The New Sudoku Players' Forum

by **shye** » Wed Mar 18, 2026 8:39 am

Code: Select all: +-------+-------+-------+ | 9 8 . | 7 . 6 | 5 4 . | | 7 . . | . . . | . . 8 | | . . . | . 8 . | 3 . 7 | +-------+-------+-------+ | 4 . . | . . 7 | . . 5 | | . . . | . 9 . | 4 . . | | 6 . . | 5 . . | . . 9 | +-------+-------+-------+ | 5 . 6 | . . . | . . . | | 8 2 . | . . . | . . 4 | | . 4 7 | 9 . 8 | . 5 6 | +-------+-------+-------+ 98.7.654.7.......8....8.3.74....7..5....9.4..6..5....95.6......82......4.479.8.56

estimated rating: 8.5
hi again everyone!

by **jco** » Thu Mar 19, 2026 9:07 pm

After a NT,

Code: Select all: ,-----------------------------------------------------------------------------, | 9 8 123 | 7 123 6 | 5 4 12 | | 7 1356 12345 | 1234 12345 123459 | 1269 1269 8 | | 12 156 1245 | 124 8 12459 | 3 1269 7 | |-------------------------+-------------------------+-------------------------| | 4 139 12389 | 12368 1236 7 | 1268 12368 5 | | 123 57 58 | 68 9 123 | 4 678 123 | | 6 137 1238 | 5 1234 1234 | 1278 12378 9 | |-------------------------+-------------------------+-------------------------| | 5 139 6 | 1234 12347 1234 | 12789 123789 123 | | 8 2 139 | 136 13567 135 | 179 1379 4 | | 13 4 7 | 9 123 8 | 12 5 6 | '-----------------------------------------------------------------------------'

Step 1: Let (A)r1c3, (B)r1c5, (C)r1c9

Code: Select all: ,-----------------------------------------------------------------------------, | 9 8 *A | 7 *B 6 | 5 4 *C | | 7 1356 12345 | 1234 AC45 123459 | 1269 1269 8 | | BC 156 1245 | 124 8 12459 | 3 1269 7 | |-------------------------+-------------------------+-------------------------| | 4 139 12389 | 12368 AC6 7 | 1268 12368 5 | | ABC 57 58 | 68 9 ABC | 4 678 AB | | 6 137 1238 | 5 AC4 1234 | 1278 12378 9 | |-------------------------+-------------------------+-------------------------| | 5 139 6 | 1234 AC47 1234 | 12789 123789 AB | | 8 2 139 | 136 13567 135 | 179 1379 4 | | ABC 4 7 | 9 AC 8 | ABC 5 6 | '-----------------------------------------------------------------------------'

1.1 We cannot have (A)r5c9 because that implies an empty cell (r9c7) [see below]

Code: Select all: ,-----------------------------------------------------------------------------, | 9 8 *A | 7 *B 6 | 5 4 *C | | 7 1356 12345 | 1234 AC45 123459 | 1269 1269 8 | | BC 156 1245 | 124 8 12459 | 3 1269 7 | |-------------------------+-------------------------+-------------------------| | 4 139 12389 | 12368 AC6 7 | 1268 12368 5 | | BC 57 58 | 68 9 123 | 4 678 **A | | 6 137 1238 | 5 AC4 1234 | 1278 12378 9 | |-------------------------+-------------------------+-------------------------| | 5 139 6 | 1234 AC47 1234 | 12789 123789 *B | | 8 2 139 | 136 13567 135 | 179 1379 4 | |**A 4 7 | 9 *C 8 | ABC!!! 5 6 | '-----------------------------------------------------------------------------'

So, (B)r5c9:

Code: Select all: ,-----------------------------------------------------------------------------, | 9 8 *A | 7 *B 6 | 5 4 *C | | 7 1356 12345 | 1234 AC45 123459 | 1269 1269 8 | |*BC 156 1245 | 124 8 12459 | 3 1269 7 | |-------------------------+-------------------------+-------------------------| | 4 139 12389 | 12368 6-AC 7 | 1268 12368 5 | |*AC 57 58 | 68 9 *AC | 4 678 *B | | 6 137 1238 | 5 4-AC 1234 | 1278 12378 9 | |-------------------------+-------------------------+-------------------------| | 5 139 6 | 1234 C47 1234 | 12789 123789 *A | | 8 2 139 | 136 13567 135 | 179 1379 4 | |*ABC 4 7 | 9 *AC 8 |*BC 5 6 | '-----------------------------------------------------------------------------'

Step 2: Either (AC) r5c6 = r5c1 = r9c1 = r9c5> -AC r46c5, or

(B-A)r9c1 = (A)r9c5,r5c9 & (C)r5c6 => -(AC)r46c5

=> +4 r6c5, +6 r4c5 [13 placements and basics]

Step 3:

Code: Select all: ,--------------------------------------------------------------------, | 9 8 123 | 7 123 6 | 5 4 12 | | 7 5 1234 | 1234 123 9 | 6 12 8 | | 12 6 124 | 124 8 5 | 3 9 7 | |----------------------+----------------------+----------------------| | 4 139 1289 | 13 6 7 | 128 1238 5 | |*123 7 5 | 8 9 123 | 4 6 *123 | | 6 13 128 | 5 4 123 | 1278 12378 9 | |----------------------+----------------------+----------------------| | 5 19-3 6 | 123 7 4 | 1289 1238 *123 | | 8 2 19 | 6 5 13 | 179 137 4 | |*13 4 7 | 9 123 8 | 12 5 6 | '--------------------------------------------------------------------'

SS (3)r9c1 = r5c1 - r5c9 = r7c9 => -3 r7c2 [1 placement and basics]

Step 4:

Code: Select all: ,--------------------------------------------------------------------, | 9 8 123 | 7 123 6 | 5 4 12 | | 7 5 1234 | 124 123 9 | 6 12 8 | | 12 6 124 | 124 8 5 | 3 9 7 | |----------------------+----------------------+----------------------| | 4 139 1289 | 13 6 7 | 128 128-3 5 | | 12 7 5 | 8 9 *123 | 4 6 *123 | | 6 13 128 | 5 4 123 | 1278 1278-3 9 | |----------------------+----------------------+----------------------| | 5 19 6 | 123 7 4 | 1289 1238 12-3 | | 8 2 19 | 6 5 *13 | 179 *137 4 | | 3 4 7 | 9 12 8 | 12 5 6 | '--------------------------------------------------------------------'

SS (3) r8c8 = r8c5 - r5c6 = r5c9 => -3 r7c9, r46c8 [1 placement & basics]

Step 5:

Code: Select all: ,-----------------------------------------------------------, | 9 8 123 | 7 3-12 6 | 5 4 *12 | | 7 5 1234 | 124 123 9 | 6 12 8 | | 12 6 124 | 124 8 5 | 3 9 7 | |-------------------+-------------------+-------------------| | 4 139 1289 | 13 6 7 | 128 128 5 | | 12 7 5 | 8 9 12 | 4 6 3 | | 6 13 128 | 5 4 123 | 1278 1278 9 | |-------------------+-------------------+-------------------| | 5 19 6 | 123 7 4 | 89 38 *12 | | 8 2 19 | 6 5 13 | 79 37 4 | | 3 4 7 | 9 *12 8 |*12 5 6 | '-----------------------------------------------------------'

Remote Pair: (12) r9c5 = r9c7 - r7c9 = r1c9 => -12 r1c5 [4 placements]

Step 6:

Code: Select all: ,-----------------------------------------------------------, | 9 8 12 | 7 3 6 | 5 4 12 | | 7 5 3 | 4 12 9 | 6 12 8 | |*12 6 4 |*12 8 5 | 3 9 7 | |-------------------+-------------------+-------------------| | 4 139 1289 | 3-1 6 7 | 128 128 5 | |*12 7 5 | 8 9 *12 | 4 6 3 | | 6 13 128 | 5 4 123 | 1278 1278 9 | |-------------------+-------------------+-------------------| | 5 19 6 | 123 7 4 | 89 38 12 | | 8 2 19 | 6 5 13 | 79 37 4 | | 3 4 7 | 9 12 8 | 12 5 6 | '-----------------------------------------------------------'

SS (1): r3c4 = r3c1 - r5c1 = r5c6 => -1 r4c4; ste

@shye: Nice to see you back !

by **Cenoman** » Fri Mar 20, 2026 11:25 am

jco wrote:@shye: Nice to see you back !

Confirmed !
A few days ago, while using your Fireworks pattern, I just thought we had not read you for long !

FWIW, my solution:

Code: Select all: +-----------------------+---------------------------+-------------------------+ | 9 8 d123 | 7 c123 6 | 5 4 12 | | 7 1356 12345 | 1234 12345 123459 | 1269 1269 8 | | 12 156 1245 | 124 8 12459 | 3 1269 7 | +-----------------------+---------------------------+-------------------------+ | 4 139 12389 | 12368 1236 7 | 1268 12368 5 | | b123 57 58 | 68 9 123 | 4 678 c123 | | 6 137 1238 | 5 1234 1234 | 1278 12378 9 | +-----------------------+---------------------------+-------------------------+ | 5 19-3 6 | 1234 12347 1234 | 12789 123789 d123 | | 8 2 19-3 | 136 13567 135 | 179 1379 4 | | a13 4 7 | 9 b123 8 | 12 5 6 | +-----------------------+---------------------------+-------------------------+

(3)r9c1 = (r9c5&r5c1) - (r1c5|r5c9) = (r1c3&r7c9) => -3 r8c3, r7c2

Code: Select all: +----------------------+---------------------------+-------------------------+ | 9 8 123* | 7 123^ 6 | 5 4 12*^ | | 7 1356 12345 | 1234 12345 123459 | 1269 1269 8 | | 12* 156 1245 | 124 8 12459 | 3 1269 7 | +----------------------+---------------------------+-------------------------+ | 4 139 12389 | 12368 1236 7 | 1268 12368 5 | | 12* 57 58 | 68 9 123 | 4 678 123* | | 6 137 1238 | 5 1234 1234 | 1278 12378 9 | +----------------------+---------------------------+-------------------------+ | 5 19 6 | 1234 12347 1234 | 12789 123789 123^ | | 8 2 19 | 136 13567 135 | 179 1379 4 | | 3 4 7 | 9 12^ 8 | 12^ 5 6 | +----------------------+---------------------------+-------------------------+

Bivalue oddagon (12)r15, c19, b1 (*) having two guardians => 3r1c3 = 3r5c9
Bivalue oddagon (12)r19, c59, b9 (^) having two guardians => 3r1c5 = 3r7c9
None can have its two guardians True (otherwise other oddagon: contradiction)
=> +3 r1c3&r7c9 OR +3 r1c5&r5c9 = True

(3)r1c3&r7c9 creating NPs (12)r5c19 And (12)r19c5 leads to a contradiction (with basics):

Code: Select all: +--------------------+---------------------------+-----------------------+ | 9 8 3 | 7 12 6 | 5 4 12 | | 7 16+5 1245 | 3-124 3-1245 123459 | 1269 1269 8 | | 12 15+6 1245 | 124 8 12459 | 3 129-6 7 | +--------------------+---------------------------+-----------------------+ | 4 139 1289 | 12-368 6-123 7 | 1268 12368 5 | | 12 7-5 5-8 | 8-6 9 3-12 | 4 6-78 12 | | 6 137 128 | 5 4-123 12-34 | 1278 12378 9 | +--------------------+---------------------------+-----------------------+ | 5 19 6 | 124 1247 124 | 12789 12789 3 | | 8 2 19 | 136 13567 135 | 179 179 4 | | 3 4 7 | 9 12 8 | 12 5 6 | +--------------------+---------------------------+-----------------------+

NP(12)r5c19 And NP(12)r19c5 => +3 r5c6; +4 r6c5, +6 r4c5, +12 b5p19; +8r5c4, +5r5c3, +7r5c2; +6 r3c2, +5r2c2 => +3 r2c5
NT(124)r346c4 => +3 r2c4 : Contradiction => (3)r1c3&r7c9 = False

Therefore +3 r1c5 And +3 r5c9

Code: Select all: +----------------------+-------------------------+------------------------+ | 9 8 b12 | 7 3 6 | 5 4 a12 | | 7 1356 12345 | 124 1245 12459 | 1269 1269 8 | | 12 156 1245 | 124 8 12459 | 3 1269 7 | +----------------------+-------------------------+------------------------+ | 4 139 12389 | 12368 126 7 | 1268 1268 5 | | 12 57 58 | 68 9 12 | 4 678 3 | | 6 137 1238 | 5 124 1234 | 1278 1278 9 | +----------------------+-------------------------+------------------------+ | 5 d19 6 | 1234 1247 1234 | 12789 123789 2-1 | | 8 2 c19 | 136 1567 135 | 179 1379 4 | | 3 4 7 | 9 12 8 | 12 5 6 | +----------------------+-------------------------+------------------------+

End with X-chain (1)r1c9 = r1c3 - r8c3 = r7c2 => -1 r7c9; ste

by **shye** » Fri Mar 20, 2026 9:21 pm

thank you for your lovely solutions! (and greetings

)

my own path

Code: Select all: +----------+----------+----------+ | . . # | . A# . | . . # | | . . . | . . . | . . . | | # . . | . . . | . . . | +----------+----------+----------+ | . . . | . . . | . . . | | B# . . | . . . | . . C# | | . . . | . . . | . . . | +----------+----------+----------+ | . . . | . . . | . . # | | . . . | . . . | . . . | | # . . | . D# . | # . . | +----------+----------+----------+

consider the # cells
A and B cannot be the same digit (placements in r9 & c9, clash in b9)
C and D cannot be the same digit (placements in r1 & c1, clash in b1)
ABCD cannot all be different because these cells are trivalue
if A and C are different, B and D are the same (call it E), and r1c9 must be E. the remaining structure becomes a bivalue oddagon on AC, contradiction
if B and D are different, A and C are the same (call it E), and r9c1 must be E. the remaining structure becomes a bivalue oddagon on BD, contradiction
so AC are the same, and BD are the same

Code: Select all: +----------+----------+----------+ | . . # | . A# . | . . # | | . . . | . . . | . . . | | # . . | . . . | . . . | +----------+----------+----------+ | . . . | ab . 7 | . . . | | B# . . | . 9 . | . . A# | | . . . | 5 . ab | . . . | +----------+----------+----------+ | . . . | . . . | . . # | | . . . | . . . | . . . | | # . . | . B# . | # . . | +----------+----------+----------+

label similar digits AB
AB hidden pair in b5
singles

Code: Select all: +----------------+----------------+------------------+ | 9 8 b123 | 7 A123 6 | 5 4 c12 | | 7 5 1234 | 1234 123 9 | 6 12 8 | | 12 6 124 | 124 8 5 | 3 9 7 | +----------------+----------------+------------------+ | 4 139 1289 |a13 6 7 | 128 1238 5 | |B123 7 5 | 8 9 c123 | 4 6 A123 | | 6 13 128 | 5 4 b123 | 1278 12378 9 | +----------------+----------------+------------------+ | 5 139 6 |c123 7 4 | 1289 1238 b123 | | 8 b2 19 | 6 5 a13 | 179 137 4 | | 13 4 7 | 9 B123 8 | 12 5 6 | +----------------+----------------+------------------+

ABC can be placed as singles as shown above
B in b7 is placed in p5, making B = 2
stte

Cenoman wrote:
jco wrote:@shye: Nice to see you back !
Confirmed !
A few days ago, while using your Fireworks pattern, I just thought we had not read you for long !

thats a lovely solution for that puzzle too!

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