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|Rule 1 - Twice in a Cell
There are six different ways we eliminate - six contradictions. The first is in the example to the right. It doesn't matter where you start on the grid. In this example I've started with the 4s in row B. By colouring one green and the other yellow we mentally draw a line between them, done graphically on the diagram. Going into the third dimension in B7 when we colour the 4 yellow we can colour the 9 green - since there are only two values left in the cell.
Continue to look for bi-value and bi-location candidates and you soon build up a web of connections. This is where the image of Medusa was perhaps attached to this strategy - her head being a tangle of snakes.
3D Medusa Rule 1: Load Example or : From the Start
Rule 2 - Twice in a Unit
This rule is shared with Simple Colouring. Its the same principle as the first rule but we are looking for two coloured occurrences of X in the same unit (row, column or box) as opposed the two of the colour in the same cell.
The example shows most of the links between bi-value and bi-location candidates, coloured between green and yellow. Ringed in red are two 7s in column 7. Since both cannot be true neither can be true and all yellow coloured candidates can be removed.
(Example requires three Medusa Rules 6 before Rules 2 comes into play)
3D Medusa Rule 2: Load Example or : From the Start
|Rule 3 - Two colours in a cell
If you had unticked 3D Medusa in the solver this example would have been found by a number of later strategies, particularly Alternating Inference Chains as the pattern is a classic Nice Loop. 3 and 7 alternate. It doesn't matter where you start in a Nice Loop but you can trace the on / off or green/blue round the loop. 3s and 7s neatly occur twice in units and cells.
But 3D Medusa is not about loops, its about the network of links. This example just happens to be the same formation. We know that either ALL the blue candidates will be true, or ALL the green ones. If there are any another candidates in any cell with two colours, they cannot be solutions. Hence the 8 can be removed from C2. In Nice Loop terms, this is an off-chain elimination.
3D Medusa Rule 3: Load Example or : From the Start
|Rule 4 - Two colours in a unit
If we can eliminate "off chain" in a cell we can certainly do so off-chain in a unit. In this example there are quite a few links between 3s, 5s and 6s. Most have been drawn on the diagram. We are certain than ALL blues are the solution or ALL greens. Therefore where there are candidates that can see both colours they can be removed. By 'see' we mean any candidates that are the same number as members of the blue/green links.
The 5s in column 7 are these. The 5s in ACG7 are removed because of the coloured 5s in column 7. The 5s in box 6 can see blue and green 5s in D7 and F7.
This rule is shared with Simple Colouring.
3D Medusa Rule 4: Load Example or : From the Start
|Rule 5 - Two colours 'elsewhere'
Given the above possibilities, its tempting to generalise. The example to the right we have some 6s in row F. Both can see a green 6 in E9 and a blue 6 F5. This is not exactly like Rule 4 but very close. The 6s can be removed because they can see two different coloured candidates of 6 elsewhere.
3D Medusa Rule 5: Load Example or : From the Start
|Rule 6 - Two colours Unit + Cell
The last type of elimination looks to be the most complex - but inconveniently it is the most common. It's well worth looking out for. The rule says
If an uncoloured candidate can see a coloured candidate elsewhere (it shares a unit) and an opposite coloured candidate in its own cell, it can be removed..
So its a combination of unit and cell - the colours green and blue are found looking along a unit and within the same cell. The example to the right demonstrates this with four eliminations.
The logic is very appealing. Consider 1 in E5. If 1 were the solution to the cell it would remove a green 1 from E6 AND a blue 7 from its own cell in E5. Since we know ALL blue or ALL green must be solutions we have a contradiction.
3D Medusa Rule 6: Load Example or : From the Start
|Rule 7 - Cell Emptied by Color
Anton Delprado in the comments below has discovered another way we can use 3D Medusa and I'm pleased to include it in the solver. It's almost a reflection of Rule 5. Take any cell that doesn't have any colors from the coloring and see if all the candidates can see the same color. If that color were the solution (and remember, all of one or all of the other will be) then all the candidates in that cell would be removed - leaving an empty cell!
Rule 7 highlights in cyan the cell that catches this. You can see the 2 and 5 in cell C1 can see the yellow 2 in C9 and 5 in C1. (Yellow is used to show eliminated cells).
Rule 7 : Load Example or : From the Start
This puzzle has an amazing series of Medusa calls, using many different rules. It ends with Rule 7. I wanted to show a second puzzle to emphasise that the cell we are comparing the Medusa net to can have any number of candidates. These are 4,6 and 9 in C6. Green candidates have been turned yellow because they are eliminated, but you can see that the 4, 6 and 9 can all see the same color somewhere along the column or row.
You can be certain that it will be one color or the other, never equally both. Because this strategy is easier to spot and somewhat follows on from Rule 5, the solver looks for it before Rule 6. But too late to re-number them now. Well spotted Anton!
Rule 7 using 3 candidates in a cell: Load Example or : From the Start
|To end this article I want to show you some special puzzles discovered by Klaus Brenner starting with this 37 elimination Rule 1 Medusa that completely solves the puzzle from that point. We go from 35 known numbers to 70 and the rest is trivial!
There are two candidates in A1 with the same color, 5 and 7. So All of those of that color can be removed.
However, the initial puzzle is not trivial and a very large number of steps are required before this mega medusa. Certainly an extreme grade.
37 Eliminations by Rule 1: Load Example or : From the Start
A slightly different puzzle, 22 clues, gives us these 10 eliminations using Rule 6. I don't know another with more in one step.
Rule 6: Load Example