I have found an interesting sub-species of Sudoku grids, based on the cycle structure. The term "cycle" is described here, and the following examples should serve to illustrate the concept.
![Image](https://mathimagics.github.io//images/SI-002.png)
On the left we have the cycles for {AB} = {36}. There are 3 cycles, of lengths 2, 3 and 4.
On the right we have {AB} = {59}, for which there is a single cycle of length 9.
Some essential facts:
- there are 36 digit pair {AB} possibilities
- for each AB, there are from 1 to 4 cycles, and the lengths of these cycles will sum to 9
- for any AB, there are 8 different combinations of cycle lengths
- 9
- 7 + 2, 6 + 3, 5 + 4
- 5 + 2 + 2, 4 + 3 + 2, 3 + 3 + 3
- 3 + 2 + 2 + 2
- the total number of cycles (TNC) in any grid is thus in the range [36, 144]
A monocyclic grid is one which every AB pair has cycle length 9, and thus the grid has only 36 cycles in total. There are only 9 grids with this property. In this list they are tagged with band# and automorphism count (7 of the 9 are automorphic):
- Code: Select all
123456789456789123798231564275943618369815472841627395587362941612594837934178256 # 14
123456789456789123897231645231564978645978231978312564369147852582693417714825396 # 17 NA = 6
123456789456789123897231645231564978645978231978312564369825417582147396714693852 # 17 NA = 6
123456789456789231789123645231564897564897312897231456375618924618942573942375168 # 27 NA =36
123456789456789231789123645231564897564897312897231456375942168618375924942618573 # 27 NA =18
123456789456789231789123645231564897564897312978312564395248176617935428842671953 # 27 NA = 6
123456789456789231789123645231564897564897312978312564395671428617248953842935176 # 27 NA = 6
123456789456789231789123645231564978564897123897231564375942816618375492942618357 # 27 NA =36
123456789457289163698713254261847935579362841834591627315928476786134592942675318 # 354
There only 8 cases of "almost" monocyclic grids, these have 35 cycles of length 9, with one {AB} pair divided into 2 or 3 cycles, so TNC = 37 or 38:
- Code: Select all
123456789456789123798213564247138956319675248685942317572364891834591672961827435 # 12
123456789456789123798213564249561837637894251815327496372945618564138972981672345 # 12
123456789456789123798231645267198534581643972934572816349867251672315498815924367 # 15 NA = 2
123456789456789123798231645267843591815697234934125867372918456581364972649572318 # 15
123456789456789132789231564218647395394518627675923841531894276847162953962375418 # 25 NA = 2
123456789456789132789231564234678915861925473975143628317894256592367841648512397 # 25
123456789456789132789231564235697418671842395948315627394178256567923841812564973 # 25
123456789457189623689732145236578914795214836841963257378625491564891372912347568 # 250