Sudoku ForceChain

Forth

All Force-based algorithms use the current phase link. The links are used to build a chain and logically derive the truth of the cell candidates.

A set X has one element true and the rest false. Which element is true is undetermined.
A chain that starts with a true element determines the true/false of the elements led by the concatenation.
In a chain starting with false element, the value of the derived element is uncertain (it can be true or false).
For a chain that starts by assuming each element of set X to be true, if the truth value of element A that all chains lead to is the same, the truth of element A is determined.
In a chain that starts by assuming that one element B of set X is true, the starting element B is determined to be false if multiple roots lead to unequal truth values for element C.

Set X is specifically rows, columns, blocks, and cells, as follows:

The (extended) link and chain structure of the Force method is relatively simple. It is also an efficient method of analysis. In other words, a good analysis algorithm for Sudoku. Forth algorithm is a logical algorithm.
Sudoku is a logical puzzle

ForceChain

ForceChain is an analysis algorithm that uses.super link

(1)ForceChain_Cell

Focused one unconfirmed cell A. Even if any of the candidate digits of cell A is assumed to be true, when the digit#n of cell B to be linked becomes true, In B cell #n is determined to be true.

When following the link assuming that the candidate digit#a is true, some digit in the cell may lead to both true and false. This is the case of (3)ForceChain_Contradiction below, the first proposition "#a is true" is an error, Therefore, there is no link starting with "#a is true". In the GNPX implementation, this case is not included in the solution of ForceChain_Cell, but is solved by ForceChain_Contradiction.

An sample of ForceChain_Cell. Cell r6c6 becomes #2 even if the digit in cell r2c8 is #1 or #2. The link string at this time is shown.

ForceChain_Cell

r6c6#2 is true

✿[W r2c8/+1 -> r2c4/-1] => [S r2c4/-1 -> r3c5/+1] => [W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [W r8c4/+2 -> r6c4/-2] => [S r6c4/-2 -> r6c6/+2]
✿[W r2c8/+2 -> r2c6/-2] => [S r2c6/-2 -> r6c6/+2]

1526.7.893...5...4...9.3..75.8...2.6.6.....9.9.3...4.14..5.6...6...3...573.4.1962

ForceChain_Cell

r6c6#2 is true
r2c6#8 is true
r6c4#8 is true

ForceChain_Cell r6c6#2 is true
✿[W r2c8/+1 -> r2c4/-1] => [S r2c4/-1 -> r3c5/+1] => [W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [W r8c4/+2 -> r6c4/-2] => [S r6c4/-2 -> r6c6/+2]
✿[W r2c8/+2 -> r2c6/-2] => [S r2c6/-2 -> r6c6/+2]

ForceChain_Cell r2c6#8 is true
✿[W r2c8/+1 -> r2c4/-1] => [S (ALS:r2468c4) r2c4/-1 -> r4c4/+3] => [W r4c4/+3 -> r4c8/-3] => [S r4c8/-3 -> r7c8/+3] => [S r7c8/-7 -> r7c5/+7] => [S r7c5/-2 -> r3c5/+2] => [W r3c5/+2 -> r2c6/-2] => [S r2c6/-2 -> r2c6/+8]
✿[W r2c8/+2 -> r3c8/-2] => [S r3c8/-2 -> r3c5/+2] => [W r3c5/+2 -> r2c6/-2] => [S r2c6/-2 -> r2c6/+8]

ForceChain_Cell r6c4#8 is true
✿[W r2c8/+1 -> r2c4/-1] => [S r2c4/-1 -> r3c5/+1] => [W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [W r8c4/+2 -> r6c4/-2] => [S r6c4/-2 -> r6c4/+8]
✿[W r2c8/+2 -> r2c6/-2] => [S r2c6/-2 -> r6c6/+2] => [S r6c6/-8 -> r6c4/+8]

1526.7.893...5...4...9.3..75.8...2.6.6.....9.9.3...4.14..5.6...6...3...573.4.1962

(2)ForceChain_House

In one House, focused cells having a candidate digit #a. Whichever cell #a is assumed to be true, when the digit #n of the linked cell B becomes true, the B cell is determined to be #n true.

In ForceChain_House as well as ForceChain_Cell, when following the link assuming that the candidate digit#a is true, some digit in the cell may lead to both true and false. This is the case of (3)ForceChain_Contradiction below, the first proposition "#a is true" is an error, Therefore, there is no link starting with "#a is true". In the GNPX implementation, this case is not included in the solution of ForceChain_House, but is solved by ForceChain_Contradiction.

An sample of ForceChain_House. Whichever cell (r2c48) with candidate digit #1 in row 2 is assumed to be true, r6c6 is confirmed as #2.

ForceChain_House(row2#1)

r6c6#2 is true

✿[W r2c4/+1 -> r2c4/-8] => [S r2c4/-8 -> r2c6/+8] => [S r2c6/-2 -> r6c6/+2]
✿[W r2c8/+1 -> r2c4/-1] => [S r2c4/-1 -> r3c5/+1] => [W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [W r8c4/+2 -> r6c4/-2] => [S r6c4/-2 -> r6c6/+2]

1526.7.893...5...4...9.3..75.8...2.6.6.....9.9.3...4.14..5.6...6...3...573.4.1962

ForceChain_House(row2#1)

r6c6#2 is true
r2c6#8 is true
r6c4#8 is true

ForceChain_House(row2#1) r6c6#2 is true
✿[W r2c4/+1 -> r2c4/-8] => [S r2c4/-8 -> r2c6/+8] => [S r2c6/-2 -> r6c6/+2]
✿[W r2c8/+1 -> r2c4/-1] => [S r2c4/-1 -> r3c5/+1] => [W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [W r8c4/+2 -> r6c4/-2] => [S r6c4/-2 -> r6c6/+2]

ForceChain_House(row2#1) r2c6#8 is true
✿[W r2c4/+1 -> r2c8/-1] => [W r2c8/+2 -> r2c6/-2] => [S r2c6/-2 -> r2c6/+8]
✿[W r2c8/+1 -> r2c4/-1] => [S (ALS:r2468c4) r2c4/-1 -> r4c4/+3] => [W r4c4/+3 -> r4c8/-3] => [S r4c8/-3 -> r7c8/+3] => [S r7c8/-7 -> r7c5/+7] => [S r7c5/-2 -> r3c5/+2] => [W r3c5/+2 -> r2c6/-2] => [S r2c6/-2 -> r2c6/+8]

ForceChain_House(row2#1) r6c4#8 is true
✿[W r2c4/+1 -> r2c4/-8] => [S r2c4/-8 -> r2c6/+8] => [S r2c6/-2 -> r6c6/+2] => [S r6c6/-8 -> r6c4/+8]
✿[W r2c8/+1 -> r2c4/-1] => [S r2c4/-1 -> r3c5/+1] => [W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [W r8c4/+2 -> r6c4/-2] => [S r6c4/-2 -> r6c4/+8]

1526.7.893...5...4...9.3..75.8...2.6.6.....9.9.3...4.14..5.6...6...3...573.4.1962

(3)ForceChain_Contradiction

Assuming that a cell/digit is true, it may be possible for both the true and false to lead the digit(#n) of other cell in the superlink concatenation. This is a contradiction, the first assumption is incorrect.

ForceChain_Contradiction

r2c4#2 is false

ForceChain_Contradiction r2c4#2 is false
✿[W r2c4/+2 -> r2c8/-2] => [S (ALS:r247c8) r2c8/-2 -> r2c8/+1]
✿[W r2c4/+2 -> r8c4/-2] => [W r8c4/+7 -> r8c7/-7] => [S r8c7/-7 -> r5c7/+7] => [W r5c7/+7 -> r5c5/-7] => [W r5c5/+1 -> r3c5/-1] => [S r3c5/-1 -> r3c8/+1] => [W r3c8/+1 -> r2c8/-1] 1526.7.893...5...4...9.3..75.8...2.6.6.....9.9.3...4.14..5.6...6...3...573.4.1962

ForceChain_Contradiction

The following cells/digits are false.
r2c4#2, r2c6#2, r2c8#1,
r3c5#1, r3c8#2,
r4c4#7, r4c8#3,
r5c4#1, r5c5#7, r5c7#7, r5c9#8,
r6c4#2, r6c6#8,
r7c2#8, r7c5#2, r7c8#7, r7c9#3,
r8c2#2 is, r8c4#7, r8c7#8

pForceChain_Contradiction r2c4#2 is false
✿[W r2c4/+2 -> r2c8/-2] => [S (ALS:r247c8) r2c8/-2 -> r2c8/+1]
✿[W r2c4/+2 -> r8c4/-2] => [W r8c4/+7 -> r8c7/-7] => [S r8c7/-7 -> r5c7/+7] => [W r5c7/+7 -> r5c5/-7] => [W r5c5/+1 -> r3c5/-1] => [S r3c5/-1 -> r3c8/+1] => [W r3c8/+1 -> r2c8/-1]

ForceChain_Contradiction r2c6#2 is false
✿[W r2c6/+2 -> r3c5/-2] => [W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [S (ALS:r268c4) r8c4/-7 -> r2c4/+1]
✿[W r2c6/+2 -> r2c8/-2] => [S r2c8/-2 -> r3c8/+2] => [S r3c8/-1 -> r3c5/+1] => [W r3c5/+1 -> r2c4/-1]

ForceChain_Contradiction r2c8#1 is false
✿[W r2c8/+1 -> r2c4/-1] => [S r2c4/-1 -> r3c5/+1]
✿[W r2c8/+1 -> r2c4/-1] => [S (ALS:r268c4) r2c4/-1 -> r8c4/+7] => [W r8c4/+7 -> r8c7/-7] => [S r8c7/-7 -> r5c7/+7] => [W r5c7/+7 -> r5c5/-7] => [W r5c5/+1 -> r3c5/-1]

ForceChain_Contradiction r3c5#1 is false
✿[W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [S (ALS:r268c4) r8c4/-7 -> r2c4/+1]
✿[W r3c5/+1 -> r2c4/-1]

ForceChain_Contradiction r3c8#2 is false
✿[W r3c8/+2 -> r3c5/-2] => [W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [S (ALS:r268c4) r8c4/-7 -> r2c4/+1]
✿[W r3c8/+2 -> r2c8/-2] => [S (ALS:r247c8) r2c8/-2 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r4c4#7 is false
✿[W r4c4/+7 -> r8c4/-7] => [S (ALS:r268c4) r8c4/-7 -> r2c4/+1]
✿[W r4c4/+7 -> r4c8/-7] => [S r4c8/-7 -> r7c8/+7] => [W r7c8/+7 -> r7c5/-7] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r4c8#3 is false
✿[W r4c8/+3 -> r4c4/-3] => [S (ALS:r2468c4) r4c4/-3 -> r2c4/+1]
✿[W r4c8/+3 -> r7c8/-3] => [W r7c8/+7 -> r7c5/-7] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r5c4#1 is false
✿[W r5c4/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [S (ALS:r268c4) r8c4/-7 -> r2c4/+1]
✿[W r5c4/+1 -> r2c4/-1]

ForceChain_Contradiction r5c5#7 is false
✿[W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [S (ALS:r268c4) r8c4/-7 -> r2c4/+1]
✿[W r5c5/+7 -> r7c5/-7] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r5c7#7 is false
✿[W r5c7/+7 -> r5c5/-7] => [W r5c5/+1 -> r3c5/-1] => [S r3c5/-1 -> r2c4/+1]
✿[W r5c7/+7 -> r4c8/-7] => [S r4c8/-7 -> r7c8/+7] => [W r7c8/+7 -> r7c5/-7] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r5c9#8 is false
✿[W r5c9/+8 -> r5c7/-8] => [W r5c7/+7 -> r5c5/-7] => [W r5c5/+1 -> r3c5/-1] => [S r3c5/-1 -> r2c4/+1]
✿[W r5c9/+8 -> r5c7/-8] => [S r5c7/-8 -> r8c7/+8] => [W r8c7/+8 -> r7c9/-8] => [S (ALS:r7c589) r7c9/-8 -> r7c5/+2] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r6c4#2 is false
✿[W r6c4/+2 -> r8c4/-2] => [W r8c4/+7 -> r8c7/-7] => [S r8c7/-7 -> r5c7/+7] => [S r5c7/-8 -> r5c9/+8] => [S r5c9/-3 -> r5c4/+3] => [S r5c4/-1 -> r2c4/+1]
✿[W r6c4/+2 -> r6c6/-2] => [S r6c6/-2 -> r2c6/+2] => [W r2c6/+2 -> r3c5/-2] => [W r3c5/+1 -> r2c4/-1]

ForceChain_Contradiction r6c6#8 is false
✿[W r6c6/+8 -> r2c6/-8] => [W r2c6/+2 -> r3c5/-2] => [W r3c5/+1 -> r5c5/-1] => [W r5c5/+7 -> r5c7/-7] => [S r5c7/-7 -> r8c7/+7] => [W r8c7/+7 -> r8c4/-7] => [S (ALS:r268c4) r8c4/-7 -> r2c4/+1]
✿[W r6c6/+8 -> r2c6/-8] => [W r2c6/+2 -> r3c5/-2] => [W r3c5/+1 -> r2c4/-1]

ForceChain_Contradiction r7c2#8 is false
✿[W r7c2/+8 -> r7c9/-8] => [S r7c9/-8 -> r5c9/+8] => [S r5c9/-3 -> r5c4/+3] => [S r5c4/-1 -> r2c4/+1]
✿[W r7c2/+8 -> r7c9/-8] => [S (ALS:r7c589) r7c9/-8 -> r7c5/+2] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r7c5#2 is false
✿[W r7c5/+2 -> r7c2/-2] => [W r7c2/+8 -> r7c9/-8] => [S r7c9/-8 -> r5c9/+8] => [S r5c9/-3 -> r5c4/+3] => [S r5c4/-1 -> r2c4/+1]
✿[W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r7c8#7 is false
✿[W r7c8/+7 -> r4c8/-7] => [S r4c8/-7 -> r4c4/+7] => [S r4c4/-3 -> r5c4/+3] => [S r5c4/-1 -> r2c4/+1]
✿[W r7c8/+7 -> r7c5/-7] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r7c9#3 is false
✿[W r7c9/+3 -> r5c9/-3] => [S r5c9/-3 -> r5c4/+3] => [S r5c4/-1 -> r2c4/+1]
✿[W r7c9/+3 -> r7c8/-3] => [W r7c8/+7 -> r7c5/-7] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r8c2#2 is false
✿[W r8c2/+2 -> r7c2/-2] => [W r7c2/+8 -> r7c9/-8] => [S r7c9/-8 -> r5c9/+8] => [S r5c9/-3 -> r5c4/+3] => [S r5c4/-1 -> r2c4/+1]
✿[W r8c2/+2 -> r7c2/-2] => [S r7c2/-2 -> r7c5/+2] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r8c4#7 is false
✿[W r8c4/+7 -> r7c5/-7] => [S (ALS:r7c259) r7c5/-7 -> r7c9/+3] => [W r7c9/+3 -> r5c9/-3] => [S r5c9/-3 -> r5c4/+3] => [S r5c4/-1 -> r2c4/+1]
✿[W r8c4/+7 -> r7c5/-7] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

ForceChain_Contradiction r8c7#8 is false
✿[W r8c7/+8 -> r5c7/-8] => [S r5c7/-8 -> r5c9/+8] => [S r5c9/-3 -> r5c4/+3] => [S r5c4/-1 -> r2c4/+1]
✿[W r8c7/+8 -> r7c9/-8] => [S (ALS:r7c589) r7c9/-8 -> r7c5/+2] => [W r7c5/+2 -> r3c5/-2] => [S r3c5/-2 -> r3c8/+2] => [S r3c8/-1 -> r2c8/+1] => [W r2c8/+1 -> r2c4/-1]

1526.7.893...5...4...9.3..75.8...2.6.6.....9.9.3...4.14..5.6...6...3...573.4.1962