## OVERLAID patterns -- pairs, trips and quads

Advanced methods and approaches for solving Sudoku puzzles

### OVERLAID patterns -- pairs, trips and quads

I thought I'd take time off of nit-picking others and put some of my own ideas out there.

Scott H use of words like "almost pairs" got be thinking about overlapping and overlaid patterns. Most of this post is looking at previously covered ground by others in several recent threads, in a different way.

The idea is that if an exclusion can be made because of pattern A and and pattern B separately, then if the patterns are combined, the exclusion remains.

When combining patterns, candidate lists are formed from the UNION of the overlaid cells. For example, r1c1=[12] in pattern A is combined with r1c1=[23] in pattern B, giving r1c1=[123] in the overlaid pattern.

Resulting exlcusions lists are formed from the INTERSECTION of the overlaid cells. For example r1c1=[<>123] in pattern A combined with r1c1=[<>345] in pattern B gives r1c1=[<>3] in the overlaid pattern.
Below are some specifics.

Notation:
I'm using [abc] to denote the CANDIDATE LISTS of cells that are GIVEN.
I'm using [<>ab] to denote the exclusions that are DEDUCED, that is, "this cell cannot contain an 'a' or a 'b' based on the given candidate lists."

This is an XY WING, the shortest possible pair of FORCING CHAINS. No matter what value r3c1 has, the candidate '1' is excluded from r12c1. (r3c2 as well, but that's just a bonus.)

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`[<>1][   ][   ]|[<>1][   ][   ]|[ 23][ 12][   ]|---------------+[ 13][<>1][   ]|[   ][   ][   ]|[   ][   ][   ]|`

We can create this pattern by OVERLAYING the following two simple patterns:

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`[<>12][<>12][<>12]|    [<>13][ <>3][ <>3]|    [<>12][<>12][<>12]|    [<>13][ <>3][ <>3]|    [   2][   1][    ]|    [   3][  12][    ]|    ------------------+    ------------------+    [  13][ <>1][    ]|    [   1][ <>1][    ]|    [ <>2][    ][    ]|    [<>13][    ][    ]|    [ <>2][    ][    ]|    [<>13][    ][    ]|    `

Here's an XYZ WING, shown to be a two sets of NAKED PAIRS, overlaid:
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`r3c1=any => r12c1<>1{<>1][   ][   ]|[<>1][   ][   ]|[123][ 12][   ]|---------------+[ 13][   ][   ]|[   ][   ][   ]|[   ][   ][   ]|`

This pattern is created by overlaying these two:
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`[<>12][<>12][<>12]|    [<>13][    ][    ]|[<>12][<>12][<>12]|    [<>13][    ][    ]|[  12][  12][<>12]|    [  13][  12][    ]|------------------+    ------------------+[  13][    ][    ]|    [  13][    ][    ]|[    ][    ][    ]|    [<>13][    ][    ]|[    ][    ][    ]|    [<>13][    ][    ]|`

Six overlaid NAKED PAIRS:
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`Any digit in r3c1 leads to r12c1<>1[    <>1][     12][     13]|[    <>1][     14][       ]|[1234567][       ][       ]|---------------------------+[     15][       ][       ]|[     16][       ][       ]|[     17][       ][       ]|`

Three overlaid NAKED TRIPLES:
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`Any digit in r3c1 leads to r12c1<>1[      <>1][      123][      123]|[      <>1][      145][      145]|[123456789][      189][      189]|---------------------------------+[      167][         ][         ]|[      167][         ][         ]|[         ][         ][         ]|`

Code: Select all
`Any digit in r3c1 leads to r12c1<>1[    <>1][   1234][   1234]|[    <>1][   1234][       ]|[1234567][       ][       ]|---------------------------+[   1567][       ][       ]|[   1567][       ][       ]|[   1567][       ][       ]|`

A NAKED PAIR, TRIPLE and QUAD overlaid:
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`Any digit in r3c1 leads to r12c1<>1[    <>1][     12][    134]|[    <>1][       ][    134]|[1234567][       ][       ]|---------------------------+[   1567][       ][       ]|[   1567][       ][       ]|[   1567][       ][       ]|`

TWO different ways that two NAKED TRIPLES can be overlaid:

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`Any digit in r3c1 leads to r12c1<>1[  <>1][  123][     ]|[  <>1][     ][  123]|[12345][     ][     ]|---------------------+[  145][     ][     ]|[  145][     ][     ]|[     ][     ][     ]|Any digit in r3c1 leads to r12c1<>12[<>12][ 123][    ]|[<>12][    ][ 123]|[1234][    ][    ]|------------------+[ 124][    ][    ]|[ 124][    ][    ]|[    ][    ][    ]|`

THREE ways two NAKED QUADS can be overlaid:

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`Any digit in r3c1 leads to r12c1<>1[    <>1][   1234][   1234]|[    <>1][   1234][       ]|[1234567][       ][       ]|---------------------------+[   1567][       ][       ]|[   1567][       ][       ]|[   1567][       ][       ]|Any digit in r3c1 leads to r12c1<>1 or 2[   <>12][   1234][   1234]|[   <>12][   1234][       ]|[ 123456][       ][       ]|---------------------------+[   1256][       ][       ]|[   1256][       ][       ]|[   1256][       ][       ]|Any digit in r3c1 leads to r12c1<>1, 2 or 3[  <>123][   1234][   1234]| [  <>123][   1234][       ]| [  12345][       ][       ]| ---------------------------+ [   1235][       ][       ]| [   1235][       ][       ]| [   1235][       ][       ]| `

[previous two diagrams corrected, added following example]

Of course, the candidates lists need not be full. For example, this is two overlapping naked quads:

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`Any digit in r3c1 leads to r12c1<>1[    <>1][     34][    123]|[    <>1][     24][       ]|[1234567][       ][       ]|---------------------------+[    567][       ][       ]|[    167][       ][       ]|[    157][       ][       ]|`
tso

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Joined: 22 June 2005