Dominoes variants
Dominoes variants
Larger set (sold in toy stores) runs from 0-0 to 9-9 and contains 55 tiles.
Even larger set contains pieces with up to 12 dots and is really huge.
Once I printed 24-dot set and had been solving it for a couple of months, but never finished.
On another side you may try 5 dots or even 4 dots sets. They are easier (and much faster) to solve.
Bigger 7-dot set can tile not only familiar 8x9 but also 4x18, 2x36, 3x24 and 6x12.
Using only subset of tiles
Another possibility to alter the puzzle is to throw away any 4 pieces
from standard set and then tile 8x6
rectangle with the rest of them. Though the rectangle is smaller, the puzzle
may be more difficult. I never tried this one.
Use the form from previous section to choose you own size. Just choose the rectangle whose area is smaller than the area of standard rectangle for specified number of dots. (That is smaller than (n+1)*(n+2)).
You may use the same form to genetate you own variant. Just choose the rectangle whose area is twice (or trice or any other whole number) bigger than the area of standard rectangle for the specified number of dots.
I never tried none of those. It seems like too many possibilities to choose one or another way to tile a rectangle when some of the tiles are absent and/or some of them may be present more than once. This reduces the number of reducing rules which allow you to filter out certain possibilities using logical considerations that make the original puzzle interesting.
....555.... ....632.... ....632.... ....632.... 71110003334 75550005554 74440002224 ....764.... ....764.... ....764.... ....111.... ....222.... ....176.... ....176.... ....176.... ....333....
Often the shape is tilable when it is connected
(consits of one
piece) and the number of black squares is equal to the number of
white squares (provided that shape is colored like checkerboard).
These 2 conditions are are not always sufficient as the example
in the picture shows.
You may tile 3-dimensional parallelepipeds with 1x1x2 blocks, each unit cube of it containing a number. 4-, 5-, and n-dimensional hyperparallelepipeds can be used as well.
Even in 2-dimensional case the square grid is not the only possible. Triangle grid and hexagonal grid may be used.
On the triangle grid there can be 3 possible regular shapes to tile:
equilateral triangles, parallelograms and hexagons. Triangles cannot be
tiled with any set of rhombic dominoes (the number of black triangles
is not equal to the number of white triangles). Parallelograms are similar
to regtangles of a square grid.
Hexagons on a triangular grid lead to an interesting problem:
which regular hexagons can be
tiled by a full set of rhombic dominoes? The simplest case is hexagon with
a unit side: it can be tiled with 3 dominoes of a full 1-dot set.
Hexagon with side
2 consists of 24 unit triangles and thus requires 12 rhombic dominoes
to cover it. There is no full set of dominoes with such number of elements.
What is the next hexagon that can be tiled? What is the general formula
for the side of such hexagon?
Hexagonal grid allows triangle-like, parallelogram-like and hexagon-like
shapes.
Playing without a computer
Let us return to the regular case.
In fact, Martin Gardner's book [1] suggests two
people build a rectangle from standard set of dominoes each, then write digits
(but not layout) in a grid and exchange their puzzles. Whoever solves his
puzzle first wins.
You need a computer (or a human partner) only to generate a puzzle, not to play it. For example if you print the rectangle with numbers in its cells on a piece of paper you may use tools as simple as a pencil to solve the puzzle.
I played paper version of the puzzle during commute in a bus, in a long flight and once in a boring meeting.