Dominoes variants

This page describes variants of the dominoe tiling puzzle.

Different pieces set

The standard Western set of dominoes has always consisted of 28 tiles that display all possible pairs of digits from 0 to 6. There is nothing magic in number 6. You may wish to try this puzzle with different full set.

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.

Finally, you may choose your own size (the number of dots):

Different rectangle dimensions

Again, 7x8 is not the only rectangle that may be tiled with the standard 6-dot set. 14x4 and 28x2. (I do not even mention trivial 56x1 here.)

Bigger 7-dot set can tile not only familiar 8x9 but also 4x18, 2x36, 3x24 and 6x12.

Choose your own number of dots , width and height of a rectangle you wish to tile and .

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)).

Using more than one full set of tiles

Exactly the same way as we use subset to tile smaller rectangle, we may use two sets to tile bigger one. E.g. use 2 1-dot sets to tile 3x4 rectangle. Or use 2 7-dot sets to tile a 12x12 square.

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.

Combinations

The generalization of this principle is: take any number of full sets of any number of dots, then tile a rectangle using any subset of it. (The same form can be used to generate these variants.)

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.

Triplet Dominosa

2 Dominosa puzzles from booklet by O.S.Adler and Fritz Jahn:

Puzzle 1

Using a domino set of range 7 (36 pieces), can you make a rectangle which is entirely made out of triplets?

Puzzle 2

How was this cross formed? 
....555....
....632....
....632....
....632....
71110003334
75550005554
74440002224
....764....
....764....
....764....
....111....
....222....
....176....
....176....
....176....
....333....

More exotic variants

Finally, you may tile not only rectangles, but any other shape drawn on a square grid. 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.

Use standard set or you own.

Number of dots:
Width:
Height:
Puzzles in a row:
Puzzle rows:
.
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