katherine_china wrote:- Code: Select all
`.--.--.--.--.--.--.--.--.--.`

|9 |0 |0 |0 |0 |0 |0 |0 |5 |

:--+--+--+--+--+--+--+--+--:

|0 |4 |0 |3 |0 |0 |0 |2 |0 |

:--+--+--+--+--+--+--+--+--:

|0 |0 |8 |0 |0 |0 |1 |0 |0 |

:--+--+--+--+--+--+--+--+--:

|0 |7 |0 |6 |0 |3 |0 |0 |0 |

:--+--+--+--+--+--+--+--+--:

|0 |0 |0 |0 |8 |0 |0 |0 |0 |

:--+--+--+--+--+--+--+--+--:

|0 |0 |0 |7 |0 |9 |0 |6 |0 |

:--+--+--+--+--+--+--+--+--:

|0 |0 |1 |0 |0 |0 |9 |0 |0 |

:--+--+--+--+--+--+--+--+--:

|0 |3 |0 |0 |0 |6 |0 |4 |0 |

:--+--+--+--+--+--+--+--+--:

|5 |0 |0 |0 |0 |0 |0 |0 |8 |

'--'--'--'--'--'--'--'--'--'

This is a very interesting puzzle.

As indicated by Ronk, it is in gsf's file, below EasterMonster (its SER is 10.8 while EM's is 11.4). Here's the full gsf line:

95106,0,1282,900000005040300020008000100070603000000080000000709060001000900030006040500000008,tarek-0014,2.00s,0,C21.m/S4.da/F20901.36324/N90627.203399/P5.12.46219.5.6.138447.24.3.2.236/M2.1.590490/V2,C21.m/S4.da/F10.34/N6.23/B3.9.7.2/H2.4.2/X2.5/Y2.20/G4.0.1/M1.9.5

Here's also the output from Sudoku Explainer (thanks Mike for your post in this thread

http://forum.enjoysudoku.com/viewtopic.php?t=5995&start=0 on the various solvers available and how to use them).

- Code: Select all
`54 Hidden Single`

2 Direct Hidden Pair

3 Naked Single

6 Pointing

2 Claiming

3 Naked Pair

2 Hidden Pair

1 Swordfish

1 XYZ-Wing

2 Turbot Fish

2 Bidirectional Y-Cycle

6 Forcing Chain

2 Nishio Forcing Chains

7 Region Forcing Chains

3 Cell Forcing Chains

3 Dynamic Cell Forcing Chains

22 Dynamic Contradiction Forcing Chains

8 Dynamic Region Forcing Chains

1 Dynamic Cell Forcing Chains (+)

3 Dynamic Contradiction Forcing Chains (+)

15 Dynamic Contradiction Forcing Chains (+ Forcing Chains)

Hardest technique: Dynamic Contradiction Forcing Chains (+ Forcing Chains)

Difficulty: 10.8

"Dynamic forcing chains" are restricted forms of T&E.

This puzzle cannot be solved within any of the NRCZT theories, even after Steve's eliminations.

What's interesting is why it can't be solved within these theories: because there are "not enough" nrczt partial chains. The longest such chains have length 7.

Here's the dilemma: if a puzzle has too many chains, it is hard to find the useful ones among the useless ones, but if it has not enough this makes it less likely that some of them will be useful.

This puzzle also illustrates my notion of a strong T-backdoor (defined near the end of this page:

http://forum.enjoysudoku.com/viewtopic.php?t=5600&start=75):

r2c7=8 is a strong NRCZT10+X2Y2belts4 backdoor for tarek-0014Remember that X2Y2-belts are my interpretation of Steve's pattern in EasterMonster (see the exact definition here:

http://forum.enjoysudoku.com/viewtopic.php?t=5894)

In the present case,

- there is a conjugacy pair r2n8 = c6 or c7;

- it can be proven within NRCZT10+X2Y2belts4 that r2n8 = c6 is impossible (this is the unique T&E step);

- then it can be proven within NRCZT6 (a subtheory of NRCZT10+X2Y2belts) that the only remaining possibility r2n8=c7 leads to the solution.

***** SudoRules version 13 *****

9.......5

.4.3...2.

..8...1..

.7.6.3...

....8....

...7.9.6.

..1...9..

.3...6.4.

5.......

;;; The first move (Steve's pattern) eliminates 15 candidates:

x2y2-belt4 in blocks b1, b3, b9, b7, with rows r1 and r7, columns c1 and c7 ==> r6c2 <> 2, r5c2 <> 6, r5c2 <> 2, r2c6 <> 7, r2c5 <> 7, r2c5 <> 6, r5c8 <> 7, r5c8 <> 3, r3c9 <> 9, r1c7 <> 8, r9c3 <> 9, r8c5 <> 7, r8c5 <> 2, r8c4 <> 2, r7c1 <> 8

;;; As no rule can be applied, we fall into T&E (which will eliminate hypothesis r2n8=c6 using only rules in NRCZT10)

GENERATING CONTEXT cont-1, SON OF cont-0 FROM HYPOTHESIS r2n8=c6 (AMONG THE TWO HIDDEN HYPOTHESES: COLUMNS c6 AND c7.)

hidden-single-in-a-row ==> r1c8 = 8

nrczt2-chain n5{r2c3 r2c5} - n5{r4c5 r5c4} ==> r5c3 <> 5

nrczt2-chain n1{r2c1 r2c5} - n1{r4c5 r5c4} ==> r5c1 <> 1

nrc3-chain n7{r5c9 r5c7} - {n7 n6}r2c7 - n6{r9c7 r7c9} ==> r7c9 <> 7

nrczt6-chain n1{r8c9 r9c8} - n1{r9c6 r1c6} - n1{r1c4 r8c4} - n8{r8c4 r7c4} - n8{r7c2 r6c2} - n1{r6c2 r5c2} ==> r5c9 <> 1

nrczt6-chain n5{r8c7 r7c8} - n5{r7c6 r3c6} - n5{r3c4 r8c4} - n8{r8c4 r7c4} - n8{r7c2 r6c2} - n5{r6c2 r5c2} ==> r5c7 <> 5

nrczt10-chain n8{r4c7 r4c1} - n8{r8c1 r8c4} - n5{r8c4 r8c5} - n9{r8c5 r8c3} - n9{r9c2 r5c2} - {n9 n1}r5c8 - n1{r4c9 r4c5} - {n1 n9}r2c5 - n9{r3c5 r3c8} - {n9 n5}r4c8 ==> r4c7 <> 5

nrczt3-chain n5{r8c7 r7c8} - n5{r4c8 r4c3} - n5{r2c3 r2c5} ==> r8c5 <> 5

nrczt5-chain n9{r2c9 r2c5} - {n9 n1}r8c5 - n1{r8c9 r9c8} - {n1 n5}r5c8 - {n5 n9}r4c8 ==> r5c9 <> 9

nrczt5-chain n9{r2c9 r2c5} - {n9 n1}r8c5 - n1{r2c5 r2c1} - n1{r4c1 r4c8} - n1{r6c9 r4c9} ==> r4c9 <> 9

hidden-single-in-a-column ==> r2c9 = 9

nrc3-chain n9{r3c4 r3c5} - {n9 n1}r8c5 - {n1 n5}r2c5 ==> r3c4 <> 5

nrczt3-chain n6{r9c3 r9c7} - n6{r2c7 r2c3} - n6{r1c2 r7c2} ==> r7c1 <> 6

nrczt3-chain n3{r1c3 r3c1} - {n3 n7}r3c8 - n7{r2c7 r2c1} ==> r1c3 <> 7

nrct4-chain n6{r7c9 r9c7} - {n6 n7}r2c7 - {n7 n3}r3c8 - n3{r7c8 r7c9} ==> r7c9 <> 2

nrczt6-chain {n5 n1}r2c5 - n1{r1c4 r1c2} - {n1 n8}r6c2 - n8{r7c2 r7c4} - n5{r7c4 r8c4} - n5{r8c7 r6c7} ==> r6c5 <> 5

nrczt6-chain n5{r7c4 r5c4} - n5{r5c8 r4c8} - n9{r4c8 r5c8} - {n9 n1}r5c2 - n1{r6c1 r2c1} - {n1 n5}r2c5 ==> r7c5 <> 5

nrczt6-lr-lasso n5{r5c6 r4c5} - {n5 n1}r2c5 - n1{r1c4 r1c2} - {n1 n9}r5c2 - n9{r4c3 r8c3} - {n9 n1}r8c5 ==> r5c8 <> 5

nrc4-chain {n9 n1}r8c5 - n1{r8c9 r9c8} - {n1 n9}r5c8 - n9{r4c8 r4c3} ==> r8c3 <> 9

hidden-single-in-a-block ==> r9c2 = 9

nrczt2-chain n6{r9c3 r9c7} - n6{r2c7 r2c1} ==> r1c3 <> 6

nrczt2-chain n6{r1c2 r7c2} - n6{r7c9 r3c9} ==> r3c1 <> 6

swordfish-in-columns n6{r7 r1 r3}{c2 c5 c9} ==> r1c7 <> 6

nrc3-chain n9{r4c3 r5c3} - {n9 n1}r5c8 - {n1 n5}r5c2 ==> r4c3 <> 5

nrc3-chain n4{r7c1 r9c3} - n6{r9c3 r7c2} - n8{r7c2 r7c4} ==> r7c4 <> 4

nrct3-chain {n7 n2}r8c3 - n2{r8c9 r9c7} - n6{r9c7 r9c3} ==> r9c3 <> 7

nrc3-chain n7{r8c3 r2c3} - n5{r2c3 r6c3} - n5{r6c7 r8c7} ==> r8c7 <> 7

nrc4-chain n7{r8c3 r2c3} - {n7 n6}r2c7 - n6{r9c7 r9c3} - n4{r9c3 r7c1} ==> r7c1 <> 7

block b7 interaction-with-row r8 ==> r8c9 <> 7

nrc4-chain {n1 n5}r5c2 - n5{r3c2 r2c3} - {n5 n1}r2c5 - n1{r2c1 r1c2} ==> r6c2 <> 1

nrczt2-chain n1{r1c2 r5c2} - n1{r5c4 r4c5} ==> r1c5 <> 1

nrczt3-chain n1{r9c8 r8c9} - n1{r6c9 r6c1} - n1{r2c1 r2c5} ==> r9c5 <> 1

nrczt3-chain n1{r2c5 r2c1} - n1{r6c1 r6c9} - n1{r4c8 r4c5} ==> r8c5 <> 1

naked-single ==> r8c5 = 9

hidden-single-in-a-block ==> r3c4 = 9

nrczt3-chain n1{r5c2 r1c2} - n1{r2c1 r2c5} - n1{r4c5 r5c4} ==> r5c8 <> 1

naked-single ==> r5c8 = 9

hidden-single-in-a-row ==> r4c3 = 9

nrct3-chain {n1 n5}r5c2 - n5{r5c4 r4c5} - {n5 n1}r4c8 ==> r4c1 <> 1

xyt4-chain {n4 n2}r7c1 - {n2 n7}r8c3 - {n7 n8}r8c1 - {n8 n4}r4c1 ==> r5c1 <> 4, r6c1 <> 4

nrc4-chain n5{r6c3 r2c3} - {n5 n1}r2c5 - n1{r2c1 r6c1} - {n1 n5}r5c2 ==> r6c2 <> 5

naked and hidden singles ==> r6c2 = 8, r8c1 = 8, r7c4 = 8, r8c3 = 7, r4c7 = 8

row r8 interaction-with-block b9 ==> r9c7 <> 2

naked-pairs-in-a-column {n2 n4}{r4 r7}c1 ==> r3c1 <> 2

naked-pairs-in-a-row {n3 n7}r3{c1 c8} ==> r3c5 <> 7, r3c6 <> 7

block b2 interaction-with-row r1 ==> r1c7 <> 7

naked-pairs-in-a-row {n3 n7}r3{c1 c8} ==> r3c9 <> 3, r3c9 <> 7

hidden-single-in-a-column ==> r5c9 = 7

naked-pairs-in-a-column {n2 n4}{r4 r7}c1 ==> r5c1 <> 2, r6c1 <> 2

hidden-pairs-in-a-column {n6 n7}{r2 r9}c7 ==> r9c7 <> 3

nrc2-chain n5{r4c5 r4c8} - n5{r7c8 r7c6} ==> r5c6 <> 5

nrc2-chain n5{r3c2 r5c2} - n5{r5c4 r4c5} ==> r3c5 <> 5

nrct2-chain n3{r1c3 r1c7} - n3{r6c7 r6c9} ==> r6c3 <> 3

xyz3-chain {n6 n5}r2c3 - {n5 n2}r3c2 - {n2 n6}r7c2 ==> r1c2 <> 6

hidden singles ==> r1c5 = 6, r1c6 = 7

hidden-pairs-in-a-block {n3 n7}{r7c5 r9c5} ==> r7c5 <> 2, r7c5 <> 4, r9c5 <> 2, r9c5 <> 4

nrc2-chain n1{r5c6 r9c6} - n1{r9c8 r4c8} ==> r4c5 <> 1

row r4 interaction-with-block b6 ==> r6c9 <> 1

xy3-chain {n3 n7}r9c5 - {n7 n6}r9c7 - {n6 n3}r7c9 ==> r7c5 <> 3

naked singles ==> r7c5 = 7, r9c5 = 3

nrc3-chain n5{r6c7 r4c8} - {n5 n3}r7c8 - n3{r3c8 r1c7} ==> r6c7 <> 3

x-wing-in-columns n3{r1 r5}{c3 c7} ==> r5c1 <> 3

naked-single ==> r5c1 = 6

hidden-pairs-in-a-block {n5 n6}{r2c3 r3c2} ==> r3c2 <> 2

row r3 interaction-with-block b2 ==> r1c4 <> 2

xyz3-chain {n2 n3}r1c3 - {n3 n4}r5c3 - {n4 n2}r4c1 ==> r6c3 <> 2

nrc3-chain {n4 n5}r6c3 - {n5 n1}r5c2 - n1{r6c1 r6c5} ==> r6c5 <> 4

xy3-chain {n1 n2}r6c5 - {n2 n4}r3c5 - {n4 n1}r1c4 ==> r2c5 <> 1

naked singles ==> r2c5 = 5, r2c3 = 6, r3c2 = 5, r5c2 = 1, r6c1 = 3, r3c1 = 7, r2c1 = 1, r3c8 = 3, r1c7 = 4, r3c9 = 6, r7c9 = 3, r1c4 = 1, r8c4 = 5, r8c7 = 2, r6c7 = 5,> r4c8 = 1, r9c8 = 7, r9c7 = 6, r6c3 = 4, r4c1 = 2

NO SOLUTION IN CONTEXT cont-1. RETRACTING cont-1 AND HYPOTHESIS r2c6=n8.

;;; After one candidate (r2n8=c6) has been eliminated by T&E, the solution can be found within NRCZT6:

BACK IN CONTEXT cont-0

hidden-single-in-a-row ==> r2c7 = 8

hidden-single-in-a-column ==> r4c8 = 8

nrczt2-chain n5{r7c8 r5c8} - n5{r5c6 r6c5} ==> r7c5 <> 5

nrczt2-chain n1{r9c8 r5c8} - n1{r5c6 r6c5} ==> r9c5 <> 1

nrc3-chain n9{r2c5 r2c9} - n9{r3c8 r5c8} - n9{r5c2 r9c2} ==> r9c5 <> 9

nrczt3-chain n6{r9c7 r7c9} - n6{r2c9 r2c1} - n6{r5c1 r5c3} ==> r9c3 <> 6

nrczt3-chain n3{r3c1 r1c3} - {n3 n7}r1c8 - n7{r2c9 r2c3} ==> r3c1 <> 7

nrct6-chain n6{r9c7 r7c9} - n6{r7c1 r9c2} - n9{r9c2 r5c2} - n9{r5c9 r4c9} - {n9 n7}r2c9 - n7{r5c9 r5c7} ==> r9c7 <> 7

nrczt6-chain {n5 n1}r2c6 - n1{r2c1 r1c2} - {n1 n9}r5c2 - n9{r9c2 r9c4} - n1{r9c4 r9c8} - {n1 n5}r5c8 ==> r5c6 <> 5

nrczt3-chain n5{r4c5 r5c4} - n5{r5c8 r7c8} - n5{r8c7 r8c5} ==> r2c5 <> 5

nrczt3-chain n5{r4c5 r5c4} - n5{r5c8 r7c8} - n5{r8c7 r8c5} ==> r3c5 <> 5

nrczt6-chain {n9 n1}r2c5 - {n1 n5}r8c5 - n5{r4c5 r5c4} - n1{r5c4 r5c6} - {n1 n9}r5c2 - n9{r5c8 r3c8} ==> r3c5 <> 9

nrczt6-chain n1{r2c6 r2c1} - n1{r4c1 r4c9} - n1{r6c9 r6c2} - n8{r6c2 r7c2} - n8{r7c4 r8c4} - n1{r8c4 r8c5} ==> r1c5 <> 1

nrczt6-lr-lasso n9{r2c5 r2c9} - n9{r5c9 r5c8} - n5{r5c8 r7c8} - n5{r7c4 r8c4} - n1{r8c4 r8c9} - n1{r4c9 r5c8} ==> r8c5 <> 9

hidden singles ==> r2c5 = 9, r3c8 = 9

nrc4-chain {n1 n5}r8c5 - n5{r8c7 r7c8} - {n5 n1}r5c8 - n1{r9c8 r8c9} ==> r8c4 <> 1

nrczt2-chain n1{r8c9 r8c5} - n1{r6c5 r5c6} ==> r5c9 <> 1

nrczt3-chain n1{r2c1 r2c6} - n1{r1c4 r9c4} - n1{r9c8 r5c8} ==> r5c1 <> 1

nrczt3-chain n1{r5c8 r9c8} - n1{r9c4 r1c4} - n1{r2c6 r5c6} ==> r5c2 <> 1

nrc3-chain n1{r1c2 r6c2} - n8{r6c2 r7c2} - n8{r7c6 r1c6} ==> r1c6 <> 1

nrc3-chain {n1 n5}r5c8 - {n5 n9}r5c2 - n9{r5c9 r4c9} ==> r4c9 <> 1

nrc2-chain n1{r2c6 r2c1} - n1{r4c1 r4c5} ==> r5c6 <> 1

nrc4-chain n5{r8c7 r7c8} - {n5 n1}r5c8 - n1{r6c9 r8c9} - {n1 n5}r8c5 ==> r8c4 <> 5

nrct2-chain n5{r8c7 r8c5} - n5{r6c5 r5c4} ==> r5c7 <> 5

nrct3-chain n5{r5c8 r7c8} - n5{r7c6 r8c5} - n5{r6c5 r5c4} ==> r5c3 <> 5, r5c2 <> 5

naked and hidden singles ==> r5c2 = 9, r8c3 = 9, r8c4 = 8, r1c6 = 8, r7c2 = 8, r6c1 = 8, r9c4 = 9, r4c9 = 9

hidden-pairs-in-a-row {n1 n5}r5{c4 c8} ==> r5c4 <> 4, r5c4 <> 2

nrc2-chain n1{r6c2 r1c2} - n1{r1c4 r5c4} ==> r6c5 <> 1

nrczt2-chain n6{r7c1 r7c9} - n6{r2c9 r2c3} ==> r3c1 <> 6

nrczt2-chain n6{r1c2 r9c2} - n6{r9c7 r1c7} ==> r1c3 <> 6

swordfish-in-columns n6{r9 r3 r1}{c2 c5 c7} ==> r3c9 <> 6

nrc3-chain {n5 n1}r5c4 - n1{r1c4 r1c2} - {n1 n5}r6c2 ==> r6c5 <> 5

naked-pairs-in-a-block {n2 n4}{r5c6 r6c5} ==> r4c5 <> 4, r4c5 <> 2

nrczt3-chain n2{r8c9 r8c1} - {n2 n6}r9c2 - n6{r9c7 r7c9} ==> r7c9 <> 2

xyzt4-chain {n6 n2}r9c2 - {n2 n7}r8c1 - {n7 n1}r2c1 - {n1 n6}r1c2 ==> r3c2 <> 6

hidden-single-in-a-row ==> r3c5 = 6

nrc3-chain {n7 n6}r2c9 - n6{r1c7 r1c2} - n1{r1c2 r2c1} ==> r2c1 <> 7

column c1 interaction-with-block b7 ==> r9c3 <> 7

nrc3-chain n5{r2c6 r2c3} - n7{r2c3 r1c3} - n7{r1c5 r3c6} ==> r3c6 <> 5

nrc3-chain n5{r7c6 r2c6} - n1{r2c6 r1c4} - {n1 n5}r5c4 ==> r7c4 <> 5

nrc3-chain n7{r8c1 r7c1} - n6{r7c1 r7c9} - {n6 n7}r2c9 ==> r8c9 <> 7

nrc4-chain n1{r9c6 r2c6} - n5{r2c6 r2c3} - n7{r2c3 r1c3} - n7{r1c5 r3c6} ==> r9c6 <> 7

nrc3-chain {n3 n7}r1c8 - n7{r9c8 r9c5} - n3{r9c5 r7c5} ==> r7c8 <> 3

nrc4-chain n7{r8c1 r8c7} - n7{r5c7 r5c9} - {n7 n6}r2c9 - n6{r7c9 r7c1} ==> r7c1 <> 7

hidden-single-in-a-block ==> r8c1 = 7

row r8 interaction-with-block b9 ==> r9c7 <> 2

xyzt5-chain {n4 n2}r6c5 - {n2 n7}r1c5 - {n7 n3}r1c8 - {n3 n2}r1c3 - {n2 n4}r9c3 ==> r9c5 <> 4

nrc2-chain n4{r9c3 r9c6} - n4{r5c6 r6c5} ==> r6c3 <> 4

nrc4-chain n4{r9c3 r9c6} - n1{r9c6 r2c6} - {n1 n6}r2c1 - n6{r5c1 r5c3} ==> r5c3 <> 4

nrczt4-chain {n2 n5}r8c7 - {n5 n4}r4c7 - n4{r4c3 r5c1} - {n4 n2}r5c6 ==> r5c7 <> 2

nrct5-chain {n2 n4}r5c6 - n4{r9c6 r9c3} - n4{r7c1 r4c1} - n1{r4c1 r2c1} - n1{r2c6 r9c6} ==> r9c6 <> 2

nrc3-chain n1{r1c4 r2c6} - {n1 n4}r9c6 - {n4 n2}r7c4 ==> r1c4 <> 2

nrczt3-chain n2{r5c6 r6c5} - n2{r1c5 r1c2} - n2{r9c2 r9c3} ==> r5c3 <> 2

nrct5-chain {n5 n2}r3c2 - n2{r3c4 r7c4} - n2{r7c6 r5c6} - n2{r5c1 r4c1} - n1{r4c1 r6c2} ==> r6c2 <> 5

naked and hidden singles ==> r6c2 = 1, r2c1 = 1, r2c6 = 5, r5c4 = 5, r4c5 = 1, r8c5 = 5, r8c7 = 2, r8c9 = 1, r5c8 = 1, r9c6 = 1, r9c3 = 4, r7c8 = 5, r1c4 = 1, r3c2 = 5

row r4 interaction-with-block b4 ==> r6c3 <> 2, r5c1 <> 2

nrc2-chain n4{r1c7 r1c5} - n4{r6c5 r5c6} ==> r5c7 <> 4

nrczt2-chain n4{r1c7 r1c5} - n4{r6c5 r6c9} ==> r4c7 <> 4

naked and hidden singles ==> r4c7 = 5, r4c3 = 2, r4c1 = 4, r6c3 = 5

row r6 interaction-with-block b6 ==> r5c9 <> 3, r5c7 <> 3

naked-single ==> r5c7 = 7

naked-pairs-in-a-row {n3 n7}r1{c3 c8} ==> r1c7 <> 3, r1c5 <> 7

hidden-single-in-a-block ==> r3c6 = 7

naked-pairs-in-a-block {n2 n4}{r7c4 r7c6} ==> r9c5 <> 2

...(naked and hidden singles)...

963128475

147395826

258467193

472613589

396582714

815749362

681234957

739856241

524971638