 20 2 6 21 4 15 3 0 19 17 8 9 11 12 22 23 24 25 13 16 18 6 13 0 4 19 1 5 20 3 14 17 2 21 18 16 9 7 8 10 11 22 23 24 25 12 15 12 21 3 18 0 4 19 2 13 16 14 1 20 17 15 5 8 6 7 9 10 22 23 24 25 11 n=27, a(27)>=23 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=117&postid=1226, Natalia Makarova, Jan 17 2021 Way of finding: from known cliques (full MOLS of order n=3^3) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24 6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23 8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22 7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21 3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20 5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19 4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18 18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14 26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13 25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12 21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11 23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10 22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8 11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7 10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6 15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5 17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4 16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3 12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2 14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1 13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0 n=28 a(28)>=3 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2015, Natalia Makarova, May 14 2011 Way of finding: ? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 2 3 4 5 6 0 1 9 10 11 12 13 7 8 16 17 18 19 20 14 15 23 24 25 26 27 21 22 4 5 6 0 1 2 3 11 12 13 7 8 9 10 18 19 20 14 15 16 17 25 26 27 21 22 23 24 6 0 1 2 3 4 5 13 7 8 9 10 11 12 20 14 15 16 17 18 19 27 21 22 23 24 25 26 1 2 3 4 5 6 0 8 9 10 11 12 13 7 15 16 17 18 19 20 14 22 23 24 25 26 27 21 3 4 5 6 0 1 2 10 11 12 13 7 8 9 17 18 19 20 14 15 16 24 25 26 27 21 22 23 5 6 0 1 2 3 4 12 13 7 8 9 10 11 19 20 14 15 16 17 18 26 27 21 22 23 24 25 14 15 16 17 18 19 20 21 22 23 24 25 26 27 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17 18 19 20 14 15 23 24 25 26 27 21 22 2 3 4 5 6 0 1 9 10 11 12 13 7 8 18 19 20 14 15 16 17 25 26 27 21 22 23 24 4 5 6 0 1 2 3 11 12 13 7 8 9 10 20 14 15 16 17 18 19 27 21 22 23 24 25 26 6 0 1 2 3 4 5 13 7 8 9 10 11 12 15 16 17 18 19 20 14 22 23 24 25 26 27 21 1 2 3 4 5 6 0 8 9 10 11 12 13 7 17 18 19 20 14 15 16 24 25 26 27 21 22 23 3 4 5 6 0 1 2 10 11 12 13 7 8 9 19 20 14 15 16 17 18 26 27 21 22 23 24 25 5 6 0 1 2 3 4 12 13 7 8 9 10 11 21 22 23 24 25 26 27 14 15 16 17 18 19 20 7 8 9 10 11 12 13 0 1 2 3 4 5 6 23 24 25 26 27 21 22 16 17 18 19 20 14 15 9 10 11 12 13 7 8 2 3 4 5 6 0 1 25 26 27 21 22 23 24 18 19 20 14 15 16 17 11 12 13 7 8 9 10 4 5 6 0 1 2 3 27 21 22 23 24 25 26 20 14 15 16 17 18 19 13 7 8 9 10 11 12 6 0 1 2 3 4 5 22 23 24 25 26 27 21 15 16 17 18 19 20 14 8 9 10 11 12 13 7 1 2 3 4 5 6 0 24 25 26 27 21 22 23 17 18 19 20 14 15 16 10 11 12 13 7 8 9 3 4 5 6 0 1 2 26 27 21 22 23 24 25 19 20 14 15 16 17 18 12 13 7 8 9 10 11 5 6 0 1 2 3 4 7 8 9 10 11 12 13 0 1 2 3 4 5 6 21 22 23 24 25 26 27 14 15 16 17 18 19 20 9 10 11 12 13 7 8 2 3 4 5 6 0 1 23 24 25 26 27 21 22 16 17 18 19 20 14 15 11 12 13 7 8 9 10 4 5 6 0 1 2 3 25 26 27 21 22 23 24 18 19 20 14 15 16 17 13 7 8 9 10 11 12 6 0 1 2 3 4 5 27 21 22 23 24 25 26 20 14 15 16 17 18 19 8 9 10 11 12 13 7 1 2 3 4 5 6 0 22 23 24 25 26 27 21 15 16 17 18 19 20 14 10 11 12 13 7 8 9 3 4 5 6 0 1 2 24 25 26 27 21 22 23 17 18 19 20 14 15 16 12 13 7 8 9 10 11 5 6 0 1 2 3 4 26 27 21 22 23 24 25 19 20 14 15 16 17 18 n=29, a(29)>=25 Announcement: - (well known) Way of finding: cyclic MODLS 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 a(29)>=112? Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=132&postid=2213, Natalia Makarova, May 30 2021 Way of finding: semi-cyclic squares 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 17 18 19 20 21 22 23 24 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 n=30, a(30)>=1 Announcement: https://boinc.progger.info/odlk/forum_thread.php?id=162&postid=5798, Natalia Makarova, Jun 20 2020 Way of finding: Pellegrino-Lancellotti method 10 9 4 1 0 8 2 3 5 7 6 11 12 13 14 15 16 17 18 22 28 29 27 24 21 26 25 20 23 19 1 17 2 7 9 5 8 6 0 4 13 3 16 19 11 18 15 10 21 12 24 20 25 22 29 23 26 28 14 27 0 6 13 9 3 2 5 4 7 8 14 10 1 11 17 16 18 27 19 15 21 28 24 29 20 25 23 12 22 26 7 8 3 18 1 4 6 9 2 0 19 15 17 5 13 12 28 11 16 14 22 26 20 23 24 27 10 29 25 21 9 7 5 2 15 3 4 0 6 1 11 19 18 16 8 20 12 14 10 13 29 22 23 25 26 17 27 21 28 24 4 2 0 3 5 19 1 8 9 6 12 16 14 17 23 7 13 15 11 10 27 25 21 20 18 22 24 26 29 28 5 4 7 8 6 9 14 1 3 2 18 12 19 26 10 11 0 13 17 16 23 27 22 15 28 29 21 24 20 25 8 5 9 6 4 0 7 16 1 3 15 18 29 10 12 14 19 2 13 17 26 23 11 28 27 21 22 25 24 20 3 1 8 0 2 6 9 7 12 5 17 24 15 14 16 10 11 19 4 18 20 13 26 21 25 28 29 22 27 23 2 0 6 4 7 1 3 5 8 11 25 14 10 12 19 13 17 18 15 9 16 21 28 27 22 24 20 23 26 29 17 10 19 16 24 12 13 18 11 14 20 21 22 23 15 25 26 6 28 29 9 2 1 5 7 0 8 27 4 3 16 18 26 14 10 11 12 17 15 19 23 27 13 29 21 28 25 20 24 1 5 6 8 3 2 4 0 9 7 22 15 20 16 10 18 13 11 19 14 12 24 17 23 21 27 26 4 22 29 25 7 9 5 2 6 8 28 1 3 0 14 12 18 11 16 22 17 10 13 15 29 25 27 28 5 19 20 21 26 24 3 0 6 4 23 1 9 2 8 7 21 14 11 13 12 18 19 15 17 16 10 29 28 8 25 27 22 24 20 23 2 3 4 26 0 6 1 7 9 5 19 13 15 27 11 14 16 12 10 17 22 26 24 18 28 29 23 25 2 20 1 8 7 6 4 3 5 0 21 9 11 19 14 12 17 10 15 23 18 13 28 22 3 25 20 21 24 16 27 26 4 1 29 0 9 2 7 5 6 8 12 11 10 17 19 15 29 13 16 18 0 28 21 20 22 24 14 26 23 27 25 4 2 9 1 7 3 8 5 6 18 16 12 15 13 17 10 14 19 28 27 23 25 24 26 9 21 29 22 11 6 5 0 7 8 20 2 3 1 4 13 15 17 19 14 16 18 11 25 10 26 7 20 22 29 23 27 28 12 21 8 24 9 1 3 5 6 4 0 2 29 26 23 20 25 27 21 22 24 9 16 1 2 3 4 5 6 7 8 28 0 19 14 11 10 18 12 13 15 17 20 22 21 28 26 24 27 29 4 23 3 13 6 9 1 8 5 0 25 2 11 7 12 17 19 15 18 16 10 14 25 29 20 26 22 21 24 2 28 27 4 0 11 1 7 6 8 23 9 5 10 16 3 19 13 12 15 14 17 18 28 27 22 24 20 23 0 26 21 25 9 5 7 15 3 2 29 1 6 4 17 18 13 8 11 14 16 19 12 10 26 28 24 21 27 7 23 25 29 20 1 9 8 6 18 22 2 4 0 3 19 17 15 12 5 13 14 10 16 11 23 21 25 22 8 28 20 27 26 29 2 6 4 7 24 17 3 5 1 0 14 12 10 13 15 9 11 18 19 16 24 23 28 5 29 26 25 20 22 21 8 2 9 27 0 1 10 3 7 6 15 14 17 18 16 19 4 11 13 12 27 24 1 29 23 25 28 21 20 22 5 8 26 0 2 4 9 12 3 7 18 15 19 16 14 10 17 6 11 13 22 3 27 25 21 29 26 28 23 24 7 20 5 4 6 0 1 9 14 8 13 11 18 10 12 16 19 17 2 15 6 25 29 23 28 20 22 24 27 26 21 4 0 2 9 3 7 8 5 19 12 10 16 14 17 11 13 15 18 1 n=31, a(31)>=27 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=1995, Natalia Makarova, May 13 2021 Way of finding: ? ? a(31)>=58 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=132&postid=2231, Natalia Makarova, May 31 2021 Way of finding: semi-cyclic squares 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 27 28 29 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 2 3 4 5 6 7 8 n=32, a(32)>=29 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=1995, Natalia Makarova, May 13 2021 Way of finding: ? ? n=33, a(33)>=1 Announcement: https://boinc.progger.info/odlk/forum_thread.php?id=162&postid=5803, Natalia Makarova, Jun 21 2020 Way of finding: Linder method 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 16 3 0 2 6 7 4 5 10 11 8 9 14 15 12 13 1 19 20 17 18 23 24 21 22 27 28 25 26 31 32 29 30 1 2 16 0 7 6 5 4 11 10 9 8 15 14 13 12 3 20 19 18 17 24 23 22 21 28 27 26 25 32 31 30 29 2 16 3 1 5 4 7 6 9 8 11 10 13 12 15 14 0 18 17 20 19 22 21 24 23 26 25 28 27 30 29 32 31 4 5 6 7 32 30 2 29 17 18 19 20 8 9 10 11 31 21 22 23 24 25 26 27 28 3 1 16 0 12 13 14 15 6 7 4 5 31 29 32 2 19 20 17 18 10 11 8 9 30 23 24 21 22 27 28 25 26 16 0 3 1 14 15 12 13 7 6 5 4 30 2 31 32 20 19 18 17 11 10 9 8 29 24 23 22 21 28 27 26 25 0 16 1 3 15 14 13 12 5 4 7 6 2 31 29 30 18 17 20 19 9 8 11 10 32 22 21 24 23 26 25 28 27 1 3 0 16 13 12 15 14 21 22 23 24 8 9 10 11 28 26 2 25 3 1 16 0 27 29 30 31 32 17 18 19 20 12 13 14 15 4 5 6 7 23 24 21 22 10 11 8 9 27 25 28 2 16 0 3 1 26 31 32 29 30 19 20 17 18 14 15 12 13 6 7 4 5 24 23 22 21 11 10 9 8 26 2 27 28 0 16 1 3 25 32 31 30 29 20 19 18 17 15 14 13 12 7 6 5 4 22 21 24 23 9 8 11 10 2 27 25 26 1 3 0 16 28 30 29 32 31 18 17 20 19 13 12 15 14 5 4 7 6 29 30 31 32 12 13 14 15 21 22 23 24 20 18 2 17 19 25 26 27 28 3 1 16 0 4 5 6 7 8 9 10 11 31 32 29 30 14 15 12 13 23 24 21 22 19 17 20 2 18 27 28 25 26 16 0 3 1 6 7 4 5 10 11 8 9 32 31 30 29 15 14 13 12 24 23 22 21 18 2 19 20 17 28 27 26 25 0 16 1 3 7 6 5 4 11 10 9 8 30 29 32 31 13 12 15 14 22 21 24 23 2 19 17 18 20 26 25 28 27 1 3 0 16 5 4 7 6 9 8 11 10 3 0 1 16 29 32 30 31 25 28 26 27 17 20 18 19 2 12 15 13 14 4 7 5 6 8 11 9 10 21 24 22 23 25 26 27 28 17 18 19 20 29 30 31 32 4 5 6 7 14 15 13 2 12 8 9 10 11 21 22 23 24 3 1 16 0 27 28 25 26 19 20 17 18 31 32 29 30 6 7 4 5 13 14 12 15 2 10 11 8 9 23 24 21 22 16 0 3 1 28 27 26 25 20 19 18 17 32 31 30 29 7 6 5 4 12 13 2 14 15 11 10 9 8 24 23 22 21 0 16 1 3 26 25 28 27 18 17 20 19 30 29 32 31 5 4 7 6 15 2 14 12 13 9 8 11 10 22 21 24 23 1 3 0 16 17 18 19 20 3 1 16 0 12 13 14 15 21 22 23 24 6 8 9 10 11 7 5 2 4 29 30 31 32 25 26 27 28 19 20 17 18 16 0 3 1 14 15 12 13 23 24 21 22 5 10 11 8 9 6 4 7 2 31 32 29 30 27 28 25 26 20 19 18 17 0 16 1 3 15 14 13 12 24 23 22 21 4 11 10 9 8 5 2 6 7 32 31 30 29 28 27 26 25 18 17 20 19 1 3 0 16 13 12 15 14 22 21 24 23 7 9 8 11 10 2 6 4 5 30 29 32 31 26 25 28 27 12 13 14 15 21 22 23 24 3 1 16 0 25 26 27 28 10 4 5 6 7 29 30 31 32 11 9 2 8 17 18 19 20 14 15 12 13 23 24 21 22 16 0 3 1 27 28 25 26 9 6 7 4 5 31 32 29 30 10 8 11 2 19 20 17 18 15 14 13 12 24 23 22 21 0 16 1 3 28 27 26 25 8 7 6 5 4 32 31 30 29 9 2 10 11 20 19 18 17 13 12 15 14 22 21 24 23 1 3 0 16 26 25 28 27 11 5 4 7 6 30 29 32 31 2 10 8 9 18 17 20 19 8 9 10 11 25 26 27 28 4 5 6 7 29 30 31 32 23 3 1 16 0 12 13 14 15 17 18 19 20 24 22 2 21 10 11 8 9 27 28 25 26 6 7 4 5 31 32 29 30 22 16 0 3 1 14 15 12 13 19 20 17 18 23 21 24 2 11 10 9 8 28 27 26 25 7 6 5 4 32 31 30 29 21 0 16 1 3 15 14 13 12 20 19 18 17 22 2 23 24 9 8 11 10 26 25 28 27 5 4 7 6 30 29 32 31 24 1 3 0 16 13 12 15 14 18 17 20 19 2 23 21 22 n=34, a(34)>=1 Announcement: http://www.natalimak1.narod.ru/dlk.htm, Natalia Makarova, 2009 Way of finding: ? 1 19 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 33 32 31 30 29 28 27 26 25 24 23 22 21 20 2 18 2 3 21 5 6 7 8 9 10 11 12 13 14 15 16 17 1 18 0 33 32 31 30 29 28 27 26 25 24 23 22 4 20 19 3 4 5 23 7 8 9 10 11 12 13 14 15 16 17 1 2 19 18 0 33 32 31 30 29 28 27 26 25 24 6 22 21 20 4 5 6 7 25 9 10 11 12 13 14 15 16 17 1 2 3 20 19 18 0 33 32 31 30 29 28 27 26 8 24 23 22 21 5 6 7 8 9 27 11 12 13 14 15 16 17 1 2 3 4 21 20 19 18 0 33 32 31 30 29 28 10 26 25 24 23 22 6 7 8 9 10 11 29 13 14 15 16 17 1 2 3 4 5 22 21 20 19 18 0 33 32 31 30 12 28 27 26 25 24 23 7 8 9 10 11 12 13 31 15 16 17 1 2 3 4 5 6 23 22 21 20 19 18 0 33 32 14 30 29 28 27 26 25 24 8 9 10 11 12 13 14 15 33 17 1 2 3 4 5 6 7 24 23 22 21 20 19 18 0 16 32 31 30 29 28 27 26 25 9 10 11 12 13 14 15 16 17 18 2 3 4 5 6 7 8 25 24 23 22 21 20 19 1 0 33 32 31 30 29 28 27 26 10 11 12 13 14 15 16 17 1 2 20 4 5 6 7 8 9 26 25 24 23 22 21 3 19 18 0 33 32 31 30 29 28 27 11 12 13 14 15 16 17 1 2 3 4 22 6 7 8 9 10 27 26 25 24 23 5 21 20 19 18 0 33 32 31 30 29 28 12 13 14 15 16 17 1 2 3 4 5 6 24 8 9 10 11 28 27 26 25 7 23 22 21 20 19 18 0 33 32 31 30 29 13 14 15 16 17 1 2 3 4 5 6 7 8 26 10 11 12 29 28 27 9 25 24 23 22 21 20 19 18 0 33 32 31 30 14 15 16 17 1 2 3 4 5 6 7 8 9 10 28 12 13 30 29 11 27 26 25 24 23 22 21 20 19 18 0 33 32 31 15 16 17 1 2 3 4 5 6 7 8 9 10 11 12 30 14 31 13 29 28 27 26 25 24 23 22 21 20 19 18 0 33 32 16 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 32 15 31 30 29 28 27 26 25 24 23 22 21 20 19 18 0 33 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 33 0 18 19 20 21 22 23 24 25 26 27 28 29 30 31 15 32 14 13 12 11 10 9 8 7 6 5 4 3 2 1 17 16 32 33 0 18 19 20 21 22 23 24 25 26 27 28 29 13 31 14 30 12 11 10 9 8 7 6 5 4 3 2 1 17 16 15 31 32 33 0 18 19 20 21 22 23 24 25 26 27 11 29 30 13 12 28 10 9 8 7 6 5 4 3 2 1 17 16 15 14 30 31 32 33 0 18 19 20 21 22 23 24 25 9 27 28 29 12 11 10 26 8 7 6 5 4 3 2 1 17 16 15 14 13 29 30 31 32 33 0 18 19 20 21 22 23 7 25 26 27 28 11 10 9 8 24 6 5 4 3 2 1 17 16 15 14 13 12 28 29 30 31 32 33 0 18 19 20 21 5 23 24 25 26 27 10 9 8 7 6 22 4 3 2 1 17 16 15 14 13 12 11 27 28 29 30 31 32 33 0 18 19 3 21 22 23 24 25 26 9 8 7 6 5 4 20 2 1 17 16 15 14 13 12 11 10 26 27 28 29 30 31 32 33 0 1 19 20 21 22 23 24 25 8 7 6 5 4 3 2 18 17 16 15 14 13 12 11 10 9 25 26 27 28 29 30 31 32 16 0 18 19 20 21 22 23 24 7 6 5 4 3 2 1 17 33 15 14 13 12 11 10 9 8 24 25 26 27 28 29 30 14 32 33 0 18 19 20 21 22 23 6 5 4 3 2 1 17 16 15 31 13 12 11 10 9 8 7 23 24 25 26 27 28 12 30 31 32 33 0 18 19 20 21 22 5 4 3 2 1 17 16 15 14 13 29 11 10 9 8 7 6 22 23 24 25 26 10 28 29 30 31 32 33 0 18 19 20 21 4 3 2 1 17 16 15 14 13 12 11 27 9 8 7 6 5 21 22 23 24 8 26 27 28 29 30 31 32 33 0 18 19 20 3 2 1 17 16 15 14 13 12 11 10 9 25 7 6 5 4 20 21 22 6 24 25 26 27 28 29 30 31 32 33 0 18 19 2 1 17 16 15 14 13 12 11 10 9 8 7 23 5 4 3 19 20 4 22 23 24 25 26 27 28 29 30 31 32 33 0 18 1 17 16 15 14 13 12 11 10 9 8 7 6 5 21 3 2 18 2 20 21 22 23 24 25 26 27 28 29 30 31 32 33 0 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 19 1 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 n=35, a(35)>=3 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2010, Natalia Makarova, May 14 2021 Way of finding: ? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 2 3 4 5 6 0 1 9 10 11 12 13 7 8 16 17 18 19 20 14 15 23 24 25 26 27 21 22 30 31 32 33 34 28 29 4 5 6 0 1 2 3 11 12 13 7 8 9 10 18 19 20 14 15 16 17 25 26 27 21 22 23 24 32 33 34 28 29 30 31 6 0 1 2 3 4 5 13 7 8 9 10 11 12 20 14 15 16 17 18 19 27 21 22 23 24 25 26 34 28 29 30 31 32 33 1 2 3 4 5 6 0 8 9 10 11 12 13 7 15 16 17 18 19 20 14 22 23 24 25 26 27 21 29 30 31 32 33 34 28 3 4 5 6 0 1 2 10 11 12 13 7 8 9 17 18 19 20 14 15 16 24 25 26 27 21 22 23 31 32 33 34 28 29 30 5 6 0 1 2 3 4 12 13 7 8 9 10 11 19 20 14 15 16 17 18 26 27 21 22 23 24 25 33 34 28 29 30 31 32 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17 18 19 20 14 15 23 24 25 26 27 21 22 30 31 32 33 34 28 29 2 3 4 5 6 0 1 9 10 11 12 13 7 8 18 19 20 14 15 16 17 25 26 27 21 22 23 24 32 33 34 28 29 30 31 4 5 6 0 1 2 3 11 12 13 7 8 9 10 20 14 15 16 17 18 19 27 21 22 23 24 25 26 34 28 29 30 31 32 33 6 0 1 2 3 4 5 13 7 8 9 10 11 12 15 16 17 18 19 20 14 22 23 24 25 26 27 21 29 30 31 32 33 34 28 1 2 3 4 5 6 0 8 9 10 11 12 13 7 17 18 19 20 14 15 16 24 25 26 27 21 22 23 31 32 33 34 28 29 30 3 4 5 6 0 1 2 10 11 12 13 7 8 9 19 20 14 15 16 17 18 26 27 21 22 23 24 25 33 34 28 29 30 31 32 5 6 0 1 2 3 4 12 13 7 8 9 10 11 28 29 30 31 32 33 34 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 30 31 32 33 34 28 29 2 3 4 5 6 0 1 9 10 11 12 13 7 8 16 17 18 19 20 14 15 23 24 25 26 27 21 22 32 33 34 28 29 30 31 4 5 6 0 1 2 3 11 12 13 7 8 9 10 18 19 20 14 15 16 17 25 26 27 21 22 23 24 34 28 29 30 31 32 33 6 0 1 2 3 4 5 13 7 8 9 10 11 12 20 14 15 16 17 18 19 27 21 22 23 24 25 26 29 30 31 32 33 34 28 1 2 3 4 5 6 0 8 9 10 11 12 13 7 15 16 17 18 19 20 14 22 23 24 25 26 27 21 31 32 33 34 28 29 30 3 4 5 6 0 1 2 10 11 12 13 7 8 9 17 18 19 20 14 15 16 24 25 26 27 21 22 23 33 34 28 29 30 31 32 5 6 0 1 2 3 4 12 13 7 8 9 10 11 19 20 14 15 16 17 18 26 27 21 22 23 24 25 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 0 1 2 3 4 5 6 9 10 11 12 13 7 8 16 17 18 19 20 14 15 23 24 25 26 27 21 22 30 31 32 33 34 28 29 2 3 4 5 6 0 1 11 12 13 7 8 9 10 18 19 20 14 15 16 17 25 26 27 21 22 23 24 32 33 34 28 29 30 31 4 5 6 0 1 2 3 13 7 8 9 10 11 12 20 14 15 16 17 18 19 27 21 22 23 24 25 26 34 28 29 30 31 32 33 6 0 1 2 3 4 5 8 9 10 11 12 13 7 15 16 17 18 19 20 14 22 23 24 25 26 27 21 29 30 31 32 33 34 28 1 2 3 4 5 6 0 10 11 12 13 7 8 9 17 18 19 20 14 15 16 24 25 26 27 21 22 23 31 32 33 34 28 29 30 3 4 5 6 0 1 2 12 13 7 8 9 10 11 19 20 14 15 16 17 18 26 27 21 22 23 24 25 33 34 28 29 30 31 32 5 6 0 1 2 3 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 23 24 25 26 27 21 22 30 31 32 33 34 28 29 2 3 4 5 6 0 1 9 10 11 12 13 7 8 16 17 18 19 20 14 15 25 26 27 21 22 23 24 32 33 34 28 29 30 31 4 5 6 0 1 2 3 11 12 13 7 8 9 10 18 19 20 14 15 16 17 27 21 22 23 24 25 26 34 28 29 30 31 32 33 6 0 1 2 3 4 5 13 7 8 9 10 11 12 20 14 15 16 17 18 19 22 23 24 25 26 27 21 29 30 31 32 33 34 28 1 2 3 4 5 6 0 8 9 10 11 12 13 7 15 16 17 18 19 20 14 24 25 26 27 21 22 23 31 32 33 34 28 29 30 3 4 5 6 0 1 2 10 11 12 13 7 8 9 17 18 19 20 14 15 16 26 27 21 22 23 24 25 33 34 28 29 30 31 32 5 6 0 1 2 3 4 12 13 7 8 9 10 11 19 20 14 15 16 17 18 n=36, a(36)>=61 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=1991, Natalia Makarova, May 13 2021 Way of finding: Pelegrino-Lancelotti method 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3 18 1 4 11 2 9 12 7 10 5 8 15 6 13 16 35 14 21 0 19 22 29 20 27 30 25 28 23 26 33 24 31 34 17 32 1 4 35 2 5 8 3 6 9 12 7 10 13 16 11 14 17 20 15 18 21 24 19 22 25 28 23 26 29 32 27 30 33 0 31 34 2 3 4 17 10 11 12 1 6 7 8 9 14 15 16 5 22 35 0 13 30 19 20 21 26 27 28 29 34 23 24 25 18 31 32 33 11 10 5 12 19 6 1 8 3 2 9 4 35 22 17 0 7 18 13 32 15 14 21 16 23 34 29 24 31 30 25 20 27 26 33 28 8 5 6 11 12 21 10 3 4 1 2 7 20 17 18 35 0 9 34 15 16 13 14 19 32 29 30 23 24 33 22 27 28 25 26 31 6 11 12 9 2 7 20 5 10 3 4 1 18 35 0 21 14 31 8 17 22 15 16 13 30 23 24 33 26 19 32 29 34 27 28 25 5 12 7 6 9 4 11 22 1 8 3 2 17 0 19 18 33 16 35 10 13 20 15 14 29 24 31 30 21 28 23 34 25 32 27 26 10 7 8 1 6 3 4 9 14 11 12 5 22 19 20 25 18 15 16 21 2 35 0 17 34 31 32 13 30 27 28 33 26 23 24 29 9 8 3 10 1 12 7 2 5 16 11 6 21 20 27 22 13 0 19 14 17 4 35 18 33 32 15 34 25 24 31 26 29 28 23 30 7 2 9 8 3 10 5 4 11 6 13 12 19 26 21 20 15 22 17 16 35 18 1 0 31 14 33 32 27 34 29 28 23 30 25 24 4 9 10 7 8 1 2 11 12 5 6 15 28 21 22 19 20 13 14 35 0 17 18 3 16 33 34 31 32 25 26 23 24 29 30 27 13 23 17 22 21 35 0 20 19 15 16 14 33 18 24 26 25 30 29 8 31 27 28 34 9 11 12 2 1 6 5 32 7 3 4 10 22 0 18 21 14 17 15 13 20 28 35 19 26 29 23 1 34 24 27 33 32 16 30 31 2 5 11 25 10 12 3 9 8 4 6 7 18 21 14 24 35 19 22 0 15 13 20 16 23 25 34 17 30 7 26 29 27 33 32 28 11 1 10 12 6 31 2 5 3 9 8 4 17 22 21 35 16 14 13 18 0 20 19 27 24 2 25 30 28 34 33 23 29 32 31 15 12 26 1 6 4 10 9 11 5 8 7 3 14 16 30 13 20 0 18 19 22 17 15 21 34 28 35 33 32 29 23 31 26 12 27 25 10 4 6 9 8 5 11 7 2 24 3 1 19 35 0 14 13 15 16 26 21 18 17 20 31 30 29 34 9 27 28 22 25 23 24 32 7 6 5 10 33 3 4 2 1 11 12 8 29 14 13 15 17 20 19 35 16 22 21 18 0 34 33 27 24 32 31 30 28 26 25 11 5 10 9 3 12 8 7 6 4 2 1 23 35 13 20 0 15 25 14 16 18 19 22 17 30 33 32 29 27 21 10 28 23 31 26 24 6 9 8 5 3 1 34 4 11 7 2 12 16 20 19 18 0 22 21 15 17 14 33 35 4 32 31 23 29 26 25 27 24 34 13 30 28 8 7 11 5 2 1 3 12 10 9 6 15 19 22 16 18 13 32 17 35 21 14 0 27 31 26 28 23 33 20 24 6 25 34 29 3 7 2 4 11 9 8 12 30 1 10 5 20 17 15 19 22 16 35 21 34 0 18 13 32 24 3 31 26 28 30 25 14 29 23 33 8 12 27 7 2 4 6 1 10 5 11 9 21 15 16 20 31 18 17 14 13 35 0 22 25 27 28 32 19 23 24 34 33 30 5 26 1 3 4 8 7 11 12 10 9 6 29 2 28 24 23 30 29 33 34 25 26 31 32 3 16 5 6 11 12 1 2 9 10 7 8 27 4 17 18 35 0 13 14 21 22 19 20 15 30 34 24 29 27 23 31 28 25 32 1 26 11 14 5 12 3 6 7 4 9 8 33 10 35 2 17 0 15 18 19 16 21 20 13 22 24 29 27 23 33 26 30 34 31 4 25 32 5 12 15 6 1 10 11 2 7 28 9 8 17 0 3 18 13 22 35 14 19 16 21 20 23 30 29 33 32 27 28 24 2 25 26 31 6 11 12 13 8 3 4 5 34 9 10 7 18 35 0 1 20 15 16 17 14 21 22 19 27 32 33 28 25 34 24 10 30 23 31 29 3 8 1 4 21 2 5 26 11 6 7 12 15 20 13 16 9 14 17 22 35 18 19 0 26 33 34 27 28 31 8 30 29 24 23 25 10 1 2 3 4 19 32 11 12 5 6 9 22 13 14 15 16 7 20 35 0 17 18 21 34 27 28 31 23 9 26 33 32 30 29 24 2 3 4 7 6 25 22 1 8 11 12 5 14 15 16 19 18 21 10 13 20 35 0 17 33 28 25 34 7 29 27 32 24 26 30 23 1 4 9 2 31 12 3 20 5 10 11 6 13 16 21 14 19 0 15 8 17 22 35 18 32 25 26 5 34 30 29 31 23 27 28 33 8 9 10 24 2 11 12 7 18 3 4 1 20 21 22 17 14 35 0 19 6 15 16 13 31 26 11 32 24 28 25 23 33 29 27 34 7 10 30 8 5 4 9 6 1 0 3 2 19 22 35 20 17 16 21 18 13 12 15 14 25 6 31 26 30 32 33 29 27 34 24 28 9 23 7 10 11 8 1 12 3 2 17 4 21 18 19 22 35 20 13 0 15 14 5 16 12 31 32 25 26 24 23 27 28 33 34 30 29 7 8 9 10 5 6 3 4 1 2 35 0 19 20 21 22 17 18 15 16 13 14 11 n=37, a(37)>=33 Announcement: - (well known) Way of finding: cyclic MODLS ? n=38, a(38)>=1 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2016, Natalia Makarova, May 14 2021 Way of finding: ? ? n=39, a(39)>=1 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2018, Natalia Makarova, May 14 2021 Way of finding: ? ? n=40, a(40)>=10 Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2025, Natalia Makarova, May 15 2021 Way of finding: ? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 1 2 0 4 5 3 7 9 6 8 11 12 10 14 15 13 17 19 16 18 21 22 20 24 25 23 27 29 26 28 31 32 30 34 35 33 37 39 36 38 9 7 6 2 3 8 5 1 4 0 19 17 16 12 13 18 15 11 14 10 29 27 26 22 23 28 25 21 24 20 39 37 36 32 33 38 35 31 34 30 7 5 1 8 6 9 2 0 3 4 17 15 11 18 16 19 12 10 13 14 27 25 21 28 26 29 22 20 23 24 37 35 31 38 36 39 32 30 33 34 3 9 8 6 7 4 1 2 0 5 13 19 18 16 17 14 11 12 10 15 23 29 28 26 27 24 21 22 20 25 33 39 38 36 37 34 31 32 30 35 2 4 5 9 8 1 0 3 7 6 12 14 15 19 18 11 10 13 17 16 22 24 25 29 28 21 20 23 27 26 32 34 35 39 38 31 30 33 37 36 4 8 3 0 2 6 9 5 1 7 14 18 13 10 12 16 19 15 11 17 24 28 23 20 22 26 29 25 21 27 34 38 33 30 32 36 39 35 31 37 8 6 7 5 9 0 3 4 2 1 18 16 17 15 19 10 13 14 12 11 28 26 27 25 29 20 23 24 22 21 38 36 37 35 39 30 33 34 32 31 6 3 9 1 0 7 4 8 5 2 16 13 19 11 10 17 14 18 15 12 26 23 29 21 20 27 24 28 25 22 36 33 39 31 30 37 34 38 35 32 5 0 4 7 1 2 8 6 9 3 15 10 14 17 11 12 18 16 19 13 25 20 24 27 21 22 28 26 29 23 35 30 34 37 31 32 38 36 39 33 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 20 24 25 23 27 29 26 28 31 32 30 34 35 33 37 39 36 38 1 2 0 4 5 3 7 9 6 8 11 12 10 14 15 13 17 19 16 18 29 27 26 22 23 28 25 21 24 20 39 37 36 32 33 38 35 31 34 30 9 7 6 2 3 8 5 1 4 0 19 17 16 12 13 18 15 11 14 10 27 25 21 28 26 29 22 20 23 24 37 35 31 38 36 39 32 30 33 34 7 5 1 8 6 9 2 0 3 4 17 15 11 18 16 19 12 10 13 14 23 29 28 26 27 24 21 22 20 25 33 39 38 36 37 34 31 32 30 35 3 9 8 6 7 4 1 2 0 5 13 19 18 16 17 14 11 12 10 15 22 24 25 29 28 21 20 23 27 26 32 34 35 39 38 31 30 33 37 36 2 4 5 9 8 1 0 3 7 6 12 14 15 19 18 11 10 13 17 16 24 28 23 20 22 26 29 25 21 27 34 38 33 30 32 36 39 35 31 37 4 8 3 0 2 6 9 5 1 7 14 18 13 10 12 16 19 15 11 17 28 26 27 25 29 20 23 24 22 21 38 36 37 35 39 30 33 34 32 31 8 6 7 5 9 0 3 4 2 1 18 16 17 15 19 10 13 14 12 11 26 23 29 21 20 27 24 28 25 22 36 33 39 31 30 37 34 38 35 32 6 3 9 1 0 7 4 8 5 2 16 13 19 11 10 17 14 18 15 12 25 20 24 27 21 22 28 26 29 23 35 30 34 37 31 32 38 36 39 33 5 0 4 7 1 2 8 6 9 3 15 10 14 17 11 12 18 16 19 13 30 31 32 33 34 35 36 37 38 39 20 21 22 23 24 25 26 27 28 29 10 11 12 13 14 15 16 17 18 19 0 1 2 3 4 5 6 7 8 9 31 32 30 34 35 33 37 39 36 38 21 22 20 24 25 23 27 29 26 28 11 12 10 14 15 13 17 19 16 18 1 2 0 4 5 3 7 9 6 8 39 37 36 32 33 38 35 31 34 30 29 27 26 22 23 28 25 21 24 20 19 17 16 12 13 18 15 11 14 10 9 7 6 2 3 8 5 1 4 0 37 35 31 38 36 39 32 30 33 34 27 25 21 28 26 29 22 20 23 24 17 15 11 18 16 19 12 10 13 14 7 5 1 8 6 9 2 0 3 4 33 39 38 36 37 34 31 32 30 35 23 29 28 26 27 24 21 22 20 25 13 19 18 16 17 14 11 12 10 15 3 9 8 6 7 4 1 2 0 5 32 34 35 39 38 31 30 33 37 36 22 24 25 29 28 21 20 23 27 26 12 14 15 19 18 11 10 13 17 16 2 4 5 9 8 1 0 3 7 6 34 38 33 30 32 36 39 35 31 37 24 28 23 20 22 26 29 25 21 27 14 18 13 10 12 16 19 15 11 17 4 8 3 0 2 6 9 5 1 7 38 36 37 35 39 30 33 34 32 31 28 26 27 25 29 20 23 24 22 21 18 16 17 15 19 10 13 14 12 11 8 6 7 5 9 0 3 4 2 1 36 33 39 31 30 37 34 38 35 32 26 23 29 21 20 27 24 28 25 22 16 13 19 11 10 17 14 18 15 12 6 3 9 1 0 7 4 8 5 2 35 30 34 37 31 32 38 36 39 33 25 20 24 27 21 22 28 26 29 23 15 10 14 17 11 12 18 16 19 13 5 0 4 7 1 2 8 6 9 3 10 11 12 13 14 15 16 17 18 19 0 1 2 3 4 5 6 7 8 9 30 31 32 33 34 35 36 37 38 39 20 21 22 23 24 25 26 27 28 29 11 12 10 14 15 13 17 19 16 18 1 2 0 4 5 3 7 9 6 8 31 32 30 34 35 33 37 39 36 38 21 22 20 24 25 23 27 29 26 28 19 17 16 12 13 18 15 11 14 10 9 7 6 2 3 8 5 1 4 0 39 37 36 32 33 38 35 31 34 30 29 27 26 22 23 28 25 21 24 20 17 15 11 18 16 19 12 10 13 14 7 5 1 8 6 9 2 0 3 4 37 35 31 38 36 39 32 30 33 34 27 25 21 28 26 29 22 20 23 24 13 19 18 16 17 14 11 12 10 15 3 9 8 6 7 4 1 2 0 5 33 39 38 36 37 34 31 32 30 35 23 29 28 26 27 24 21 22 20 25 12 14 15 19 18 11 10 13 17 16 2 4 5 9 8 1 0 3 7 6 32 34 35 39 38 31 30 33 37 36 22 24 25 29 28 21 20 23 27 26 14 18 13 10 12 16 19 15 11 17 4 8 3 0 2 6 9 5 1 7 34 38 33 30 32 36 39 35 31 37 24 28 23 20 22 26 29 25 21 27 18 16 17 15 19 10 13 14 12 11 8 6 7 5 9 0 3 4 2 1 38 36 37 35 39 30 33 34 32 31 28 26 27 25 29 20 23 24 22 21 16 13 19 11 10 17 14 18 15 12 6 3 9 1 0 7 4 8 5 2 36 33 39 31 30 37 34 38 35 32 26 23 29 21 20 27 24 28 25 22 15 10 14 17 11 12 18 16 19 13 5 0 4 7 1 2 8 6 9 3 35 30 34 37 31 32 38 36 39 33 25 20 24 27 21 22 28 26 29 23 Nov 11 2021