Single's Chains

Single’s chains is the first and simplest in a family of powerful chaining strategies such as X-Cycles, XY-Chains, 3D Medusa, Alternating Inference Chains, etc. They all follow a similar idea - we “assume” that a given candidate in a cell will either be the value of that cell or will not be. From this assumption we make deductions to other linked candidates and sometimes we may reach a specific setting which allows to eliminate from or place candidates in specific cells.

To simplify the notation and make spotting and understanding such strategies easier, we give the first candidate a colour - either ON or OFF, i.e. we either think of the candidate as the true value for the cell (ON) or not (OFF). By following simple rules we can then turn other linked candidates ON or OFF. Such links will be represented by a line.

Single’s chains is the simplest such method since we only consider a single fixed candidate and we make “deductions” in a pretty straightforward way.

It is important to know that when we create the chains, initially, we do not know which colour is ON and which colour is OFF. We could have used other two random colours. When we reach a specific setting of candidates and colours, the correct colours can be deduced or they may not matter at all. What matters is that if one candidate is, say, red it will imply that the next linked is, say, green. The green candidate will imply that the next is red and so on.

Creating Chains

Let us first see how chains are constructed by considering the following board and the candidate 9.

Load in Solver

The chain starts with a 9 coloured OFF at A1, which we will denote (9, A1) for short. Since there are only two candidates 9 in row 1, then the 9 in I1 is ON, or we can simply write (9, A1) ⇒ (9, I1). In other words, if the first is OFF, then the second would surely be ON. Before we continue let us further clarify some of the notions, mentioned above. We could have started with (9, A1) which would then imply (9, I1). The fact is that the initial colour we choose does not matter. What is important is that both have different colours. The second observation is that we could have started with (9, I1) ⇒ (9, A1), or in other words the implication goes both ways. To emphasise that the implication works in both directions we will use the corresponding sign “⇔”.

For other chaining strategies implications do not necessarily work both ways, which will be important at a later stage. In Single’s chains implications always work both ways so let us just ignore the above comment for now and continue with the example. Since there are only two candidates 9 in the bottom-left square and the one in I1 is ON, then we have that (9, I1) ⇔ (9, H2). By following the same logic for row I, we also deduce that (9, I1) ⇔ (9, I7). Using the same rule (that two 9’s are the only ones in a given row, column or square, and one of them is coloured), we can deduce the following links:

(9, I7) ⇔ (9, H9), (9, H2) ⇔ (9, H9), (9, I7) ⇔ (9, B7), (9, B7) ⇔ (9, B5) and so on, until there are no more links possible.

Single's Chains Rule 2 - Twice in a House

Once we have created a chain, if we notice that two candidates share a house and have the same colour, then this colour must be the OFF colour. The logic here is relatively simple: if the colour was ON then a house would have two values placed which are the same. Let us look at an example illustrating this rule.

Load in Solver

The chain starts from the dark blue cell A6 with a 5 coloured ON. Tracing the chain we notice that the top-right square contains two red 5 candidates. By the rule, this means that the red colour is the OFF colour and we can eliminate all red candidates. Furthermore, all candidates coloured green are the correct values for the given cells.

To see the logic behind the rule suppose, once again, that we flip the colours, given in the picture, i.e. we start with a (5, A6) instead of (5, A6). All colours would be flipped and so the two 5 candidates in the top-right square would be ON. This is impossible as a house should only have a single value 5. Hence both can be removed.

Single's Chains Rule 4 - Seen by Two Colours

If we cannot use rule 2 we can still make eliminations from outside of the chain. Rule 4 states that if a candidate not in the chain, say X, is seen from two candidates which have different colours, then we can eliminate X. As this rule sounds slightly more complicated, let us directly see it in an example.

Load in Solver

We can create a chain starting with (4, D3). Please note that light-blue cells are not part of the chain. Rule 2 does not apply here, but we can see that candidate 4 in cells D6 and I4 (which are not from the chain and marked in light-blue) see two differently coloured candidates from the chain. Namely, (4, D6) sees (4, D3) and (4, G6). Similarly, (4, I4) sees (4, F4) and (4, G7). The rule simply states that if there are such cells as D6 and I4 we can eliminate the candidate from these cells. Let us see why this is the case. The red and green colours in this case do not show us which colour represents the correct value and which colour represents the wrong value. By tracing the chain we can see that if (4, D3) is OFF, then (4, G6) is ON and so (4, D6) can be eliminated, because it sees the ON value in G6. Similarly, if (4, D3) is ON then (4, G6) is OFF and (4, D6) can also be eliminated. Since this happens in both cases, we can simply remove the candidate 4 from the light-blue cells.