The Macmahon puzzles | THE GAME OF SCIENCE | EUROtoday
To decide Macmahon’s variety of sq. idea that Fermi would have favored):
“3 colours taken from 4 in 4 give in consequence 81 mixtures. To these 81 it’s a must to take away the mixtures that, rotating the sq., ends in equal coloured figures. We have 3 outcomes with all of the triangles of the identical coloration. These move the filter.
We have 6 outcomes with two colours organized in an alternate manner (plow, for instance), of those we should maintain 3 and despise the opposite 3 which can be generated by turning 90 °.
Of the remaining, 72, which have 3 colours organized non -symmetrically or 2 colours symmetrically not alternated (Aar, for instance), 3 out of 4 are the results of turning 90 °, 180 ° and 270 ° levels one in all them , so we stayed with 18 and despise 54.
81 – 3 – 54 = 24 = 3 + 3 + 18
That is, the variety of totally different mixtures when coloring with three colours a sq. divided into 4 equal triangles is 24 ″.

But, as Juan Zubieta factors out, if the account of the outdated (although it’s superior) appears excessively rudimentary, we will resort to Burnside’s motto (however that’s one other article).
At the second, let’s deal with the attention-grabbing combinatorial prospects of multicolored squares, which lend themselves to suggest totally different challenges. For instance, are you able to type with them a rectangle of 6˟4 such that the perimeters of the items in touch are at all times the identical coloration and your entire perimeter of the rectangle can also be of the identical coloration?
There are additionally attention-grabbing combinatorial prospects utilizing solely a part of the squares. For instance:
-With the 15 items of just one or two colours, construct a rectangle of three˟5.
-With the remaining 9 items, three colours, construct a sq. of three˟3.
As within the first puzzle, the perimeters in touch at all times should be of the identical coloration.
And along with solo puzzles just like the earlier ones, Macmahon’s squares lend themselves to suggest totally different video games for 2 gamers impressed by dominoes (do you consider any?).
In specialised shops there are wooden or plastic variations of the multicolored squares. And it’s not tough to improvise a home made model (for instance, printing a picture of the 24 squares taken from the community, hitting it on a cardboard sheet and reducing the items).
From the sq. to the dice
Going from the airplane to house, and as an extension of the theme of Macmahon’s squares, it’s attainable to ask what number of other ways we will coloration the six sides of a dice utilizing three colours. And the reply is 57 (are you able to show it?).
One solution to handle the issue is to start out with the trivial case of a single coloration (clearly, there is just one solution to coloration a dice utilizing a single coloration), proceed with two colours (what number of totally different cubes may be in white and/or black ?), Solve the case of the 57 cubes with three colours … and, to add observe, acquire a common method for n colours.
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