Forcing Chains

Forcing Chains as the name suggests, use multiple chains to force a solution in a cell or force a candidate reduction.

Forcing chains use alternating inference chains (AIC) like X Chains and Nice Loops. Forcing chains are not loops.

If there are a small set of choices somewhere in a puzzle where one of those choices has to be true, then if we start with each option in that set of choices and end up with the same solution in every case, then the solution has to be true.

The set of options might be choosing if a candidate is placed in a cell or the same candidate is removed from the cell, choosing each of the possible candidates as the solution in a cell, choosing each of the possible locations of a candidate in a unit. At this point, it does not matter which option is true, just that one has to be true.

If an AIC is created from each of the initial choices and all chains end in the same conclusion, the conclusion must be true, because one of the starting alternatives has to be true.

Cell Forcing Chains

These forcing chains use the knowledge that one candidate will eventually have to be placed in a cell.

If a chain is created by assuming each candidate, in turn, is placed in a the cell and all the chains reach the same conclusion then that conclusion must be true. One of the candidates has to be placed in the original cell so if every candidate choice reaches the same conclusion then it does not matter which candidate we eventually place in the cell the conclusion will be true.

This is easier if you start with bi value cells and only have to two chains to work with but the logic holds up no matter how many candidates are in a cell.

There are four outcomes that lead to candidate reductions.

If all the forcing chains conclude the same candidate has to be placed in a cell then that candidate can be placed as the solution for that cell and all other candidates removed.

If all the forcing chains conclude the same candidate cannot be placed in a cell then that candidate can be removed from that cell.

If each of the forcing chains conclude a different candidate has to be placed in a cell then one of those placed candidates has to be true so all other candidates can be removed from the cell.

If all the forcing chains conclude that a candidate has to placed in a different cell in the same unit then the candidate can be removed from any other cells in the unit.This Cell Forcing Chains example in Figure 1 has two chains starting at the candidates in a bi value cell (yellow), one chain colored green and the other colored blue.

The standard notation for these chains -

[r2c1] -1- [r2c3] =1= [r9c3] -1- [r7c1] -6- [r8c2] =6= [r8c6] =1= [r8c9]

[r2c1] -6- [r2c9] =6= [r1c9] =1= [r1c8] -1- [r9c8] =1= [r8c9]

The candidate 1 or the candidate 6 has to be placed in the yellow cell. Both the chain starting at candidate 1 and the chain starting at candidate 6 conclude the candidate 1 has to be placed in the grey cell.

Either the candidate 1 or the candidate 6 has to eventually be placed in the yellow cell so the value 1 can be made the solution in the grey cell.The Cell Forcing Chains example in Figure 1 shows two chains starting at the candidates in a bi value cell (yellow), one chain colored green and the other colored blue.

The standard notation for these chains -

[r6c4] -4- [r4c6] -8- [r7c6] =8= [r7c2] -8- [r6c2]

[r6c4] -8- [r6c2]

Both chains conclude that the candidate 8 can not be placed in the grey cell.

The candidate 4 and the candidate 8 are the only two candidates in the yellow cell so one of them has to eventually be placed in the yellow cell. In either case the candidate 8 is forced out of the grey cell.The Cell Forcing Chains example in Figure 3 shows two chains starting at the candidates in a bi value cell (yellow), one chain colored green and the other colored blue.

The standard notation for these chains -

[r2c7] -1- [r3c9] =1= [r8c9]

[r2c7] -8- [r8c7] =8= [r8c9]

The chains that start from the candidates in the yellow cell conclude the candidates 1 and 8 have to be in the grey cell. Either the value 1 or the value 8 has to be placed in the yellow cell, so the grey cell will have to have either the value 8 or the value 1.

All other candidates can be removed from the grey cell.The Cell Forcing Chains example in Figure 4 also shows two chains starting at the candidates in a bi value cell (yellow), one chain colored green and the other colored blue.

The standard notation for these chains -

[r1c4] -6- [r3c4] -5- [r3c5] -2- [r1c5] =2= [r1c7] -2- [r2c7] =2= [r2c1]

[r1c4] -8- [r1c5] -2- [r3c5] -5- [r3c4] =5= [r6c4] =2= [r6c6] -2- [r7c6] =2= [r8c4] -2- [r8c2] =2= [r7c1]

Both these chains conclude by placing the value 2 in a cell in column 1.

The blue chain says "if I put the value 8 in the yellow cell then I will end up with the value 2 in the blue/brown cell". The green chains says "if I put the value 6 in the yellow cell then I will end up with the value 2 in the green/brown cell".

The value 6 or the value 8 have to be placed on the yellow cell so that means that the value 2 will end up in one of the brown cells. Both brown cells are in the same container, in this instance column 1.

No matter which candidate ends up in the yellow cell there will be a 2 placed in one of the brown cells in column 1. All other instances of the value 2 in non brown cells in column 1 can be removed. The forcing chains from the yellow cell tell us the value two can only be in one of the brown cells for column 1.