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# MAX Puzzle

The MAX Puzzle asks you to fill shapes with congruent copies of the MAX tile shown at the top right corner.
A tile is either red, blue, or yellow.
There are two levels to play this game.
Beginner level: fill the areas with MAX tiles without gaps or overlaps.
Expert level: fill the areas such that no two adjacent tiles have the same color.
You draw a tile, clicking the board at the target position, set the "rotations, reflections" slider, select the desired color, and then click the "store new tile" button.
You can create, re-color, store, and delete tiles. The vertices snap to an isometric (60 degree) grid.

### DETAILS

Controls
The "challenge" popup-menu shows you various shapes to fill. Use the "<,>" buttons to page forwards and backwards. Challenge 10 is an empty board for free play.
Click the "solution" button only in challenge 8 or 9; see details below.
You draw a tile by a stamping method: Move the sample tile with the Locator (which initially is positioned at the top right corner) to the desired position on the board. Click one of the three color buttons to select its color. Use the "rotations, reflections" slider to select the orientation of the tile. Now click the "store new tile" button. The new tile will be displayed from now on. You can now use the Locator-tile to stamp the next tile.
Using the "show stored tiles" slider to display only some of the placed tiles.
After selecting "show stored tiles", click "re-color last shown" to change the coloring of a placed tile.
After selecting "show stored tiles", click "delete last shown" button to delete the tile that was shown last.
Click "delete all tile" to clear the board.
Challenge 8
In challenge 8 you are asked to fill an area of infinite size. It is the only challenge with a proper solution given
(click the "solution" button to see it). If we call this area a "strip", then we can join (horizontally) an infinite number of such strips to cover a 60-degrees wedge of the plane. The little diagram shows how we can then combine two such wedges to fill a 120-degrees wedge. Then we can combine three such 120-degrees wedges to fill the while plane,
but we can also combine three such 120-degrees wedges in a way such that a small regular triangle stays uncovered.
In a similar we can combine six 60-degrees wedges so that a small hexagon area (with unit side-length) stays uncovered.
Challenge 9
An infinite area with a blunted corner is given. This is a much harder problem that challenge 8. If you click the "solution" button, only an attempt of a solution is shown; it covers a large strip, but it is not clear whether it can be extended infinitely far.
Note that if the strip in challenge 9 can be covered totally, then the tiling must become repetitive at some stage, similar to how it happened in challenge 8.
History
The MAX Puzzle was invented by the author of this Demonstration in 1987 and implemented as a Zillions game in November 2000.
The task is to fill shapes with congruent copies of the MAX tile shown at the top right corner. The author has collected more than 200 interesting shapes that can be tiled with MAX tiles, of which only a small selection are presented here.
Using the six-pointed star (challenge 2) as a playing board, he also developed a 2-player game. The rules are:
Each player uses only one color. You win if you either connect two tips of the star or, if no player manages to connect tips, create the longest "fence" consisting of connected tiles of your color.

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