When doing any kind of puzzle, it is often useful to have an notational system to assist in solving more complicated cases. It also assist in following stages of progress, exploring various alternatives and backtrack in the case of errors. This is my system for thinking about slitherlink. It has helped me in keeping track of various permutations, not necessarily on paper, but definitely in my mind. It can easily be adopted to describe solutions that can be replicated without having to use images. And, using the terminology described here, I will describe (in a follow-up), my strategy to solve slitherlink puzzles.

In the standard slitherlink puzzles, each square contain a number, indicating how many of its four sides form part of the loop. This can be 0, 1, 2 or 3 segments. To code the different alternatives, the top segment is used as the base. The coding then move in a clockwise direction to number the alternatives.The top edge is numbered 1, the right 2, the bottom 3 and the left edge 4. For 1 and 2 segments in the loop, the number of the first segment that is filled, indicate the option for that square. For 3 segments in the loop, the open line indicate the option for that square. Two special cases exist for a cell with 2 segments in the loop. This is where two opposite edges are filled in. With the top and bottom edges filled in are defined as option 5 and with the left and right edges filled in are defined as option 6.

These are the different possibilities described in words, and the corresponding short hand:

## One edge part of the loop

Top edge filled in: 1-1

Right edge filled in: 1-2

Bottom edge filled in: 1-3

Left edge filled in: 1-4

## Two edges part of the loop

Top and right edge: 2-1

Right and bottom edge: 2-2

Bottom and left edge: 2-3

Left and top edge: 2-4

Top and bottom edge: 2-5

Left and right edge: 2-6

## Three edges part of the loop

Top edge open: 3-1

Right edge open: 3-2

Bottom edge open: 3-3

Left edge open: 3-4

## No edge part of the loop

This has no alternatives and will thus always be coded as 0-0.

To make it easier to and shorter to write, the dash can be removed and 1-1 becomes 11 and so on.

The notation can now be extended to describe a puzzle or puzzle solution and partial solutions. The puzzle grid is labelled (arbitrarily) from 1 to x for the columns and 1-y for the rows. For the content code, 4 will indicate and edge that is part of the loop and 5 will indicate an edge that is not part of the loop. In the case of the content field 4 and 5, the number at the end will indicate the edge that is part of the loop or not, respectively. A cell is then described using four numbers:

column-row-content-edge

For describing an unsolved puzzle, the edge can be left out.

This 4×4 puzzles using the notational system, can be described as follows:

2113

4100

1222

2233

4213

2313

4321

1431

2424

3421

4431

All four edges of Cell 11 is outside the loop and could be described as 1151, 1152, 1153 and 1154. One of the first steps in solving this puzzle, would be to fill in the two bottom corners (14 and 44) containing the threes. The two corner edges will, in the case of a 3, always be part of the loop and would be coded as 1443, 1444, 4442 and 4443. **Important**, this is not used for description of a puzzle, but rather an incomplete state.

Some description of the puzzle should probably also be given. This would include the number of columns and rows. To cross check for mistakes, the number of filled in cells could also be included.

Any inputs and refinements are welcome. This approach can also be used to assist in solving the slitherlink variations, as seen at the krazydad site.

For more about slitherlink puzzles see:

- Nikoli, the developer of slitherlink
- Play slitherlink online at Kwon-Tom Loop
- Slitherlink techniques
- The Parity Method of Melorama to solve slitherlinks
- A collection of slitherlinks and alternatives is available for download at krazydad
- Slitherlink on Wikipedia
- A piece on solving slitherlink variations

ps: managed to sort out the graphics problem.

Tags: notation, segment, notational system, toy, Toys