The queen and the horse | THE GAME OF SCIENCE | EUROtoday

The drawback of the 18 chips on a 6˟6 board, raised final week, has given rise to attention-grabbing variants and generalizations (see corresponding feedback), similar to Juan Zubieta: “For the 4×4 case, the chips can be placed at the ends of the first two rows and in the center of the last two.

For cases where n It is even, to obtain a solution, any valid solution can be copied 4 times for NXN. Whenever the board is at least 12˟12, each side can be divided by 3 size greater than or equal to 4, so that the 4 corners and the central area will be square, and therefore a valid solution of the corresponding size can be applied; The lateral areas, whether rectangular or square, can be filled by placing the chips such as painting a checker, because they do not affect the restriction of the diagonals. Therefore, if my reasoning is correct, it would only be verified that there is any solution for sizes 6˟6 and 10˟10. Intuitively seem to me that there must be many solutions. ”

That’s proper, there are quite a few potential options. For instance, the aforementioned resolution for case 4˟4 (see determine) can function a foundation for an answer (or a number of) on the 6˟6 board with 18 chips, are you able to see how?

And from an answer for the 8˟8 board (which in flip can begin from the 4˟4 dashboard), can we discover any resolution for the ten˟10 board?

Metaproblema of the 8 queens

As we noticed, that of the 8 queens is the “mother” of the issues that include putting on a chess board – the same – a sure variety of equal chips in order that sure necessities are met.

There are 92 options to the issue of the 8 queens, though solely 12 primarily totally different, because the others could be obtained by rotation and reflection. Each of the 12 primary options could be rotated 90º, 180º and 270º, with what we’ve 4 options that, in flip, can result in one other 4 by reflection; So we can have 12˟8 = 96 options … however, a second, we’ve simply seen that there are 92, the place are the 4 lacking? Without the necessity to look at the 12 primary options, are you able to deduce why there are solely 92 options that, theoretically, ought to be 96?

The horse’s excursions

Chess horses are as standard as queens, if no more, when posing issues with placement and mobility over the checker and its derivatives of roughly containers. One of the oldest and well-known is the issue of Guarini horsesraised within the sixteenth century, which consists of exchanging, respecting the chess guidelines, the positions of two white horses and two black horses situated within the corners of a 3˟3 board, as indicated within the determine. How many actions are a minimum of vital to realize it?

But the “horse problem” par excellence, extensively studied by Euler and different nice mathematicians, consists of discovering the excursions of a chess horse, on a NXN board, in order that it passes as soon as via all of the containers. We know that it’s potential on the 8˟8 standard board (one other factor is to seek out the options), however is it potential for n = 3, 4, 5, 6 and seven? Euler additionally studied the issue in rectangular boards, similar to 3˟4.

Can you discover the journey of a horse that ranging from the higher left field reaches the decrease proper field after having handed as soon as to all of the others?