Re: [math-fun] Cyclic Difference Sets
Are we supposed to guess what you're talking about, or would you be so kind as to tell us? —Dan ----- A perfect cyclic difference set with n values has length L = (n-1)^2 + n and works mod L. All differences up to L-1 exist uniquely. 3 values is structurally unique, 1 solution. 4 values has 2 solutions. 5 values has 1 solution. 6 values has 5 solutions. 7 values has 0 solutions. 8 values has 1 solution. Do any subsequent values have more than 1 solution? {{{7,3},{0,1,3}}, {{13,4},{0,1,3,9}},{{13,4},{0,1,4,6}}, {{21,5},{0,1,4,14,16}}, {{31,6},{0,1,3,8,12,18}},{{31,6},{0,1,3,10,14,26}},{{31,6},{0,1,4,6,13,21}},{{31,6},{0,1,4,10,12,17}},{{31,6},{0,1,8,11,13,17}}, {{57,8},{0,1,3,13,32,36,43,52}}, {{73,9},{0,1,12,20,26,30,33,35,57}},{{91,10},{0,1,5,7,27,35,44,67,77,80}},{{133,12},{0,1,3,17,21,58,65,73,100,105,111,124}},{{183,14},{0,1,3,24,41,52,57,66,70,96,102,149,164,176}},{{273,17},{0,1,22,33,83,122,135,141,145,159,175,200,226,229,231,238,246}},{{307,18},{0,1,3,30,37,50,55,76,98,117,129,133,157,189,199,222,293,299}},{{381,20},{0,1,13,16,21,45,75,82,86,92,128,142,180,206,229,231,264,286,354,363}}} -----
Here’s a database: https://www.dmgordon.org/diffset/
On Nov 10, 2020, at 3:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Are we supposed to guess what you're talking about, or would you be so kind as to tell us?
—Dan
----- A perfect cyclic difference set with n values has length L = (n-1)^2 + n and works mod L. All differences up to L-1 exist uniquely. 3 values is structurally unique, 1 solution. 4 values has 2 solutions. 5 values has 1 solution. 6 values has 5 solutions. 7 values has 0 solutions. 8 values has 1 solution. Do any subsequent values have more than 1 solution?
{{{7,3},{0,1,3}}, {{13,4},{0,1,3,9}},{{13,4},{0,1,4,6}}, {{21,5},{0,1,4,14,16}}, {{31,6},{0,1,3,8,12,18}},{{31,6},{0,1,3,10,14,26}},{{31,6},{0,1,4,6,13,21}},{{31,6},{0,1,4,10,12,17}},{{31,6},{0,1,8,11,13,17}}, {{57,8},{0,1,3,13,32,36,43,52}}, {{73,9},{0,1,12,20,26,30,33,35,57}},{{91,10},{0,1,5,7,27,35,44,67,77,80}},{{133,12},{0,1,3,17,21,58,65,73,100,105,111,124}},{{183,14},{0,1,3,24,41,52,57,66,70,96,102,149,164,176}},{{273,17},{0,1,22,33,83,122,135,141,145,159,175,200,226,229,231,238,246}},{{307,18},{0,1,3,30,37,50,55,76,98,117,129,133,157,189,199,222,293,299}},{{381,20},{0,1,13,16,21,45,75,82,86,92,128,142,180,206,229,231,264,286,354,363}}} -----
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Mod 31, the subsets of {0, 1, 3, 8, 12, 18} have differences 1 to 30 as follows. {{0,1},{1,3},{0,3},{8,12},{3,8},{12,18},{1,8},{0,8},{3,12},{8,18}, {1,12},{0,12},{18,0},{18,1},{3,18},{18,3},{1,18},{0,18},{12,0},{12,1}, {18,8},{12,3},{8,0},{8,1},{18,12},{8,3},{12,8},{3,0},{3,1},{1,0}} {0, 1, 3, 8, 12, 18} corresponds to perfect partition 1, 2, 5, 4, 6, 13 Seems my program for 8 marks was wrong, there are at least 6 solutions. 9 marks has at least 4 solutions. {{57, 8}, {0, 1, 3, 13, 32, 36, 43, 52}},{{57, 8}, {0, 1, 4, 9, 20, 22, 34, 51}},{{57, 8}, {0, 1, 4, 12, 14, 30, 37, 52}},{{57, 8}, {0, 1, 5, 7, 17, 35, 38, 49}},{{57, 8}, {0, 1, 5, 27, 34, 37, 43, 45}},{{57, 8}, {0, 1, 7, 19, 23, 44, 47, 49}},{{73, 9}, {0, 1, 3, 7, 15, 31, 36, 54, 63}},{{73, 9}, {0, 1, 5, 12, 18, 21, 49, 51, 59}},{{73, 9}, {0, 1, 7, 11, 35, 48, 51, 53, 65}},{{73, 9}, {0, 1, 12, 20, 26, 30, 33, 35, 57}}, I'm hoping to count how many distinct difference sets there are for each number of marks, up to maybe 98 marks.
How are you eliminating redundant sets? Given a perfect difference set mod N, you can generate another by multiplying each mark by any value relatively prime to N. -tom On Tue, Nov 10, 2020 at 1:13 PM ed pegg <ed@mathpuzzle.com> wrote:
Mod 31, the subsets of {0, 1, 3, 8, 12, 18} have differences 1 to 30 as follows.
{{0,1},{1,3},{0,3},{8,12},{3,8},{12,18},{1,8},{0,8},{3,12},{8,18}, {1,12},{0,12},{18,0},{18,1},{3,18},{18,3},{1,18},{0,18},{12,0},{12,1}, {18,8},{12,3},{8,0},{8,1},{18,12},{8,3},{12,8},{3,0},{3,1},{1,0}}
{0, 1, 3, 8, 12, 18} corresponds to perfect partition 1, 2, 5, 4, 6, 13
Seems my program for 8 marks was wrong, there are at least 6 solutions. 9 marks has at least 4 solutions. {{57, 8}, {0, 1, 3, 13, 32, 36, 43, 52}},{{57, 8}, {0, 1, 4, 9, 20, 22, 34, 51}},{{57, 8}, {0, 1, 4, 12, 14, 30, 37, 52}},{{57, 8}, {0, 1, 5, 7, 17, 35, 38, 49}},{{57, 8}, {0, 1, 5, 27, 34, 37, 43, 45}},{{57, 8}, {0, 1, 7, 19, 23, 44, 47, 49}},{{73, 9}, {0, 1, 3, 7, 15, 31, 36, 54, 63}},{{73, 9}, {0, 1, 5, 12, 18, 21, 49, 51, 59}},{{73, 9}, {0, 1, 7, 11, 35, 48, 51, 53, 65}},{{73, 9}, {0, 1, 12, 20, 26, 30, 33, 35, 57}},
I'm hoping to count how many distinct difference sets there are for each number of marks, up to maybe 98 marks.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- - https://cube20.org/ <http://cube20.org/> - https://golly.sf.net/ <http://golly.sf.net/> - https://experiments.cubing.net/ -
How are you eliminating redundant sets? Adding all values and sorting mod n, then picking the first.
If I've calculated them right, the number of distinct cyclic difference sets, or the number of cyclically perfect partitions starts out 1, 2, 1, 5, 6, 4, 6, 18, 20, 6, 51, 42 {{{7,3},1}, {{13,4},2}, {{21,5},1}, {{31,6},5}, {{57,8},6}, {{73,9},4}, {{91,10},6}, {{133,12},18}, {{183,14},20}, {{273,17},6}, {{307,18},51}, {{381,20},42}} For example, the four cyclically perfect partitions of length 9 are {{1,2,4,8,16,5,18,9,10}, {1,4,7,6,3,28,2,8,14}, {1,6,4,24,13,3,2,12,8}, {1,11,8,6,4,3,2,22,16}} All sub-arcs of a perfect partition have a unique total and cover all integer values up to the set total. These correspond to the following difference sets: {{{73,9},{0,1,3,7,15,31,36,54,63}}, {{73,9},{0,1,5,12,18,21,49,51,59}}, {{73,9},{0,1,7,11,35,48,51,53,65}}, {{73,9},{0,1,12,20,26,30,33,35,57}}} In case anyone is interested, here are the perfect partitions. {{1,2,4}, {1,2,6,4}, {1,3,2,7}, {1,3,10,2,5}, {1,2,5,4,6,13}, {1,2,7,4,12,5}, {1,3,2,7,8,10}, {1,3,6,2,5,14}, {1,7,3,2,4,14}, {1,2,10,19,4,7,9,5}, {1,3,5,11,2,12,17,6}, {1,3,8,2,16,7,15,5}, {1,4,2,10,18,3,11,8}, {1,4,22,7,3,6,2,12}, {1,6,12,4,21,3,2,8}, {1,2,4,8,16,5,18,9,10}, {1,4,7,6,3,28,2,8,14}, {1,6,4,24,13,3,2,12,8}, {1,11,8,6,4,3,2,22,16}, {1,2,6,18,22,7,5,16,4,10}, {1,3,9,11,6,8,2,5,28,18}, {1,4,2,20,8,9,23,10,3,11}, {1,4,3,10,2,9,14,16,6,26}, {1,5,4,13,3,8,7,12,2,36}, {1,6,9,11,29,4,8,2,3,18}, {1,2,9,8,14,4,43,7,6,10,5,24}, {1,2,12,31,25,4,9,10,7,11,16,5}, {1,2,14,4,37,7,8,27,5,6,13,9}, {1,2,14,12,32,19,6,5,4,18,13,7}, {1,3,8,9,5,19,23,16,13,2,28,6}, {1,3,12,34,21,2,8,9,5,6,7,25}, {1,3,23,24,6,22,10,11,18,2,5,8}, {1,4,7,3,16,2,6,17,20,9,13,35}, {1,4,16,3,15,10,12,14,17,33,2,6}, {1,4,19,20,27,3,6,25,7,8,2,11}, {1,4,20,3,40,10,9,2,15,16,6,7}, {1,5,12,21,29,11,3,16,4,22,2,7}, {1,7,13,12,3,11,5,18,4,2,48,9}, {1,8,10,5,7,21,4,2,11,3,26,35}, {1,14,3,2,4,7,21,8,25,10,12,26}, {1,14,10,20,7,6,3,2,17,4,8,41}, {1,15,5,3,25,2,7,4,6,12,14,39}, {1,22,14,20,5,13,8,3,4,2,10,31}, {1,2,13,7,5,14,34,6,4,33,18,17,21,8}, {1,2,21,17,11,5,9,4,26,6,47,15,12,7}, {1,2,28,14,5,6,9,12,48,18,4,13,16,7}, {1,3,5,6,25,32,23,10,18,2,17,7,22,12}, {1,3,12,7,20,14,44,6,5,24,2,28,8,9}, {1,3,24,6,12,14,11,55,7,2,8,5,16,19}, {1,4,6,31,3,13,2,7,14,12,17,46,8,19}, {1,4,8,52,3,25,18,2,9,24,6,10,7,14}, {1,4,20,2,12,3,6,7,33,11,8,10,35,31}, {1,5,2,24,15,29,14,21,13,4,33,3,9,10}, {1,5,23,27,42,3,4,11,2,19,12,10,16,8}, {1,6,8,22,4,5,33,21,3,20,32,16,2,10}, {1,8,3,10,23,5,56,4,2,14,15,17,7,18}, {1,8,21,45,6,7,11,17,3,2,10,4,23,25}, {1,9,5,40,3,4,21,35,16,18,2,6,11,12}, {1,9,14,26,4,2,11,5,3,12,27,34,7,28}, {1,9,21,25,3,4,8,5,6,16,2,36,14,33}, {1,10,22,34,27,12,3,4,2,14,24,5,8,17}, {1,10,48,9,19,4,8,6,7,17,3,2,34,15}, {1,12,48,6,2,38,3,22,7,10,11,5,4,14}, {1,2,4,8,16,32,27,26,11,9,45,13,10,29,5,17,18}, {1,3,12,10,31,7,27,2,6,5,19,20,62,14,9,28,17}, {1,7,3,15,33,5,24,68,2,14,6,17,4,9,19,12,34}, {1,7,12,44,25,41,9,17,4,6,22,33,13,2,3,11,23}, {1,7,31,2,11,3,9,36,17,4,22,6,18,72,5,10,19}, {1,21,11,50,39,13,6,4,14,16,25,26,3,2,7,8,27}, {1,2,18,4,6,37,9,14,79,7,8,11,16,13,32,12,5,33}, {1,2,20,6,12,47,15,30,27,21,4,39,7,10,34,19,5,8}, {1,2,24,15,4,16,12,25,38,50,14,17,5,8,10,11,49,6}, {1,2,27,7,13,5,21,22,19,12,4,24,32,10,23,71,6,8}, {1,2,41,12,5,6,8,7,9,4,29,36,18,22,10,27,25,45}, {1,2,42,13,4,11,23,14,12,20,7,29,18,71,10,9,16,5}, {1,3,8,45,9,18,10,6,7,17,2,29,15,5,59,38,14,21}, {1,3,11,13,5,20,10,7,51,6,16,34,26,19,2,44,31,8}, {1,3,11,20,7,33,10,9,17,22,6,2,16,5,32,12,13,88}, {1,3,20,9,58,19,2,25,36,13,22,31,12,5,11,26,8,6}, {1,3,25,7,20,30,46,15,38,10,6,5,13,47,2,17,14,8}, {1,3,34,2,45,7,18,5,48,16,10,57,6,13,14,8,9,11}, {1,4,2,16,12,13,8,24,38,44,15,27,10,9,17,3,11,53}, {1,4,3,36,23,6,18,22,9,2,19,63,14,34,16,10,15,12}, {1,4,6,17,13,8,16,9,20,2,12,72,15,3,32,7,19,51}, {1,4,16,7,17,25,8,26,3,6,13,39,2,12,18,64,36,10}, {1,4,27,13,70,17,6,29,22,2,9,3,7,18,30,8,26,15}, {1,4,27,43,6,11,2,7,3,18,16,8,14,39,25,33,15,35}, {1,4,36,35,32,49,7,6,3,14,8,12,27,19,2,24,18,10}, {1,5,2,25,4,12,44,34,18,20,19,26,21,3,11,40,13,9}, {1,5,4,45,11,27,41,7,24,26,13,8,43,16,2,12,3,19}, {1,5,9,8,4,7,32,24,18,2,35,3,10,49,29,25,16,30}, {1,5,14,4,22,12,9,50,27,51,3,8,2,15,16,32,7,29}, {1,5,22,24,12,13,8,26,9,30,2,29,19,53,37,3,4,10}, {1,5,24,4,32,17,54,15,7,9,26,52,3,8,10,2,25,13}, {1,6,9,32,18,5,14,21,4,13,11,91,2,8,12,31,3,26}, {1,6,11,3,2,28,34,35,4,38,10,27,19,24,12,13,32,8}, {1,6,36,9,16,13,26,2,3,15,4,8,65,21,35,14,23,10}, {1,6,52,16,50,3,31,5,9,15,4,13,8,2,20,35,26,11}, {1,7,3,27,6,12,23,5,19,2,51,9,4,16,34,57,17,14}, {1,7,15,24,20,9,2,3,38,13,6,26,4,12,21,79,10,17}, {1,7,35,32,36,44,4,12,17,5,23,2,24,15,31,6,3,10}, {1,9,2,3,23,7,43,4,20,16,6,19,34,51,13,8,31,17}, {1,9,2,4,43,36,5,26,8,22,46,28,7,17,3,18,19,13}, {1,9,12,16,14,3,15,8,27,31,59,2,4,7,36,39,5,19}, {1,9,29,4,20,11,45,16,65,2,3,12,6,7,19,8,14,36}, {1,12,3,5,9,27,4,2,66,10,24,25,28,11,7,19,32,22}, {1,12,9,5,38,16,28,3,32,37,2,18,58,8,7,4,6,23}, {1,12,23,3,7,21,16,18,22,2,6,11,32,20,5,4,90,14}, {1,12,34,5,16,60,36,14,6,3,15,4,7,37,17,8,2,30}, {1,12,36,19,25,7,11,4,5,30,3,21,2,8,6,72,17,28}, {1,14,39,19,2,10,28,8,35,6,3,13,4,7,18,5,32,63}, {1,15,28,6,12,11,3,5,2,20,36,35,13,4,38,9,45,24}, {1,17,2,10,4,7,38,5,8,24,15,42,6,22,3,68,9,26}, {1,17,9,31,6,2,12,16,5,47,3,4,15,10,13,11,62,43}, {1,20,14,59,19,5,3,9,33,7,4,2,16,10,15,23,37,30}, {1,22,20,7,9,10,5,29,3,8,17,13,48,4,2,12,21,76}, {1,23,26,5,11,9,18,21,13,6,22,35,71,2,8,4,3,29}, {1,27,3,13,2,6,50,5,9,11,12,10,7,19,41,53,4,34}, {1,37,11,24,26,16,29,23,18,33,6,8,13,4,3,2,10,43}, {1,39,18,4,6,76,15,11,9,3,31,2,14,5,8,17,7,41}, {1,2,9,5,48,10,19,23,7,8,13,18,4,33,71,26,6,34,20,24}, {1,2,10,15,23,14,17,4,18,8,33,56,11,5,24,20,46,32,36,6}, {1,2,40,6,14,47,5,7,9,8,10,77,31,33,11,26,4,15,13,22}, {1,3,5,12,11,26,61,24,10,56,6,7,33,2,36,15,14,16,25,18}, {1,3,18,25,14,20,30,5,31,2,11,29,12,51,32,26,48,8,9,6}, {1,3,23,19,2,14,17,24,36,13,7,8,10,12,59,11,54,29,34,5}, {1,3,48,10,2,20,47,23,8,18,7,86,21,6,9,19,11,5,24,13}, {1,4,3,38,42,20,6,48,16,9,47,19,11,33,32,2,15,12,10,13}, {1,4,14,10,7,6,3,48,2,32,43,12,15,46,8,30,22,11,47,20}, {1,4,16,6,43,7,37,3,11,17,24,12,30,32,57,2,46,15,10,8}, {1,4,17,52,15,10,13,3,8,19,12,6,14,44,2,70,9,47,7,28}, {1,5,19,7,40,28,8,12,10,23,16,18,3,14,27,2,9,4,50,85}, {1,5,24,2,21,15,18,43,28,65,7,12,8,14,3,57,9,4,35,10}, {1,6,3,28,14,4,12,17,24,36,50,26,39,5,20,2,13,8,11,62}, {1,6,4,12,26,21,20,5,9,91,19,8,29,2,13,30,32,3,33,17}, {1,7,3,57,9,17,22,5,35,2,21,15,14,4,16,12,13,6,24,98}, {1,7,24,5,21,14,9,2,52,13,3,17,10,12,6,41,15,4,34,91}, {1,7,33,3,26,16,14,9,12,25,27,11,20,96,10,5,13,4,2,47}, {1,7,45,48,47,17,3,13,26,12,18,10,4,5,6,35,27,34,2,21}, {1,7,69,48,6,18,10,27,19,41,25,4,11,2,3,33,14,12,9,22}, {1,9,14,18,11,5,3,17,13,26,44,2,4,65,15,12,28,7,66,21}, {1,10,22,13,25,18,12,39,3,37,4,20,96,6,2,7,14,5,31,16}, {1,11,26,24,32,8,13,6,9,5,2,23,42,4,70,10,44,3,31,17}, {1,12,3,5,24,30,7,4,6,36,14,38,26,23,2,33,22,68,9,18}, {1,12,51,35,14,65,4,5,6,22,34,25,23,7,17,3,16,2,8,31}, {1,13,15,31,39,11,68,23,21,3,2,7,10,8,37,16,4,32,6,34}, {1,14,7,5,19,10,6,2,36,39,11,9,23,25,3,30,72,4,13,52}, {1,14,16,20,17,7,3,2,6,46,38,4,19,9,13,21,48,39,33,25}, {1,16,22,33,91,2,7,5,6,41,15,28,8,24,3,10,21,19,4,25}, {1,16,35,23,4,15,11,29,10,18,3,64,6,66,34,2,7,5,8,24}, {1,18,7,2,8,6,71,5,33,22,29,37,3,12,49,4,30,13,11,20}, {1,18,7,6,16,38,24,3,9,5,23,33,2,8,34,11,4,89,30,20}, {1,18,15,10,7,23,8,4,24,13,16,46,22,47,9,2,3,87,6,20}, {1,18,28,32,52,6,55,11,14,2,13,22,54,5,4,8,26,7,3,20}, {1,18,35,9,2,14,13,82,43,7,8,22,4,6,49,12,5,28,3,20}, {1,22,7,3,6,15,19,26,11,14,27,88,4,8,5,42,2,18,28,35}, {1,22,9,8,64,12,43,25,2,19,7,30,5,6,4,14,20,13,3,74}, {1,22,18,46,17,2,28,3,6,5,10,38,7,13,12,4,72,43,8,26}, {1,23,17,2,8,3,25,20,14,12,6,15,16,60,5,4,35,22,7,86}, {1,28,11,23,8,36,16,9,57,7,72,12,2,4,15,5,17,10,3,45}, {1,28,50,23,13,6,2,3,9,18,16,15,22,4,55,7,10,35,25,39}, {1,66,9,8,33,2,3,18,7,19,13,29,11,34,14,6,4,12,15,77}}
So, all four of your 9/73 solutions are identical when we permit multiplication by values coprime to 73. If we start with the first one: {0,1,3,7,15,31,36,54,63} multiplication by 49 gives us {0,49,1,51,5,59,12,18,21} which sorted is the second {0,1,5,12,18,21,49,51,59} multiplication by 51 gives us the third {0,1,7,11,35,48,51,53,65} and multiplication by 33 gives us the last. {0,1,12,20,26,30,33,35,57} On Wed, Nov 11, 2020 at 1:54 PM ed pegg <ed@mathpuzzle.com> wrote:
How are you eliminating redundant sets? Adding all values and sorting mod n, then picking the first.
If I've calculated them right, the number of distinct cyclic difference sets, or the number of cyclically perfect partitions starts out 1, 2, 1, 5, 6, 4, 6, 18, 20, 6, 51, 42 {{{7,3},1}, {{13,4},2}, {{21,5},1}, {{31,6},5}, {{57,8},6}, {{73,9},4}, {{91,10},6}, {{133,12},18}, {{183,14},20}, {{273,17},6}, {{307,18},51}, {{381,20},42}}
For example, the four cyclically perfect partitions of length 9 are {{1,2,4,8,16,5,18,9,10}, {1,4,7,6,3,28,2,8,14}, {1,6,4,24,13,3,2,12,8}, {1,11,8,6,4,3,2,22,16}} All sub-arcs of a perfect partition have a unique total and cover all integer values up to the set total. These correspond to the following difference sets: {{{73,9},{0,1,3,7,15,31,36,54,63}}, {{73,9},{0,1,5,12,18,21,49,51,59}}, {{73,9},{0,1,7,11,35,48,51,53,65}}, {{73,9},{0,1,12,20,26,30,33,35,57}}}
In case anyone is interested, here are the perfect partitions. {{1,2,4}, {1,2,6,4}, {1,3,2,7}, {1,3,10,2,5}, {1,2,5,4,6,13}, {1,2,7,4,12,5}, {1,3,2,7,8,10}, {1,3,6,2,5,14}, {1,7,3,2,4,14}, {1,2,10,19,4,7,9,5}, {1,3,5,11,2,12,17,6}, {1,3,8,2,16,7,15,5}, {1,4,2,10,18,3,11,8}, {1,4,22,7,3,6,2,12}, {1,6,12,4,21,3,2,8}, {1,2,4,8,16,5,18,9,10}, {1,4,7,6,3,28,2,8,14}, {1,6,4,24,13,3,2,12,8}, {1,11,8,6,4,3,2,22,16}, {1,2,6,18,22,7,5,16,4,10}, {1,3,9,11,6,8,2,5,28,18}, {1,4,2,20,8,9,23,10,3,11}, {1,4,3,10,2,9,14,16,6,26}, {1,5,4,13,3,8,7,12,2,36}, {1,6,9,11,29,4,8,2,3,18}, {1,2,9,8,14,4,43,7,6,10,5,24}, {1,2,12,31,25,4,9,10,7,11,16,5}, {1,2,14,4,37,7,8,27,5,6,13,9}, {1,2,14,12,32,19,6,5,4,18,13,7}, {1,3,8,9,5,19,23,16,13,2,28,6}, {1,3,12,34,21,2,8,9,5,6,7,25}, {1,3,23,24,6,22,10,11,18,2,5,8}, {1,4,7,3,16,2,6,17,20,9,13,35}, {1,4,16,3,15,10,12,14,17,33,2,6}, {1,4,19,20,27,3,6,25,7,8,2,11}, {1,4,20,3,40,10,9,2,15,16,6,7}, {1,5,12,21,29,11,3,16,4,22,2,7}, {1,7,13,12,3,11,5,18,4,2,48,9}, {1,8,10,5,7,21,4,2,11,3,26,35}, {1,14,3,2,4,7,21,8,25,10,12,26}, {1,14,10,20,7,6,3,2,17,4,8,41}, {1,15,5,3,25,2,7,4,6,12,14,39}, {1,22,14,20,5,13,8,3,4,2,10,31}, {1,2,13,7,5,14,34,6,4,33,18,17,21,8}, {1,2,21,17,11,5,9,4,26,6,47,15,12,7}, {1,2,28,14,5,6,9,12,48,18,4,13,16,7}, {1,3,5,6,25,32,23,10,18,2,17,7,22,12}, {1,3,12,7,20,14,44,6,5,24,2,28,8,9}, {1,3,24,6,12,14,11,55,7,2,8,5,16,19}, {1,4,6,31,3,13,2,7,14,12,17,46,8,19}, {1,4,8,52,3,25,18,2,9,24,6,10,7,14}, {1,4,20,2,12,3,6,7,33,11,8,10,35,31}, {1,5,2,24,15,29,14,21,13,4,33,3,9,10}, {1,5,23,27,42,3,4,11,2,19,12,10,16,8}, {1,6,8,22,4,5,33,21,3,20,32,16,2,10}, {1,8,3,10,23,5,56,4,2,14,15,17,7,18}, {1,8,21,45,6,7,11,17,3,2,10,4,23,25}, {1,9,5,40,3,4,21,35,16,18,2,6,11,12}, {1,9,14,26,4,2,11,5,3,12,27,34,7,28}, {1,9,21,25,3,4,8,5,6,16,2,36,14,33}, {1,10,22,34,27,12,3,4,2,14,24,5,8,17}, {1,10,48,9,19,4,8,6,7,17,3,2,34,15}, {1,12,48,6,2,38,3,22,7,10,11,5,4,14}, {1,2,4,8,16,32,27,26,11,9,45,13,10,29,5,17,18}, {1,3,12,10,31,7,27,2,6,5,19,20,62,14,9,28,17}, {1,7,3,15,33,5,24,68,2,14,6,17,4,9,19,12,34}, {1,7,12,44,25,41,9,17,4,6,22,33,13,2,3,11,23}, {1,7,31,2,11,3,9,36,17,4,22,6,18,72,5,10,19}, {1,21,11,50,39,13,6,4,14,16,25,26,3,2,7,8,27}, {1,2,18,4,6,37,9,14,79,7,8,11,16,13,32,12,5,33}, {1,2,20,6,12,47,15,30,27,21,4,39,7,10,34,19,5,8}, {1,2,24,15,4,16,12,25,38,50,14,17,5,8,10,11,49,6}, {1,2,27,7,13,5,21,22,19,12,4,24,32,10,23,71,6,8}, {1,2,41,12,5,6,8,7,9,4,29,36,18,22,10,27,25,45}, {1,2,42,13,4,11,23,14,12,20,7,29,18,71,10,9,16,5}, {1,3,8,45,9,18,10,6,7,17,2,29,15,5,59,38,14,21}, {1,3,11,13,5,20,10,7,51,6,16,34,26,19,2,44,31,8}, {1,3,11,20,7,33,10,9,17,22,6,2,16,5,32,12,13,88}, {1,3,20,9,58,19,2,25,36,13,22,31,12,5,11,26,8,6}, {1,3,25,7,20,30,46,15,38,10,6,5,13,47,2,17,14,8}, {1,3,34,2,45,7,18,5,48,16,10,57,6,13,14,8,9,11}, {1,4,2,16,12,13,8,24,38,44,15,27,10,9,17,3,11,53}, {1,4,3,36,23,6,18,22,9,2,19,63,14,34,16,10,15,12}, {1,4,6,17,13,8,16,9,20,2,12,72,15,3,32,7,19,51}, {1,4,16,7,17,25,8,26,3,6,13,39,2,12,18,64,36,10}, {1,4,27,13,70,17,6,29,22,2,9,3,7,18,30,8,26,15}, {1,4,27,43,6,11,2,7,3,18,16,8,14,39,25,33,15,35}, {1,4,36,35,32,49,7,6,3,14,8,12,27,19,2,24,18,10}, {1,5,2,25,4,12,44,34,18,20,19,26,21,3,11,40,13,9}, {1,5,4,45,11,27,41,7,24,26,13,8,43,16,2,12,3,19}, {1,5,9,8,4,7,32,24,18,2,35,3,10,49,29,25,16,30}, {1,5,14,4,22,12,9,50,27,51,3,8,2,15,16,32,7,29}, {1,5,22,24,12,13,8,26,9,30,2,29,19,53,37,3,4,10}, {1,5,24,4,32,17,54,15,7,9,26,52,3,8,10,2,25,13}, {1,6,9,32,18,5,14,21,4,13,11,91,2,8,12,31,3,26}, {1,6,11,3,2,28,34,35,4,38,10,27,19,24,12,13,32,8}, {1,6,36,9,16,13,26,2,3,15,4,8,65,21,35,14,23,10}, {1,6,52,16,50,3,31,5,9,15,4,13,8,2,20,35,26,11}, {1,7,3,27,6,12,23,5,19,2,51,9,4,16,34,57,17,14}, {1,7,15,24,20,9,2,3,38,13,6,26,4,12,21,79,10,17}, {1,7,35,32,36,44,4,12,17,5,23,2,24,15,31,6,3,10}, {1,9,2,3,23,7,43,4,20,16,6,19,34,51,13,8,31,17}, {1,9,2,4,43,36,5,26,8,22,46,28,7,17,3,18,19,13}, {1,9,12,16,14,3,15,8,27,31,59,2,4,7,36,39,5,19}, {1,9,29,4,20,11,45,16,65,2,3,12,6,7,19,8,14,36}, {1,12,3,5,9,27,4,2,66,10,24,25,28,11,7,19,32,22}, {1,12,9,5,38,16,28,3,32,37,2,18,58,8,7,4,6,23}, {1,12,23,3,7,21,16,18,22,2,6,11,32,20,5,4,90,14}, {1,12,34,5,16,60,36,14,6,3,15,4,7,37,17,8,2,30}, {1,12,36,19,25,7,11,4,5,30,3,21,2,8,6,72,17,28}, {1,14,39,19,2,10,28,8,35,6,3,13,4,7,18,5,32,63}, {1,15,28,6,12,11,3,5,2,20,36,35,13,4,38,9,45,24}, {1,17,2,10,4,7,38,5,8,24,15,42,6,22,3,68,9,26}, {1,17,9,31,6,2,12,16,5,47,3,4,15,10,13,11,62,43}, {1,20,14,59,19,5,3,9,33,7,4,2,16,10,15,23,37,30}, {1,22,20,7,9,10,5,29,3,8,17,13,48,4,2,12,21,76}, {1,23,26,5,11,9,18,21,13,6,22,35,71,2,8,4,3,29}, {1,27,3,13,2,6,50,5,9,11,12,10,7,19,41,53,4,34}, {1,37,11,24,26,16,29,23,18,33,6,8,13,4,3,2,10,43}, {1,39,18,4,6,76,15,11,9,3,31,2,14,5,8,17,7,41}, {1,2,9,5,48,10,19,23,7,8,13,18,4,33,71,26,6,34,20,24}, {1,2,10,15,23,14,17,4,18,8,33,56,11,5,24,20,46,32,36,6}, {1,2,40,6,14,47,5,7,9,8,10,77,31,33,11,26,4,15,13,22}, {1,3,5,12,11,26,61,24,10,56,6,7,33,2,36,15,14,16,25,18}, {1,3,18,25,14,20,30,5,31,2,11,29,12,51,32,26,48,8,9,6}, {1,3,23,19,2,14,17,24,36,13,7,8,10,12,59,11,54,29,34,5}, {1,3,48,10,2,20,47,23,8,18,7,86,21,6,9,19,11,5,24,13}, {1,4,3,38,42,20,6,48,16,9,47,19,11,33,32,2,15,12,10,13}, {1,4,14,10,7,6,3,48,2,32,43,12,15,46,8,30,22,11,47,20}, {1,4,16,6,43,7,37,3,11,17,24,12,30,32,57,2,46,15,10,8}, {1,4,17,52,15,10,13,3,8,19,12,6,14,44,2,70,9,47,7,28}, {1,5,19,7,40,28,8,12,10,23,16,18,3,14,27,2,9,4,50,85}, {1,5,24,2,21,15,18,43,28,65,7,12,8,14,3,57,9,4,35,10}, {1,6,3,28,14,4,12,17,24,36,50,26,39,5,20,2,13,8,11,62}, {1,6,4,12,26,21,20,5,9,91,19,8,29,2,13,30,32,3,33,17}, {1,7,3,57,9,17,22,5,35,2,21,15,14,4,16,12,13,6,24,98}, {1,7,24,5,21,14,9,2,52,13,3,17,10,12,6,41,15,4,34,91}, {1,7,33,3,26,16,14,9,12,25,27,11,20,96,10,5,13,4,2,47}, {1,7,45,48,47,17,3,13,26,12,18,10,4,5,6,35,27,34,2,21}, {1,7,69,48,6,18,10,27,19,41,25,4,11,2,3,33,14,12,9,22}, {1,9,14,18,11,5,3,17,13,26,44,2,4,65,15,12,28,7,66,21}, {1,10,22,13,25,18,12,39,3,37,4,20,96,6,2,7,14,5,31,16}, {1,11,26,24,32,8,13,6,9,5,2,23,42,4,70,10,44,3,31,17}, {1,12,3,5,24,30,7,4,6,36,14,38,26,23,2,33,22,68,9,18}, {1,12,51,35,14,65,4,5,6,22,34,25,23,7,17,3,16,2,8,31}, {1,13,15,31,39,11,68,23,21,3,2,7,10,8,37,16,4,32,6,34}, {1,14,7,5,19,10,6,2,36,39,11,9,23,25,3,30,72,4,13,52}, {1,14,16,20,17,7,3,2,6,46,38,4,19,9,13,21,48,39,33,25}, {1,16,22,33,91,2,7,5,6,41,15,28,8,24,3,10,21,19,4,25}, {1,16,35,23,4,15,11,29,10,18,3,64,6,66,34,2,7,5,8,24}, {1,18,7,2,8,6,71,5,33,22,29,37,3,12,49,4,30,13,11,20}, {1,18,7,6,16,38,24,3,9,5,23,33,2,8,34,11,4,89,30,20}, {1,18,15,10,7,23,8,4,24,13,16,46,22,47,9,2,3,87,6,20}, {1,18,28,32,52,6,55,11,14,2,13,22,54,5,4,8,26,7,3,20}, {1,18,35,9,2,14,13,82,43,7,8,22,4,6,49,12,5,28,3,20}, {1,22,7,3,6,15,19,26,11,14,27,88,4,8,5,42,2,18,28,35}, {1,22,9,8,64,12,43,25,2,19,7,30,5,6,4,14,20,13,3,74}, {1,22,18,46,17,2,28,3,6,5,10,38,7,13,12,4,72,43,8,26}, {1,23,17,2,8,3,25,20,14,12,6,15,16,60,5,4,35,22,7,86}, {1,28,11,23,8,36,16,9,57,7,72,12,2,4,15,5,17,10,3,45}, {1,28,50,23,13,6,2,3,9,18,16,15,22,4,55,7,10,35,25,39}, {1,66,9,8,33,2,3,18,7,19,13,29,11,34,14,6,4,12,15,77}}
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- - https://cube20.org/ <http://cube20.org/> - https://golly.sf.net/ <http://golly.sf.net/> - https://experiments.cubing.net/ -
participants (4)
-
Dan Asimov -
ed pegg -
Tomas Rokicki -
Veit Elser