Philip Eve
United Kingdom
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Blokus is a game played on a square board, tiled with squares 20 to a side, by 4 players. Each player has as pieces one of each of the free polyominoes* of size up to 5.

* http://en.wikipedia.org/wiki/Polyomino

This suggests the question: what would a game of Blokus be like using polyominoes of size up to 6? How big would you want the board to be in that case? Let's assume that the board would still be square.

We could start by asking what the total size in tiles of each player's pieces is in ordinary Blokus, and how much space is left on the game board if all the pieces are played. Each player has one monomino (1 tile), one domino (2 tiles), two triominoes (6 tiles), 5 tetrominoes (20 tiles), and 12 pentominoes (60 tiles), adding up to 89 tiles in total. Between all the players, then, there are 356 tiles' worth of pieces for 400 spaces on the board.

There are 35 free hexominoes, which gives us 210 tiles' worth of pieces. If we add that to the 89 we already have, we get 299 tiles' worth of pieces for each player, or 1196 between all the players.

How to arrive at a first guess for how big the new board would need to be? We could assume that the amount of leftover space on the board, if all pieces are played, should be similar to the figure for ordinary Blokus. In that case, we can estimate the side of the board as the square root of 1196 * 400 / 356. This yields 36.658, which suggests that we might try a board of side length 37 (but leaves open the possibility that a board of side length 36 might turn out to be more suitable).

I wonder whether anyone has ever tried making a Blokus set where all the players have hexominoes to play with as well.
 
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K H
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There is room for variance, and either 36x36 or 37x37 could be tolerated I think. I rather like the number 36, as it happens to be 6^2 and you are using hexominoes. There will be slighly more leftover pieces at the end of the game, but so what?

And while I have not attempted including square hexominoes in Blokus, I did include triangular hexominoes[1] in TriPakus, which is a variant of Blokus Trigon.


[1] Are they still called -ominoes if the segments are triangular?
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Philip Eve
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Slounger wrote:
There is room for variance, and either 36x36 or 37x37 could be tolerated I think. I rather like the number 36, as it happens to be 6^2 and you are using hexominoes. There will be slighly more leftover pieces at the end of the game, but so what?

And while I have not attempted including square hexominoes in Blokus, I did include triangular hexominoes[1] in TriPakus, which is a variant of Blokus Trigon.


[1] Are they still called -ominoes if the segments are triangular?


That is a nice point with the 36. Also, a 36 by 36 or 37 by 37 board would be a bit big and you would probably want it to fold to quarter size for storage; so that is another practical reason to favour an even side length.

Wikipedia uses the term "polyiamond" for shapes made by assembling equilateral triangles. So you might call them hexiamonds.
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