## X2Y2-BELTS

Advanced methods and approaches for solving Sudoku puzzles
As a better example of an x2y2-belt of crosses with spine of length 6 on blocks 1 3 6 5 8 7 (and with each of the 6 crosses centered in the physical center of its block):

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`     I----------------------I      I                      I      I                      I      I          I-----------I      I          I                    I          I                    I----------I                     `

consider the following variant of EM:

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`100030002090400050006000700040200100500090030000007006700600000030050090002001000`

After elementary propagation of constraints, we get a belt of 6 crosses with the above spine (only the contents of relevant cells are displayed):
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`1_______578_____xxx_____ | xxx_____3_______xxx_____ | xxx_____1648____2_______238_____9_______378_____ | 4_______xxx_____3_______ | 368_____5_______138_____ xxx_____258_____6_______ | xxx_____xxx_____xxx_____ | 7_______1648____1_______ _________________________|__________________________|_________________________  xxx_____4_______xxx_____ | 2_______68______xxx_____ | 1_______78______xxx_____ 5_______xxx_____xxx_____ | 18______9_______468_____ | 248_____3_______478_____ xxx_____xxx_____xxx_____ | xxx_____148_____7_______ | xxx_____248_____6_______ _________________________|__________________________|_________________________  7_______158_____xxx_____ | 6_______248_____xxx_____ | xxx_____xxx_____xxx_____ 468_____5_______148_____ | 78______5_______248_____ | xxx_____9_______xxx_____ xxx_____568_____2_______ | xxx_____478_____1_______ | xxx_____xxx_____xxx______`

Inner candidates:
- in blocks 1 6 8 : 2 and 7
- in blocks 3 5 7 : 1 and 6

Outer candidates:
- horizontal:
in blocks 1 and 3 : 3 and 8
in blocks 6 and 5: 4 and 8
in blocks 8 and 7 : 4 and 8
- vertical:
in blocks 3 and 6 : 4 and 8
in blocks 5 and 8 : 4 and 8
in blocks 7 and 1 : 5 and 8

Rule x2y2-belt can be applied immediately. It leads to 22 eliminations.

But, as I explained in a previous post, x2y2-belts should be classified after quads, and (for 3D symmetry reasons) after the level I called L4_0.
Does this x2y2-belt survive quads? Contrary to my previous example (which didn't survive singles), it does. Here are the first stepts of the resolution path in L4_0 + x2y2-belts:
(my output function for belts is still rudimentary)
***** SudoRules version 13 *****
hidden-pairs-in-a-block {n3 n9}{r7c6 r9c4} ==> r9c4 <> 8, r7c6 <> 8, r7c6 <> 4
hidden-pairs-in-a-block {n5 n9}{r4c9 r6c7} ==> r6c7 <> 8, r6c7 <> 4, r6c7 <> 2, r4c9 <> 8, r4c9 <> 7
hidden-pairs-in-a-block {n3 n5}{r4c4 r6c6} ==> r6c6 <> 8, r6c6 <> 4, r4c4 <> 8
x2y2-belt24 in blocks 1 3 6 5 8 7 ==> r6c2 <> 8, r5c2 <> 8, r3c1 <> 2, r2c6 <> 8, r2c5 <> 8, r1c3 <> 7, r9c8 <> 8, r9c8 <> 4, r7c8 <> 8, r7c8 <> 4, r3c9 <> 1, r1c7 <> 6, r5c3 <> 8, r3c5 <> 8, r9c1 <> 6, r8c9 <> 8, r8c9 <> 4, r8c7 <> 8, r8c7 <> 4
naked-quads-in-a-block {n1 n2 n6 n7}{r7c8 r8c7 r9c8 r8c9} ==> r9c9 <> 7, r9c7 <> 6
swordfish-in-columns n6{r8 r4 r2}{c1 c5 c7} ==> r2c6 <> 6

This is interesting because this belt doesn't degenerate into quads. On the contrary all the eliminations it entails are needed before a quad appears.

The only problem with this example is that this puzzle has no solution (as was the case for my previous example).

The more I try to find an x2y2-belt with spine of length 6, the more I get convinced that it is impossible, but I have no proof.
Such impossibility would mean that Steve's pattern is a very rare one (unless we dilute it in general chains of subsets - which would amount to negating its beauty) and that it is intimately related to the very particular symmetries of the EM core.
(There may be millions of puzzles based on this core, but what's a million among billions of billions of minimal puzzles?)

Coloin, as you have worked on the EM core pattern, I've a question for you about the diagonals on which it is built. Is there any reason for choosing the first or second diagonal in each block? I mean: does it have any impact on the presence of Steve's pattern? (I couldn't find any reasoon why it would).
denis_berthier
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