With equal rhombuses it is easy to obtain aperiodic tessellations (see previous article), for example, by means of spiral groupings, like the one shown in the figure.
As we saw, the irregular tessellations and the Penrose tiles were made known, in large part, thanks to the magnificent informative work of Martin Gardner in his section of mathematical games of
Scientific American
.
One of the compilations of articles in this section is entitled precisely
Penrose tiles to trapdoor ciphers
, and in reviewing it as a result of the previous article I rediscovered a curious problem that I submit to the consideration of my astute readers: Can we place 16 horses in a chess board so that each of them threatens exactly 4 others?
And an additional question that is also a clue: How is the problem of the 16 horses related to the fourth dimension?
The peculiar way of moving of the chess knight makes it especially suitable for elaborating all kinds of puzzles and interesting configurations.
The quintessential "knight problem", which we have already dealt with on occasion, consists of going around the board with the knight passing one and only once through all the squares (which is called a polygraph).
The polygraphs of the horse have been studied by great mathematicians, including Euler, who discovered one such that, by numbering the boxes according to the order in which the jumping chess visits them, a magic square of order 8 is obtained.
The oldest puzzle with the chess horse as the protagonist, or at least the most famous among the “classics”, is the Guarini problem, so named because it appears in a text by this author from the 16th century, although in reality it is much more ancient:
Two white knights and two black knights are placed in the four corners of a square board with nine squares, and the white knights must be transferred to the place occupied by the black ones and vice versa, moving them as established by the chess rules and without leaving the board.
There are also numerous, and some very interesting, conventional chess problems with the knight as the protagonist, like this one by Grigoriev from 1932: Can the white knight stop the black pawn and prevent his coronation?
And if we go to the game itself from the problems, we cannot fail to mention the “opening of the four knights”, which consists of both players, after advancing their respective king pawns two squares, place their two knights on the columns. bishop.
This opening enjoyed great popularity until the 1930s, when it fell into disuse;
but not definitely, as it has been revalued in recent times.
"Magic" horses
An Amazon is a powerful warrior on horseback, and in fantasy chess a piece that combines the movements of the queen and the horse is called this way.
It is the most popular of the so-called "magic pieces", but not the only one: the general moves like the rook and the knight at the same time, and the cardinal, like the bishop and the knight;
not forgetting the centurion, who is a super horse that, in addition to its usual movements, can move to any square located two away, both diagonally and orthogonally.
Can any of these equine hybrids checkmate on their own, without the support of other pieces, on a clear board?
Carlo Frabetti
is a writer and mathematician, member of the New York Academy of Sciences.
He has published more than 50 popular science works for adults, children and young people, including 'Damn physics', 'Damn maths' or 'The great game'.
He was a screenwriter for 'La bola de cristal'.
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