The New Sudoku Players' Forum

by **RichardGoodrich** » Thu Jul 03, 2025 2:03 pm

Continuing this as a Living Post - last update July 16, 2025

Below are two lists of 416-group ED band occupancy I got from the 1to9only github. The first list is a compact list in five columns. I edited that list to a single column list which I may edit
in comparison to older lists I have seen: particularly one by gsf (Glenn S. Fowler). Just for clarification / amplification these 416 gangster groups are for the first horizontal band (rows 1,2,3) of the 5,472,730,538 essentially different grids in minlex order. The quantities in this list should add to: 5,472,730,538. This is JUST the start so that I can talk about them either in this post or others posts that may reference it. Please comment if you find my understanding to be incorrect!.

Referencing the Wikipedia article at https://en.wikipedia.org/wiki/Mathematics_of_Sudoku The ED exists to map the number: 6,670,903,752,021,072,936,960 of possible filled grids to a set of equivalence classes which is 5,472,730,538 long and on this forum is called the ED. The list is usually shown in a minlex order (think dictionary order). Further, we speak of chutes composed of three horizontal bands and three vertical towers in reference to the 9x9 Sudoku grid. So the band referenced is the first band consisting or rows 1,2,3. So the ED can be further divided into these 416 groups. It seems there are multiple ways of generating those groups (and the ED itself). As far as I can tell the preferred one is what I will call the row minlex order. (In fact on this forum Gordon F. Royle said that an equivalence class may be represented by any item in that class. But for consistency a rule is most often used. Ordering by maxlex, blocks, or even a graph renumbering order using a nauty library could be done as well as any other consistent method)

At the very bottom of this post is Prof Gordon F. Royle's post (slightly edited by me) from much earlier on this forum of a general defintion for a Sudoku canon. He is a graph theory scientist and mathematician from Australia. His post is VERY edifying as a general response to what has become the canon on this forum. Jokingly the term canonize (even with it's Vatican overtones!) has been applied to this process. And others talked about any other rule being a normalization.

This forum has always been difficult for me to navigate. However, here are three key links to some of the development of the idea of a canon for Sudoku and the ED minlex
ordering. I will edit this as I go....

I find this link on the numerology of Sudoku very useful https://sudopedia.sudocue.net/index.php/Mathematics_of_Sudoku

Look at the following for a bit on how symmetry gives us that ED#

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by 1to9only in Five Columns

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Code: Select all: 001 1007170 101 13768854 201 3165254 301 119226 401 1 002 25502082 102 26929453 202 3064062 302 20203 402 0 003 16538087 103 26382806 203 2966309 303 62246 403 0 004 8417906 104 4359314 204 2932890 304 63613 404 0 005 48737791 105 25997296 205 2841380 305 69669 405 4 006 96229042 106 25467197 206 2701985 306 58811 406 0 007 15765443 107 24888528 207 2628788 307 21225 407 19 008 5306280 108 24423300 208 2532198 308 56942 408 0 009 8136013 109 23988326 209 2443960 309 55120 409 0 010 47174193 110 23541927 210 1243959 310 49427 410 0 011 46788396 111 23070530 211 2317171 311 91869 411 3 012 46177270 112 22609142 212 2357854 312 89983 412 0 013 15340394 113 22100458 213 1137589 313 80765 413 0 014 45397270 114 10879514 214 1083228 314 43270 414 0 015 45600758 115 21378062 215 2183311 315 74594 415 0 016 1631576 116 20985174 216 2244753 316 69012 416 1 017 15093541 117 20674972 217 2143677 317 73627 018 45101600 118 20107116 218 2100798 318 62449 019 44832423 119 19854606 219 1007465 319 59123 020 88782526 120 9732970 220 1970315 320 57580 021 44036568 121 19084488 221 1841722 321 47910 022 85627559 122 9491325 222 1873099 322 44876 023 42711122 123 18532281 223 1772301 323 46852 024 85102373 124 9142485 224 347777 324 46002 025 41847039 125 18075269 225 1968442 325 40108 026 41335391 126 17675306 226 1677704 326 37300 027 4455504 127 17545752 227 1521001 327 36969 028 41102914 128 16990098 228 1498734 328 31504 029 4591391 129 8369473 229 1515366 329 28919 030 4664261 130 16406705 230 1457098 330 27982 031 13606209 131 16189996 231 1331185 331 29202 032 40697707 132 15791769 232 1279569 332 25098 033 80468663 133 2613345 233 1262013 333 20652 034 79175610 134 15362664 234 1218744 334 10105 035 77979783 135 15272476 235 386642 335 19471 036 38536298 136 14918036 236 1182963 336 18996 037 76146967 137 7254450 237 570172 337 17212 038 74505665 138 14383075 238 1111083 338 14780 039 74154564 139 7011714 239 1076551 339 13660 040 72171447 140 13738161 240 167032 340 12324 041 36053455 141 13445152 241 533940 341 10597 042 70552290 142 6593805 242 1048083 342 9562 043 69437575 143 12918117 243 974591 343 9012 044 67978951 144 6403269 244 967788 344 8215 045 33904021 145 12568136 245 455310 345 7261 046 66337407 146 12354720 246 915249 346 3569 047 65880161 147 12036469 247 500537 347 7136 048 64996381 148 5931073 248 783336 348 455 049 63898062 149 5949060 249 822496 349 2935 050 62192220 150 11577852 250 377256 350 2990 051 61691475 151 11435633 251 408556 351 4836 052 60192385 152 11155974 252 437792 352 2156 053 29966384 153 10671486 253 387029 353 2141 054 29734495 154 10525735 254 140436 354 1959 055 58731513 155 10188634 255 361962 355 4171 056 57263818 156 10059617 256 354702 356 3376 057 57033275 157 9805813 257 675674 357 3171 058 55394556 158 9629320 258 661737 358 3150 059 55022930 159 9490222 259 313209 359 647 060 54018514 160 9280124 260 623191 360 1528 061 52964870 161 8844112 261 546083 361 2484 062 52242492 162 8628099 262 524804 362 2233 063 51245000 163 8429593 263 534167 363 1930 064 50540742 164 8227144 264 503384 364 1353 065 49644127 165 7998287 265 464985 365 1368 066 49190978 166 7813413 266 461786 366 1232 067 24077300 167 3839149 267 441645 367 1667 068 47978806 168 7548052 268 418773 368 925 069 47059527 169 7349287 269 424148 369 872 070 46231581 170 7146807 270 378441 370 928 071 22715795 171 6993422 271 361885 371 808 072 44778204 172 6828801 272 360821 372 560 073 44053469 173 6674911 273 176161 373 757 074 43401907 174 6476248 274 172023 374 451 075 21398806 175 3166465 275 165927 375 245 076 42061440 176 6205963 276 154694 376 333 077 41316125 177 6040631 277 150664 377 156 078 40571245 178 5882934 278 309399 378 193 079 40282447 179 5812748 279 144927 379 161 080 39233218 180 5615082 280 141820 380 23 081 38522319 181 5461387 281 137601 381 163 082 37881913 182 5367414 282 287667 382 154 083 37460193 183 5222068 283 246093 383 111 084 18460204 184 5072949 284 123480 384 124 085 36127803 185 4918277 285 124070 385 87 086 35584769 186 4778878 286 116970 386 49 087 34821531 187 4641003 287 117351 387 66 088 34334716 188 4539624 288 110418 388 125 089 33769162 189 4407284 289 37988 389 27 090 33174401 190 2186822 290 109351 390 59 091 32520037 191 4220821 291 211267 391 19 092 31945541 192 4158097 292 209636 392 41 093 31221072 193 4070158 293 189161 393 2 094 30579410 194 3857103 294 188766 394 16 095 29977732 195 3785628 295 171584 395 0 096 29390061 196 3693474 296 152633 396 11 097 14518368 197 3555681 297 147806 397 0 098 14372444 198 3453089 298 70955 398 0 099 28268021 199 3345667 299 133302 399 10 100 27849953 200 3252227 300 139754 400 0

by 1to9only in One Column

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Code: Select all: 001 1007170 002 25502082 003 16538087 004 8417906 005 48737791 006 96229042 007 15765443 008 5306280 009 8136013 010 47174193 011 46788396 012 46177270 013 15340394 014 45397270 015 45600758 016 1631576 017 15093541 018 45101600 019 44832423 020 88782526 021 44036568 022 85627559 023 42711122 024 85102373 025 41847039 026 41335391 027 4455504 028 41102914 029 4591391 030 4664261 031 13606209 032 40697707 033 80468663 034 79175610 035 77979783 036 38536298 037 76146967 038 74505665 039 74154564 040 72171447 041 36053455 042 70552290 043 69437575 044 67978951 045 33904021 046 66337407 047 65880161 048 64996381 049 63898062 050 62192220 051 61691475 052 60192385 053 29966384 054 29734495 055 58731513 056 57263818 057 57033275 058 55394556 059 55022930 060 54018514 061 52964870 062 52242492 063 51245000 064 50540742 065 49644127 066 49190978 067 24077300 068 47978806 069 47059527 070 46231581 071 22715795 072 44778204 073 44053469 074 43401907 075 21398806 076 42061440 077 41316125 078 40571245 079 40282447 080 39233218 081 38522319 082 37881913 083 37460193 084 18460204 085 36127803 086 35584769 087 34821531 088 34334716 089 33769162 090 33174401 091 32520037 092 31945541 093 31221072 094 30579410 095 29977732 096 29390061 097 14518368 098 14372444 099 28268021 100 27849953 101 13768854 102 26929453 103 26382806 104 4359314 105 25997296 106 25467197 107 24888528 108 24423300 109 23988326 110 23541927 111 23070530 112 22609142 113 22100458 114 10879514 115 21378062 116 20985174 117 20674972 118 20107116 119 19854606 120 9732970 121 19084488 122 9491325 123 18532281 124 9142485 125 18075269 126 17675306 127 17545752 128 16990098 129 8369473 130 16406705 131 16189996 132 15791769 133 2613345 134 15362664 135 15272476 136 14918036 137 7254450 138 14383075 139 7011714 140 13738161 141 13445152 142 6593805 143 12918117 144 6403269 145 12568136 146 12354720 147 12036469 148 5931073 149 5949060 150 11577852 151 11435633 152 11155974 153 10671486 154 10525735 155 10188634 156 10059617 157 9805813 158 9629320 159 9490222 160 9280124 161 8844112 162 8628099 163 8429593 164 8227144 165 7998287 166 7813413 167 3839149 168 7548052 169 7349287 170 7146807 171 6993422 172 6828801 173 6674911 174 6476248 175 3166465 176 6205963 177 6040631 178 5882934 179 5812748 180 5615082 181 5461387 182 5367414 183 5222068 184 5072949 185 4918277 186 4778878 187 4641003 188 4539624 189 4407284 190 2186822 191 4220821 192 4158097 193 4070158 194 3857103 195 3785628 196 3693474 197 3555681 198 3453089 199 3345667 200 3252227 201 3165254 202 3064062 203 2966309 204 2932890 205 2841380 206 2701985 207 2628788 208 2532198 209 2443960 210 1243959 211 2317171 212 2357854 213 1137589 214 1083228 215 2183311 216 2244753 217 2143677 218 2100798 219 1007465 220 1970315 221 1841722 222 1873099 223 1772301 224 347777 225 1968442 226 1677704 227 1521001 228 1498734 229 1515366 230 1457098 231 1331185 232 1279569 233 1262013 234 1218744 235 386642 236 1182963 237 570172 238 1111083 239 1076551 240 167032 241 533940 242 1048083 243 974591 244 967788 245 455310 246 915249 247 500537 248 783336 249 822496 250 377256 251 408556 252 437792 253 387029 254 140436 255 361962 256 354702 257 675674 258 661737 259 313209 260 623191 261 546083 262 524804 263 534167 264 503384 265 464985 266 461786 267 441645 268 418773 269 424148 270 378441 271 361885 272 360821 273 176161 274 172023 275 165927 276 154694 277 150664 278 309399 279 144927 280 141820 281 137601 282 287667 283 246093 284 123480 285 124070 286 116970 287 117351 288 110418 289 37988 290 109351 291 211267 292 209636 293 189161 294 188766 295 171584 296 152633 297 147806 298 70955 299 133302 300 139754 301 119226 302 20203 303 62246 304 63613 305 69669 306 58811 307 21225 308 56942 309 55120 310 49427 311 91869 312 89983 313 80765 314 43270 315 74594 316 69012 317 73627 318 62449 319 59123 320 57580 321 47910 322 44876 323 46852 324 46002 325 40108 326 37300 327 36969 328 31504 329 28919 330 27982 331 29202 332 25098 333 20652 334 10105 335 19471 336 18996 337 17212 338 14780 339 13660 340 12324 341 10597 342 9562 343 9012 344 8215 345 7261 346 3569 347 7136 348 455 349 2935 350 2990 351 4836 352 2156 353 2141 354 1959 355 4171 356 3376 357 3171 358 3150 359 647 360 1528 361 2484 362 2233 363 1930 364 1353 365 1368 366 1232 367 1667 368 925 369 872 370 928 371 808 372 560 373 757 374 451 375 245 376 333 377 156 378 193 379 161 380 23 381 163 382 154 383 111 384 124 385 87 386 49 387 66 388 125 389 27 390 59 391 19 392 41 393 2 394 16 395 0 396 11 397 0 398 0 399 10 400 0 401 1 402 0 403 0 404 0 405 4 406 0 407 19 408 0 409 0 410 0 411 3 412 0 413 0 414 0 415 0 416 1

Gordon R. Royle on "Canonical Form:

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Gordon F. Royle wrote:The basic idea is that there are a number of operations that turn a grid (partially filled or complete, valid puzzle or pseudo-puzzle) into one that is entirely equivalent. This means that each grid comes with a large collection of equivalent grids, called the "equivalence class" of that grid. In general, each of these sets of equivalent grids will contain 9! x 6^8 x 2 grids, but sometimes fewer.

For searching, counting and a variety of other mathematical purposes, we usually want to treat an entire equivalence class of grids as a single object, because every grid in that class is "essentially the same". But for programming, solving and so on, we need to actually use a specific individual grid to represent the entire equivalence class. So the concept of a CANONICAL grid is to choose SOME RULE hat picks out ONE individual grid from the entire equivalence class and DEFINE that grid to be the "canonical" one.

Wolfgang gave one example of such a rule: pick the grid from the class whose completion is the lowest 81-digit number. But there are plenty of other possible rules and they don't need to involve finding the grid's completion, and so they don't need to be restricted to valid puzzles. Here are some choices..

(a) Write each grid as a number in decimal with "0" for the empty spaces and from all of these choose the one that gives the SMALLEST number as the canonical one
[ this will agree with Wolfgang's order for complete grids ]

(b) .. as before .. but choose the LARGEST number as the canonical one [so complete canonical grids will start with 987654321

(c) Write each grid as a number, but block-by-block, rather than row-by-row... then choose the smallest, or largest or whatever.

One that I use is:

(d) Convert grid to a graph, use the graph-labelling program nauty, and then convert graph back to a grid.

The ONLY advantage of the last one is that it is very fast for puzzles with few clues. For 17 clue puzzles I can find the canonical grids at the rate of about 4000 per second (on a standardish 2.4G PIV). Given that I am processing millions of 17/18-clue puzzles, I need to work quickly with the canonical version to avoid repetitions. Its disadvantage is that the canonical version of a puzzle has no human interpretation - it just seems like an arbitrarily chosen grid.

The New Sudoku Players' Forum

Essentially Different - 416 Gangster Group Occupancy

Essentially Different - 416 Gangster Group Occupancy