Lammas

This is a simple game of logic using a number of differently coloured tiles arranged in a grid. The object is to convert the complete grid to tiles of a single colour. The rule for conversion is given by the four tiles at the top left corner as a template. The game as described below is not very difficult and not particularly exciting, but it looks as if it could be fertile ground for analysing to see if there are 'best' strategies and provable maximum number of steps required to reach a solution from any pattern. Also worth analysis is a harder version which might have positions that are insoluble.

You need to have graphics and javascript enabled.
Once loaded you can disconnect from the network to play.

SETUP

NORMAL PLAY

HARD PLAY

Random reset
Go back one turn

Turn counter

Rules in detail

Using the program

To see the operation of the rules click on �NORMAL PLAY'. Notice how the bottom two tiles in the fourth column match the pattern of the top two tiles in the first column. Since this is a match we will put the same tile into column five row six as column two row two. We expect to see the pink star replaced with a green blob... Click on Apply rule and see this happens as predicted. You may also have noticed that the rightmost tile in the top row was also changed to a green blob. (Click on Go back one turn to see this again.) This has happened because the green blob we just substituted on the bottom row is now a candidate to be above the red and yellow star on the top row and this again matches our template. Because we look for patterns in a definite sequence, left to right on working our way down the rows this is the next to last place tested, the top tiles being 'below' the bottom row. With this example you can continue applying the rule without making any changes and the pattern solves itself!

Normally the pattern needs intervention. Clicking on a tile changes it in a cycle thus you may need to click more than once to get to the tile you desire. During setup you can alter any tile. During normal play you can alter any of the four tiles in the template corner. During hard play you can't alter the very top left tile manually and may therefore need to scheme to get the application of the rules to do it for you.

If you click on apply rule and the turn counter stays the same then no substitutions could be found.

Rules

1

Set out a grid with a number of different types of tile arranged however you want.

2

Use the four tiles in the top left corner as a template (see illustration) to see if any other tiles in the grid can be changed.

a: If the pattern given by the tile A directly to the left of tile B appears elsewhere in the grid then copy tile D into the grid in the same relation as in the template.
b: Repeat the matching process for tiles A and C this time making a copy of D if A is found elsewhere in the grid above C, putting the copy of D into the grid so that the L-shaped A-C-D pattern is repeated.

3

The grid is supposed to �wrap-around' so that the right hand column of tiles has neighbours on its right comprising the left-most column. The same applies for the vertical direction where the bottom row are deemed to be sitting on top of the top row.

4

Carry out this substitution working top to bottom left to right as if reading text in a book.

5

After trying out all substitutions (the application of a pattern being one turn regardless of how many substitutions) change the template pattern and have another attempt until all tiles are the same type.

For enthusiasts

Making changes to the program.

The program is written in javascript and you can easily make alterations. Firstly make sure you have all the necessary parts ie two "arrow...", three "bu..." four "tile..." and eleven "digit..." gif image files. If your browser is a pig when it comes to conveniently keeping the parts of a web page together then I've zipped the whole thing (21Kbytes) which you can download then unzip into the directory of your choice.

If changing the number of tiles or size of the grid then the only changes required are in the first half dozen lines of the javascript and in the text body where the cells are listed. Fuller instructions are included in the source of the web page just before the definition of the cells.

The code and page as supplied are copyrighted. You can copy and adapt it freely but you must acknowledge the original authorship and may not pass it off as all your own work.

Investigations

I've been an interested observer as clever proofs are devised to show how certain game strategies are optimal or what the provable worst case might be. Also how some puzzles appear not to have very good heuristic functions. I don't think the Lammas puzzle is particularly challenging when compared with say a rubik cube but I suspect that many of the same features are there but with a lesser complexity but more potential for variation with altered conditions.

One thing I'd like to know is what percentage of 4-tile,6x6 grid patterns are self solving. Ie need no intervention required. The self-solving example provided just happened to be the first I tried.

I suspect that larger grids don't need many more goes than smaller ones. Is there a number of goes is guaranteed to be the maximum required for any sized grid (for a given number of tiles).

If you have any thoughts, trials or analytical results then please e-mail me and I'll maintain links and or a review on these pages.

Peter Fox