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Square One God's Algorithm Computed

Submitted by masonjones on Tue, 12/06/2005 - 09:04.
I reported this to the most visible Square One guy I could find on the web (Jaap Scherphius), and he sent me here. Yes, I have spent the last year or so computing God's Algorithm for Square One, in the center-flip-count domain, taking advantage of all top/bottom rotational degeneracies available. The table calculation itself is finished, but I am not finished analyzing it yet. I can report that the maximum number of flips required to get back to START is 13, with many shapes (not just square-square) contributing to that number.

The main stumbling block to reporting results at the moment is that I discovered a unexpected position-counting problem, where the shape degeneracy interferes with the 16-fold starting position degeneracy, and which requires me to do a recount on the tables. I will post the results back here in a few days when the recount is finished.

After reporting the counts by depth and shape, my first priority will be to exhaustively search for all parity-flip solutions (preserving center flip) to see if there are any less than 10 from START. Jaap reports one on his site at 9, but the center flip is not preserved.

I am not a mathematician, a programmer with an engineering doctorate who likes the Rubik-type puzzles. Square One is my alltime favorite, hands down.

Mike Masonjones
kickme@tampabay.rr.com
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10 moves max

Submitted by masonjones on Thu, 12/22/2005 - 07:23.
An interesting result: Using the God's Algorithm tables, I can say that all even parity permutations of square/square are at most 10 flips from start (which also means that if you don't care about the flip state of the center piece, even parity permutations of square/square are at most 9 from start).
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God's algorithm counts

Submitted by masonjones on Wed, 12/21/2005 - 15:22.
Here are the god's algorithm count results for Square One by depth.

1
64
1153
17050
235144
3091458
38893230
452031138
4459167504
33671064770
149502310936
183662070768
63945120032
157452752
------------
435891456000

For the hard core, I have also included a lising by shape (c=corner, e=edge, format=topshape/bottomshape) and depth at the bottom of this posting. Totals are given first for each shape, and reflect the symmetry degeneracies. i.e. Total positions = 8!*8!*2*((4+1/2)*(1/6)*2+(9+1/3)*3*2+(8+1/2+1/4)*(8+1/2+1/4)) = 435891456000. Will someone please explain to me (Mike G., this means you, as Jaap claims you have a proof on his website) why this is exactly 15!/3?

Note that the flipped state of the center piece was taken into account, and it is determined solely by whether you are at an odd or even depth.

One interesting note is that only a very few shapes are unrepresented at the maximum depth of 13. I was hoping for a handful of antipodes, but instead almost every shape is represented, and square/square is rather more underrepresented than some of the others (like kite/kite (one off start shape)). Perhaps if I had accounted for face turns like Jaap and Mike G. do in their optimal solver, instead of just flips, but that requires about another 5-10 years of Moore's Law to kick in before I can do that one, since it will require much more space and computing power.

As it was, this calculation took over a year on my old slow computer (800 MHz) and even slower hard drive. I also did not have access to or expertise in a programming language that accesses RAM memory directly - no matter what I did, the program I used always seemed to go to disk to read variables that should have, in theory, fit just fine in the available memory. The program intends to accomplish this by only keeping 2 shapes active at a time, requiring only 200MB of memory and relatively rare loads/unloads, instead of needing random access to the full complement of 170 100MB (4 bits/position) shape files. But as I said, I couldn't get my machine to do it in practice.

I realize also after the fact that I could have taken advantage of another factor of 2 in at least space savings by noting the degeneracy of the top/bottom starting position (instead of just the factor of 16 I utilized from free top/bottom rotations of the start position), but I do not think it would have saved much calculation time, since it would have been more computationally expensive to keep swapping the pieces around artificially all the time instead of just saving the redundant positions to disk. This is also why all the shape files are the same size, each redundant by a factor equal to its symmetry degeneracy - it takes more space, but it is computationally more efficient and definitely easier to program.

Perhaps I will take advantage of the extra factor of 2 (really 170/90) when I convert the full tables into smaller ones (it will be quite easy to put 5 positions in 8 bits using just depth mod 3, sufficient for retrieving an optimal solution for any position given). So in theory, I could get the entire set of tables down to 3.4 GB in this manner, small enough for a DVD.

That's enough rambling. If you're still with me, here are the full set of counts by shape and depth (in no particular order, but I have kept all top/bottom bottom/top redundant pairs together for clarity):

cceeeeeeee/cccccc: 541900800 0 0 0 0 64 2048 43008 735936 10145984 88956960 244998048 181255424 15763296 32
cccccc/cceeeeeeee: 541900800 0 0 0 0 64 2048 43008 735936 10145984 88956960 244998048 181255424 15763296 32
ceceeeeeee/cccccc: 541900800 0 0 0 0 0 0 960 43040 1152320 21737184 181502048 249156800 88295072 13376
cccccc/ceceeeeeee: 541900800 0 0 0 0 0 0 960 43040 1152320 21737184 181502048 249156800 88295072 13376
ceeceeeeee/cccccc: 541900800 0 0 0 0 64 1664 34560 647552 9684672 89058912 247442336 181242272 13788768 0
cccccc/ceeceeeeee: 541900800 0 0 0 0 64 1664 34560 647552 9684672 89058912 247442336 181242272 13788768 0
ceeeeceeee/cccccc: 270950400 0 0 0 0 64 1856 33728 515392 6555968 52421920 124122336 82536032 4763104 0
cccccc/ceeeeceeee: 270950400 0 0 0 0 64 1856 33728 515392 6555968 52421920 124122336 82536032 4763104 0
ceeeeeceee/cccccc: 541900800 0 0 0 0 0 0 608 43424 1248896 23491392 188124736 247407552 81576160 8032
cccccc/ceeeeeceee: 541900800 0 0 0 0 0 0 608 43424 1248896 23491392 188124736 247407552 81576160 8032
ccceeeeee/cccccee: 3251404800 0 0 0 0 128 6656 181632 3600576 53828992 491939328 1451892352 1130155776 119799296 64
cccccee/ccceeeeee: 3251404800 0 0 0 0 128 6656 181632 3600576 53828992 491939328 1451892352 1130155776 119799296 64
cceeceeee/cccccee: 3251404800 0 0 0 0 192 8448 215168 4114496 59773216 525977344 1461192192 1095602048 104521632 64
cccccee/cceeceeee: 3251404800 0 0 0 0 192 8448 215168 4114496 59773216 525977344 1461192192 1095602048 104521632 64
cceeeceee/cccccee: 3251404800 0 0 0 0 0 960 35776 825536 14833920 204015872 1232710208 1420843584 378122496 16448
cccccee/cceeeceee: 3251404800 0 0 0 0 0 960 35776 825536 14833920 204015872 1232710208 1420843584 378122496 16448
cceeeecee/cccccee: 3251404800 0 0 0 0 192 8448 215168 4114496 59773216 525977344 1461192192 1095602048 104521632 64
cccccee/cceeeecee: 3251404800 0 0 0 0 192 8448 215168 4114496 59773216 525977344 1461192192 1095602048 104521632 64
cceeeeece/cccccee: 3251404800 0 0 0 0 64 1664 38016 766080 13279776 185485024 1185556960 1439389632 426827584 60000
cccccee/cceeeeece: 3251404800 0 0 0 0 64 1664 38016 766080 13279776 185485024 1185556960 1439389632 426827584 60000
cececeeee/cccccee: 3251404800 0 0 0 0 128 5760 129472 2219072 32027392 348255232 1414301696 1275221952 179243712 384
cccccee/cececeeee: 3251404800 0 0 0 0 128 5760 129472 2219072 32027392 348255232 1414301696 1275221952 179243712 384
ceceeceee/cccccee: 3251404800 0 0 0 0 0 1920 56992 1140416 18574560 232922240 1259614272 1391600832 347456576 36992
cccccee/ceceeceee: 3251404800 0 0 0 0 0 1920 56992 1140416 18574560 232922240 1259614272 1391600832 347456576 36992
ceceeecee/cccccee: 3251404800 0 0 0 0 0 1920 56992 1140416 18574560 232922240 1259614272 1391600832 347456576 36992
cccccee/ceceeecee: 3251404800 0 0 0 0 0 1920 56992 1140416 18574560 232922240 1259614272 1391600832 347456576 36992
ceceeeeec/cccccee: 3251404800 0 0 0 0 64 1664 38016 766080 13279776 185485024 1185556960 1439389632 426827584 60000
cccccee/ceceeeeec: 3251404800 0 0 0 0 64 1664 38016 766080 13279776 185485024 1185556960 1439389632 426827584 60000
ceeceecee/cccccee: 1083801600 0 0 0 0 256 7808 148992 2308800 28885888 220537216 495417984 319046976 17447680 0
cccccee/ceeceecee: 1083801600 0 0 0 0 256 7808 148992 2308800 28885888 220537216 495417984 319046976 17447680 0
ccceeeeee/ccccece: 3251404800 0 0 0 0 0 384 16768 436800 8685248 136291328 1058548864 1488706304 558451520 267584
ccccece/ccceeeeee: 3251404800 0 0 0 0 0 384 16768 436800 8685248 136291328 1058548864 1488706304 558451520 267584
cceeceeee/ccccece: 3251404800 0 0 0 0 0 1536 50336 1046336 17396800 222208032 1236014464 1402378272 372240800 68224
ccccece/cceeceeee: 3251404800 0 0 0 0 0 1536 50336 1046336 17396800 222208032 1236014464 1402378272 372240800 68224
cceeeceee/ccccece: 3251404800 0 0 0 0 128 3328 68864 1175232 17354752 209394368 1209565760 1415087808 398712896 41664
ccccece/cceeeceee: 3251404800 0 0 0 0 128 3328 68864 1175232 17354752 209394368 1209565760 1415087808 398712896 41664
cceeeecee/ccccece: 3251404800 0 0 0 0 0 1536 50336 1046336 17396800 222208032 1236014464 1402378272 372240800 68224
ccccece/cceeeecee: 3251404800 0 0 0 0 0 1536 50336 1046336 17396800 222208032 1236014464 1402378272 372240800 68224
cceeeeece/ccccece: 3251404800 0 0 0 0 64 2944 68928 1250016 18988928 224473344 1203811232 1399802624 402833248 173472
ccccece/cceeeeece: 3251404800 0 0 0 0 64 2944 68928 1250016 18988928 224473344 1203811232 1399802624 402833248 173472
cececeeee/ccccece: 3251404800 0 0 0 0 128 4608 106112 1907840 28616064 322969920 1389881792 1300818944 207098304 1088
ccccece/cececeeee: 3251404800 0 0 0 0 128 4608 106112 1907840 28616064 322969920 1389881792 1300818944 207098304 1088
ceceeceee/ccccece: 3251404800 0 0 0 0 512 13824 246688 3626528 45010688 395918592 1321531072 1226080672 258913440 62784
ccccece/ceceeceee: 3251404800 0 0 0 0 512 13824 246688 3626528 45010688 395918592 1321531072 1226080672 258913440 62784
ceceeecee/ccccece: 3251404800 0 0 0 0 512 13824 246688 3626528 45010688 395918592 1321531072 1226080672 258913440 62784
ccccece/ceceeecee: 3251404800 0 0 0 0 512 13824 246688 3626528 45010688 395918592 1321531072 1226080672 258913440 62784
ceceeeeec/ccccece: 3251404800 0 0 0 0 64 2944 68928 1250016 18988928 224473344 1203811232 1399802624 402833248 173472
ccccece/ceceeeeec: 3251404800 0 0 0 0 64 2944 68928 1250016 18988928 224473344 1203811232 1399802624 402833248 173472
ceeceecee/ccccece: 1083801600 0 0 0 0 0 0 4096 133504 2831936 45162816 352244736 496498688 186820032 105792
ccccece/ceeceecee: 1083801600 0 0 0 0 0 0 4096 133504 2831936 45162816 352244736 496498688 186820032 105792
ccceeeeee/cccecce: 3251404800 0 0 0 0 0 512 16384 355520 6416704 94343808 811189184 1523908416 808080128 7094144
cccecce/ccceeeeee: 3251404800 0 0 0 0 0 512 16384 355520 6416704 94343808 811189184 1523908416 808080128 7094144
cceeceeee/cccecce: 3251404800 0 0 0 0 64 2016 41024 709376 10812960 136104576 928514272 1483515584 686334080 5370848
cccecce/cceeceeee: 3251404800 0 0 0 0 64 2016 41024 709376 10812960 136104576 928514272 1483515584 686334080 5370848
cceeeceee/cccecce: 3251404800 0 0 0 0 0 384 19968 566912 11604928 167762688 1065406144 1456606080 548671360 766336
cccecce/cceeeceee: 3251404800 0 0 0 0 0 384 19968 566912 11604928 167762688 1065406144 1456606080 548671360 766336
cceeeecee/cccecce: 3251404800 0 0 0 0 64 2016 41024 709376 10812960 136104576 928514272 1483515584 686334080 5370848
cccecce/cceeeecee: 3251404800 0 0 0 0 64 2016 41024 709376 10812960 136104576 928514272 1483515584 686334080 5370848
cceeeeece/cccecce: 3251404800 0 0 0 0 0 768 29792 689056 12301536 164059072 1040801344 1459679264 572569728 1274240
cccecce/cceeeeece: 3251404800 0 0 0 0 0 768 29792 689056 12301536 164059072 1040801344 1459679264 572569728 1274240
cececeeee/cccecce: 3251404800 0 0 0 0 0 128 11392 333824 6702720 103537856 881996160 1519426624 736992128 2403968
cccecce/cececeeee: 3251404800 0 0 0 0 0 128 11392 333824 6702720 103537856 881996160 1519426624 736992128 2403968
ceceeceee/cccecce: 3251404800 0 0 0 0 0 1472 41600 808512 12832576 158333312 989377312 1462481952 623450912 4077152
cccecce/ceceeceee: 3251404800 0 0 0 0 0 1472 41600 808512 12832576 158333312 989377312 1462481952 623450912 4077152
ceceeecee/cccecce: 3251404800 0 0 0 0 0 1472 41600 808512 12832576 158333312 989377312 1462481952 623450912 4077152
cccecce/ceceeecee: 3251404800 0 0 0 0 0 1472 41600 808512 12832576 158333312 989377312 1462481952 623450912 4077152
ceceeeeec/cccecce: 3251404800 0 0 0 0 0 768 29792 689056 12301536 164059072 1040801344 1459679264 572569728 1274240
cccecce/ceceeeeec: 3251404800 0 0 0 0 0 768 29792 689056 12301536 164059072 1040801344 1459679264 572569728 1274240
ceeceecee/cccecce: 1083801600 0 0 0 0 0 0 1984 64960 1373056 22558976 227917632 515214592 312608128 4062272
cccecce/ceeceecee: 1083801600 0 0 0 0 0 0 1984 64960 1373056 22558976 227917632 515214592 312608128 4062272
cccceeee/cccceeee: 3251404800 0 0 0 64 1856 33792 513472 7128960 85727168 654021696 1485339008 964517760 54120896 128
ccceceee/cccceeee: 3251404800 0 0 0 0 0 256 12448 339392 7122368 118361728 1003414144 1506607968 615153440 393056
cccceeee/ccceceee: 3251404800 0 0 0 0 0 256 12448 339392 7122368 118361728 1003414144 1506607968 615153440 393056
ccceecee/cccceeee: 3251404800 0 0 0 0 0 2560 108864 2683712 44538368 434644224 1417536256 1188371712 163518912 192
cccceeee/ccceecee: 3251404800 0 0 0 0 0 2560 108864 2683712 44538368 434644224 1417536256 1188371712 163518912 192
ccceeece/cccceeee: 3251404800 0 0 0 0 0 256 12448 339392 7122368 118361728 1003414144 1506607968 615153440 393056
cccceeee/ccceeece: 3251404800 0 0 0 0 0 256 12448 339392 7122368 118361728 1003414144 1506607968 615153440 393056
cceeccee/cccceeee: 1625702400 0 0 0 0 256 8832 184000 3023264 38823360 298084128 723124608 511734720 50718976 256
cccceeee/cceeccee: 1625702400 0 0 0 0 256 8832 184000 3023264 38823360 298084128 723124608 511734720 50718976 256
cceecece/cccceeee: 3251404800 0 0 0 0 0 512 27200 712544 13686080 197865120 1224497248 1427069696 387491872 54528
cccceeee/cceecece: 3251404800 0 0 0 0 0 512 27200 712544 13686080 197865120 1224497248 1427069696 387491872 54528
cceeecce/cccceeee: 3251404800 0 0 0 0 0 384 12096 284352 5674816 93529856 875636352 1530381312 744379136 1506496
cccceeee/cceeecce: 3251404800 0 0 0 0 0 384 12096 284352 5674816 93529856 875636352 1530381312 744379136 1506496
cececece/cccceeee: 812851200 0 0 0 0 0 384 14336 329696 5705808 71343056 338214144 334751760 62491312 704
cccceeee/cececece: 812851200 0 0 0 0 0 384 14336 329696 5705808 71343056 338214144 334751760 62491312 704
cececeec/cccceeee: 3251404800 0 0 0 0 0 512 27200 712544 13686080 197865120 1224497248 1427069696 387491872 54528
cccceeee/cececeec: 3251404800 0 0 0 0 0 512 27200 712544 13686080 197865120 1224497248 1427069696 387491872 54528
ceceecec/cccceeee: 3251404800 0 0 0 64 1536 25728 368448 4723872 53039296 408022912 1207836608 1212225696 364456512 704128
cccceeee/ceceecec: 3251404800 0 0 0 64 1536 25728 368448 4723872 53039296 408022912 1207836608 1212225696 364456512 704128
ccceceee/ccceceee: 3251404800 0 0 0 0 0 896 32608 672064 11245568 151416800 1048248352 1473010080 566175872 602560
ccceecee/ccceceee: 3251404800 0 0 0 0 0 64 10624 348384 7674112 127665632 1041163424 1497404288 576854240 284032
ccceceee/ccceecee: 3251404800 0 0 0 0 0 64 10624 348384 7674112 127665632 1041163424 1497404288 576854240 284032
ccceeece/ccceceee: 3251404800 0 0 0 64 1600 26880 379456 4791904 53640224 429300736 1322820160 1191491136 248860960 91680
ccceceee/ccceeece: 3251404800 0 0 0 64 1600 26880 379456 4791904 53640224 429300736 1322820160 1191491136 248860960 91680
cceeccee/ccceceee: 1625702400 0 0 0 0 0 128 7232 211584 4516976 73085408 550768736 739417248 257558256 136832
ccceceee/cceeccee: 1625702400 0 0 0 0 0 128 7232 211584 4516976 73085408 550768736 739417248 257558256 136832
cceecece/ccceceee: 3251404800 0 0 0 0 0 768 28096 628480 11063232 153656672 1066654464 1471064800 547956608 351680
ccceceee/cceecece: 3251404800 0 0 0 0 0 768 28096 628480 11063232 153656672 1066654464 1471064800 547956608 351680
cceeecce/ccceceee: 3251404800 0 0 0 0 0 384 18560 508576 10283520 153066560 1064207840 1471739200 551192480 387680
ccceceee/cceeecce: 3251404800 0 0 0 0 0 384 18560 508576 10283520 153066560 1064207840 1471739200 551192480 387680
cececece/ccceceee: 812851200 0 0 0 0 0 640 20128 397840 6212744 72149288 329850768 333869016 70341960 8816
ccceceee/cececece: 812851200 0 0 0 0 0 640 20128 397840 6212744 72149288 329850768 333869016 70341960 8816
cececeec/ccceceee: 3251404800 0 0 0 0 1152 28960 488064 6714400 74089312 491329056 1159739872 1126267360 391384000 1362624
ccceceee/cececeec: 3251404800 0 0 0 0 1152 28960 488064 6714400 74089312 491329056 1159739872 1126267360 391384000 1362624
ceceecec/ccceceee: 3251404800 0 0 0 0 0 832 28448 617536 10756800 146770784 1015659936 1477030656 599257216 1282592
ccceceee/ceceecec: 3251404800 0 0 0 0 0 832 28448 617536 10756800 146770784 1015659936 1477030656 599257216 1282592
ccceecee/ccceecee: 3251404800 0 0 0 128 3584 63232 926592 11977504 127866528 779917536 1436852896 833743936 60052800 64
ccceeece/ccceecee: 3251404800 0 0 0 0 0 64 10624 348384 7674112 127665632 1041163424 1497404288 576854240 284032
ccceecee/ccceeece: 3251404800 0 0 0 0 0 64 10624 348384 7674112 127665632 1041163424 1497404288 576854240 284032
cceeccee/ccceecee: 1625702400 0 0 0 0 0 1152 50944 1278016 21466752 211137504 700294208 600433696 91039296 832
ccceecee/cceeccee: 1625702400 0 0 0 0 0 1152 50944 1278016 21466752 211137504 700294208 600433696 91039296 832
cceecece/ccceecee: 3251404800 0 0 0 0 0 128 11968 390880 8669120 144557344 1110685600 1480492576 506335712 261472
ccceecee/cceecece: 3251404800 0 0 0 0 0 128 11968 390880 8669120 144557344 1110685600 1480492576 506335712 261472
cceeecce/ccceecee: 3251404800 0 0 0 0 352 10784 219072 3485536 44114944 363009120 1101764608 1255821664 479603424 3375296
ccceecee/cceeecce: 3251404800 0 0 0 0 352 10784 219072 3485536 44114944 363009120 1101764608 1255821664 479603424 3375296
cececece/ccceecee: 812851200 0 0 0 128 2848 44672 594768 6764104 56411824 192441176 270343184 207082312 79072976 93208
ccceecee/cececece: 812851200 0 0 0 128 2848 44672 594768 6764104 56411824 192441176 270343184 207082312 79072976 93208
cececeec/ccceecee: 3251404800 0 0 0 0 0 128 11968 390880 8669120 144557344 1110685600 1480492576 506335712 261472
ccceecee/cececeec: 3251404800 0 0 0 0 0 128 11968 390880 8669120 144557344 1110685600 1480492576 506335712 261472
ceceecec/ccceecee: 3251404800 0 0 0 0 0 1024 31168 629568 10441728 141278272 1015149760 1483176768 600079744 616768
ccceecee/ceceecec: 3251404800 0 0 0 0 0 1024 31168 629568 10441728 141278272 1015149760 1483176768 600079744 616768
ccceeece/ccceeece: 3251404800 0 0 0 0 0 896 32608 672064 11245568 151416800 1048248352 1473010080 566175872 602560
cceeccee/ccceeece: 1625702400 0 0 0 0 0 128 7232 211584 4516976 73085408 550768736 739417248 257558256 136832
ccceeece/cceeccee: 1625702400 0 0 0 0 0 128 7232 211584 4516976 73085408 550768736 739417248 257558256 136832
cceecece/ccceeece: 3251404800 0 0 0 0 1152 28960 488064 6714400 74089312 491329056 1159739872 1126267360 391384000 1362624
ccceeece/cceecece: 3251404800 0 0 0 0 1152 28960 488064 6714400 74089312 491329056 1159739872 1126267360 391384000 1362624
cceeecce/ccceeece: 3251404800 0 0 0 0 0 384 18560 508576 10283520 153066560 1064207840 1471739200 551192480 387680
ccceeece/cceeecce: 3251404800 0 0 0 0 0 384 18560 508576 10283520 153066560 1064207840 1471739200 551192480 387680
cececece/ccceeece: 812851200 0 0 0 0 0 640 20128 397840 6212744 72149288 329850768 333869016 70341960 8816
ccceeece/cececece: 812851200 0 0 0 0 0 640 20128 397840 6212744 72149288 329850768 333869016 70341960 8816
cececeec/ccceeece: 3251404800 0 0 0 0 0 768 28096 628480 11063232 153656672 1066654464 1471064800 547956608 351680
ccceeece/cececeec: 3251404800 0 0 0 0 0 768 28096 628480 11063232 153656672 1066654464 1471064800 547956608 351680
ceceecec/ccceeece: 3251404800 0 0 0 0 0 832 28448 617536 10756800 146770784 1015659936 1477030656 599257216 1282592
ccceeece/ceceecec: 3251404800 0 0 0 0 0 832 28448 617536 10756800 146770784 1015659936 1477030656 599257216 1282592
cceeccee/cceeccee: 812851200 0 0 16 400 6128 77600 892224 9323408 72924832 235458048 309914592 161565952 22687808 192
cceecece/cceeccee: 1625702400 0 0 0 0 0 0 960 80064 2767104 60680448 558566000 752050384 251517136 40304
cceeccee/cceecece: 1625702400 0 0 0 0 0 0 960 80064 2767104 60680448 558566000 752050384 251517136 40304
cceeecce/cceeccee: 1625702400 0 0 0 0 0 0 2688 102016 2412256 43421536 427908128 768607008 382528128 720640
cceeccee/cceeecce: 1625702400 0 0 0 0 0 0 2688 102016 2412256 43421536 427908128 768607008 382528128 720640
cececece/cceeccee: 406425600 0 0 0 0 0 0 448 33504 949552 18120096 143842176 185056816 58420624 2384
cceeccee/cececece: 406425600 0 0 0 0 0 0 448 33504 949552 18120096 143842176 185056816 58420624 2384
cececeec/cceeccee: 1625702400 0 0 0 0 0 0 960 80064 2767104 60680448 558566000 752050384 251517136 40304
cceeccee/cececeec: 1625702400 0 0 0 0 0 0 960 80064 2767104 60680448 558566000 752050384 251517136 40304
ceceecec/cceeccee: 1625702400 0 0 0 64 1792 35648 558144 7103424 68560864 317064416 498386080 486692960 245344320 1954688
cceeccee/ceceecec: 1625702400 0 0 0 64 1792 35648 558144 7103424 68560864 317064416 498386080 486692960 245344320 1954688
cceecece/cceecece: 3251404800 0 0 0 0 0 0 5120 215232 5712224 108580512 985470272 1516513184 634514784 393472
cceeecce/cceecece: 3251404800 0 0 0 0 0 0 2240 123328 3666880 76967904 857571840 1546843840 764461440 1767328
cceecece/cceeecce: 3251404800 0 0 0 0 0 0 2240 123328 3666880 76967904 857571840 1546843840 764461440 1767328
cececece/cceecece: 812851200 0 0 0 0 0 0 832 42752 1263104 26485216 254017248 379776336 151144416 121296
cceecece/cececece: 812851200 0 0 0 0 0 0 832 42752 1263104 26485216 254017248 379776336 151144416 121296
cececeec/cceecece: 3251404800 0 0 64 1536 24960 339200 4072800 42624288 308894304 774391520 877145504 806124000 435564768 2221856
cceecece/cececeec: 3251404800 0 0 64 1536 24960 339200 4072800 42624288 308894304 774391520 877145504 806124000 435564768 2221856
ceceecec/cceecece: 3251404800 0 0 0 0 0 0 2688 115744 3169600 66108960 777815520 1555725344 844714592 3752352
cceecece/ceceecec: 3251404800 0 0 0 0 0 0 2688 115744 3169600 66108960 777815520 1555725344 844714592 3752352
cceeecce/cceeecce: 3251404800 0 0 0 128 3456 56576 755584 8992128 91836160 577502464 1283492352 1039074560 249614848 76544
cececece/cceeecce: 812851200 0 0 0 128 2848 44672 590864 6676616 55370000 184885264 239377672 214229168 111084216 589752
cceeecce/cececece: 812851200 0 0 0 128 2848 44672 590864 6676616 55370000 184885264 239377672 214229168 111084216 589752
cececeec/cceeecce: 3251404800 0 0 0 0 0 0 2240 123328 3666880 76967904 857571840 1546843840 764461440 1767328
cceeecce/cececeec: 3251404800 0 0 0 0 0 0 2240 123328 3666880 76967904 857571840 1546843840 764461440 1767328
ceceecec/cceeecce: 3251404800 0 0 0 0 0 0 6144 276032 6992192 117638656 926066048 1505936448 692638016 1851264
cceeecce/ceceecec: 3251404800 0 0 0 0 0 0 6144 276032 6992192 117638656 926066048 1505936448 692638016 1851264
cececece/cececece: 203212800 1 32 433 4842 50424 496354 4356750 25756434 46028208 25614194 19053576 49529984 32117008 204560
cececeec/cececece: 812851200 0 0 0 0 0 0 832 42752 1263104 26485216 254017248 379776336 151144416 121296
cececece/cececeec: 812851200 0 0 0 0 0 0 832 42752 1263104 26485216 254017248 379776336 151144416 121296
ceceecec/cececece: 812851200 0 0 0 0 0 0 0 10240 490048 13576720 188391696 392390576 217543856 448064
cececece/ceceecec: 812851200 0 0 0 0 0 0 0 10240 490048 13576720 188391696 392390576 217543856 448064
cececeec/cececeec: 3251404800 0 0 0 0 0 0 5120 215232 5712224 108580512 985470272 1516513184 634514784 393472
ceceecec/cececeec: 3251404800 0 0 0 0 0 0 2688 115744 3169600 66108960 777815520 1555725344 844714592 3752352
cececeec/ceceecec: 3251404800 0 0 0 0 0 0 2688 115744 3169600 66108960 777815520 1555725344 844714592 3752352
ceceecec/ceceecec: 3251404800 0 32 576 7520 87200 937024 9483936 84080704 466269504 752979648 633919808 775750528 515941376 11946944
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> the extra factor of 2 (real

Submitted by Mike G on Tue, 02/07/2006 - 15:46.
> the extra factor of 2 (really 170/90)

It could be reduced a little further, by a factor of 170/65 in total, if mirrored shapes are regarded as equivalent.

Do you have some examples of square-square positions that require 13 flips to solve? E.g., in the notation used as input to Jaap's solver, or as a sequence of turns that could be used to generate the position from the solved state.

A curiosity: the non-square-square position 8B4GH3F57C6ED21A/ (which requires the maximum 13 flips) needs 30 turns (U + D + flips) when you make the middle slice square instead of kite-shaped. That might just be a fluke, but perhaps minor variations on other 13-flip positions would be good candidates for "deep" positions in the turn metric.
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> perhaps minor variations on

Submitted by jaap on Wed, 02/08/2006 - 06:21.
> perhaps minor variations on other 13-flip positions would be good candidates for "deep" positions in the turn metric.

That is a good point. The square-square shape is one of the most symmetric shapes, and therefore also in a sense one of the hardest to get to. Also, other symmetric shapes tend to have easy parity fixes.

Just as symmetric patterns on the cube are good candidates for being far from start, so are these 13-flip square-square patterns on the square-1.

Jaap's Puzzle Page:
http://www.geocities.com/jaapsch/puzzles/
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Of course, 7 flips is best for parity fix for Square One

Submitted by masonjones on Wed, 12/21/2005 - 14:28.
Jaap pointed me to an odd parity fix solution with 7 flips on his site that I had previously overlooked. Alas, it is so obvious, too, once you see it done. Since he has already put all his favorite positions and moves through the optimal solver, my calculation can't improve on those results. But I can however, report the counting results for the God's Algorithm calculation I mentioned in my previous post... See next post.
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Great to see this follow-up t

Submitted by Mike G on Thu, 12/08/2005 - 06:04.
Great to see this follow-up to your original Cube Lovers post (28 May 96) where you mentioned the reduction that was possible via cyclic permutation of the side colours of the U and D layers. I felt sorely tempted to try the calculation after reading that, so will be very interested to see some more details when they're available!

Mike G.
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