## SudoCue - Eureka Notation |

This page explains the use of Eureka Notation for Double Implication Chains. It was last updated at September 26, 2006. Eureka Notation is used in several advanced solving techniques explained in the solving guide. ## Introduction
Many advanced solving techniques rely on chains of candidates, linked together to propagate implications from the first node
to the last node in the chain. The purpose of such a chain is to prove that a contradiction exists for the first candidate
in the chain. A ## True or False
Logic can only be used when we can write predicates which are either P = True
Candidate P = False
Candidate ## Links
When two candidate belong to the same constraint, they interact with each other. Each cell is a constraint, because only one of its
candidates can be When there are more than 2 candidates left in a constraint, any pair of candidates in that constraint are weakly linked. They cannot both be true, because that would violate the constraint. However, they could both be false, because there are other candidates left to satisfy the constraint. A weak link is less powerful than a strong link, but they can be very useful in chains. ## Inference
In Sudoku, we use the term inference to describe the logical deductions we can make from the interaction between linked candidates.
There is strong and weak inference. A strong link can cause both types, but a weak link can only cause weak inference. Inference always
has the same effect in both directions, so we speak of inference ## Weak Inference
(P=true => Q=false) & (Q=true => P=false) The first line shows the definition in plain logic. The second line is a formal mathematical notation. Because chains are often written in plain text, these mathematical operators are not always available, and many people do not understand them. In Eureka Notation, the dash ‘-’ character is used as a symbol for weak inference. The third line tells you exactly the same as the previous, but it is more compact and easy to use in a chain. ## Strong Inference
(P=false => Q=true) & (Q=false => P=true)
Strong inference is represented by an equal sign ‘=’. This may be a little confusing at first, because a strong link causes
P and Q ## Alternating InferenceDouble Implication Chains can be constructed when we use strong and weak inference in an alternating sequence. Take this example: P-Q=R-S
Because the inference alternates, the implications can travel all the way from the first node to the last node in the chain.
The main purpose of the chain is to establish a link between the first and the last node. These are the only 2 we use
for further deductions. In many chains, the first and the last node are the same candidate. We call these chains P-Q=R-S=T-U=V-P This is a short loop, representing an XY-Wing. When P=true, Q=false, R=true, S=false, T=true, U=false, V=true, P=false. There is only one possible conclusion: P must be false, because it eliminates itself at a distance. ## Candidate Notation
We cannot use P and Q to represent specific candidates. Therefore we must use a notation that allows us to locate the candidate
in the grid. Each candidate belongs to a (4)r4c2-(4)r5c1=(6)r5c1-(6)r1c1=(3)r1c1-(3)r2c2=(4)r2c2-(4)r4c2 Even with the short inference symbols, the chain is pretty long. To overcome this problem, we use contraction to shorten the chain. When multiple candidates of the same cell are subsequent nodes in the chain, then we may place the digits for those candidates between a single set of parentheses. The inference symbol is then placed between those digits. (4)r5c1=(6)r5c1 is contracted to (4=6)r5c1 Now the XY-Wing can be written in a more compact format: (4)r4c2-(4=6)r5c1-(6=3)r1c1-(3=4)r2c2-(4)r4c2 ## Multiple CandidatesChains can often be used to provide similar proof for multiple candidates. These only appear at the beginning and end of the chain. We can add these extra candidates to the chain when we separate them with a pipe ‘|’ symbol. Here is an example: (4)r4c2|(4)r6c2-(4=6)r5c1-(6=3)r1c1-(3=4)r2c2-(4)r4c2|(4)r6c2
When alternatives are followed by a inference symbol, It is possible to contract multiple candidates, but only when they represent the same digit and share a row or column. The alternatives can be contracted in the following way: (4)r4c2|(4)r6c2 is contracted to (4)r46c2 The pipe symbol is dropped when we apply this type of contraction. It no longer has any purpose. ## ConclusionA chain is of no use if we cannot draw any conclusions from it. After all, we want to use them to solve our Sudokus. The chain itself is used as the proof for our conclusion. A double arrow separates the chain and the conclusion, which must be written as a true or false predicate. Here is our XY-Wing example again, with its conclusion: (4)r4c2-(4=6)r5c1-(6=3)r1c1-(3=4)r2c2-(4)r4c2 => r4c2<>4
This provides us with enough information to remove candidate 4 from |

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