Congratulations to Katherine Henderson, age 12, from Maidstone
Girls' Grammar School who reasoned that there are the 24 unique
domino solutions given below, where each arrangement has the 5-3
domino horizontally. Katherine gave the answer 384. She explained
that there are 3 columns of dominoes which can be arranged in six
different ways. Each of these arrangements can be varied by
swopping the rows giving 6 x 4 = 24 arrangements. Katherine then
explained that from each of these 24 arrangements you can find 8
others: 4 which are mirror images and 4 which are rotations. This
gives 192 (24x8) patterns altogether. Then if you take the 2-2 you
can also turn it round 180 
to form twice as many solutions. This gives 384
(192x2) domino square solutions.
James Page of Hethersett High School explained that ``I have
found that the domino 5-3 is the key domino, as wherever it goes it
has to be followed by two blanks.'' He explained that, from one
solution, he found different patterns by swapping the rows or
columns. Daniel Gamboa and Michael Dowden of Necton Middle School,
Norfolk found one of the solutions. Camilla Egginton of Maidstone
Girls' Grammar School discovered that in her solutions certain
blocks stayed next to one another and concluded that having found
one solution all the others are different ways of re-arranging
it.
| 5-3 |
B |
B |
| 1-2 |
3 |
2 |
| B-1 |
1 |
6 |
| 2-2 |
4 |
B |
|
| 5-3 |
B |
B |
| 1-2 |
2 |
3 |
| B-1 |
6 |
1 |
| 2-2 |
B |
4 |
|
| B |
5-3 |
B |
| 3 |
1-2 |
2 |
| 1 |
B-1 |
6 |
| 4 |
2-2 |
B |
|
| B |
5-3 |
B |
| 2 |
1-2 |
3 |
| 6 |
B-1 |
1 |
| B |
2-2 |
4 |
|
| B |
B |
5-3 |
| 3 |
2 |
1-2 |
| 1 |
6 |
B-1 |
| 4 |
B |
2-2 |
|
| B |
B |
5-3 |
| 2 |
3 |
1-2 |
| 6 |
1 |
B-1 |
| B |
4 |
2-2 |
|
| 5-3 |
B |
B |
| 1-2 |
3 |
2 |
| 2-2 |
4 |
B |
| B-1 |
1 |
6 |
|
| 5-3 |
B |
B |
| 1-2 |
2 |
3 |
| 2-2 |
B |
4 |
| B-1 |
6 |
1 |
|
| B |
5-3 |
B |
| 3 |
1-2 |
2 |
| 4 |
2-2 |
B |
| 1 |
B-1 |
6 |
|
| B |
5-3 |
B |
| 2 |
1-2 |
3 |
| B |
2-2 |
4 |
| 6 |
B-1 |
1 |
|
| B |
B |
5-3 |
| 3 |
2 |
1-2 |
| 4 |
B |
2-2 |
| 1 |
6 |
B-1 |
|
| B |
B |
5-3 |
| 2 |
3 |
1-2 |
| B |
4 |
2-2 |
| 6 |
1 |
B-1 |
|
| 1-2 |
3 |
2 |
| 5-3 |
B |
B |
| B-1 |
1 |
6 |
| 2-2 |
4 |
B |
|
| 1-2 |
2 |
3 |
| 5-3 |
B |
B |
| B-1 |
6 |
1 |
| 2-2 |
B |
4 |
|
| 3 |
1-2 |
2 |
| B |
5-3 |
B |
| 1 |
B-1 |
6 |
| 4 |
2-2 |
B |
|
| 2 |
1-2 |
3 |
| B |
5-3 |
B |
| 6 |
B-1 |
1 |
| B |
2-2 |
4 |
|
| 3 |
2 |
1-2 |
| B |
B |
5-3 |
| 1 |
6 |
B-1 |
| 4 |
B |
2-2 |
|
| 2 |
3 |
1-2 |
| B |
B |
5-3 |
| 6 |
1 |
B-1 |
| B |
4 |
2-2 |
|
| 1-2 |
3 |
2 |
| 5-3 |
B |
B |
| 2-2 |
4 |
B |
| B-1 |
1 |
6 |
|
| 1-2 |
2 |
3 |
| 5-3 |
B |
B |
| 2-2 |
B |
4 |
| B-1 |
6 |
1 |
|
| 3 |
1-2 |
2 |
| B |
5-3 |
B |
| 4 |
2-2 |
B |
| 1 |
B-1 |
6 |
|
| 2 |
1-2 |
3 |
| B |
5-3 |
B |
| B |
2-2 |
4 |
| 6 |
B-1 |
1 |
|
| 3 |
2 |
1-2 |
| B |
B |
5-3 |
| 4 |
B |
2-2 |
| 1 |
6 |
B-1 |
|
| 2 |
3 |
1-2 |
| B |
B |
5-3 |
| B |
4 |
2-2 |
| 6 |
1 |
B-1 |
|
Reflections in the diagonals give patterns with the 5-3 domino
placed vertically and reflections in the vertical mirror line give
patterns with this domino placed as 3-5. Reflections in the
horizontal mirror line turn the vertically placed dominoes upside
down.
No-one has found a different solution that is not derived from
this one basic pattern. You may like to investigate patterns such
as the one below.
