Part 39

297.--BISHOPS--UNGUARDED.

Place as few bishops as possible on an ordinary chessboard so that every square of the board shall be either occupied or attacked. It will be seen that the rook has more scope than the bishop: for wherever you place the former, it will always attack fourteen other squares; whereas the latter will attack seven, nine, eleven, or thirteen squares, according to the position of the diagonal on which it is placed. And it is well here to state that when we speak of "diagonals" in connection with the chessboard, we do not limit ourselves to the two long diagonals from corner to corner, but include all the shorter lines that are parallel to these. To prevent misunderstanding on future occasions, it will be well for the reader to note carefully this fact.

298.--BISHOPS--GUARDED.

Now, how many bishops are necessary in order that every square shall be either occupied or attacked, and every bishop guarded by another bishop? And how may they be placed?

299.--BISHOPS IN CONVOCATION.

+---+---+---+---+---+---+---+---+ | B | B | B | B | B | B | B | B | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | B | B | B | B | B | B | | +---+---+---+---+---+---+---+---+

The greatest number of bishops that can be placed at the same time on the chessboard, without any bishop attacking another, is fourteen. I show, in diagram, the simplest way of doing this. In fact, on a square chequered board of any number of squares the greatest number of bishops that can be placed without attack is always two less than twice the number of squares on the side. It is an interesting puzzle to discover in just how many different ways the fourteen bishops may be so placed without mutual attack. I shall give an exceedingly simple rule for determining the number of ways for a square chequered board of any number of squares.

300.--THE EIGHT QUEENS.

+---+---+---+---+---+---+---+---+ | | | | ..Q | | | | +---+---+---+...+---+---+---+---+ | | ..Q.. | | | | | +---+...+---+---+---+---+---+---+ | Q.. | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | Q | | +---+---+---+---+---+---+---+---+ | | Q | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | ..Q | +---+---+---+---+---+---+...+---+ | | | | | ..Q.. | | +---+---+---+---+...+---+---+---+ | | | | Q.. | | | | +---+---+---+---+---+---+---+---+

The queen is by far the strongest piece on the chessboard. If you place her on one of the four squares in the centre of the board, she attacks no fewer than twenty-seven other squares; and if you try to hide her in a corner, she still attacks twenty-one squares. Eight queens may be placed on the board so that no queen attacks another, and it is an old puzzle (first proposed by Nauck in 1850, and it has quite a little literature of its own) to discover in just how many different ways this may be done. I show one way in the diagram, and there are in all twelve of these fundamentally different ways. These twelve produce ninety-two ways if we regard reversals and reflections as different. The diagram is in a way a symmetrical arrangement. If you turn the page upside down, it will reproduce itself exactly; but if you look at it with one of the other sides at the bottom, you get another way that is not identical. Then if you reflect these two ways in a mirror you get two more ways. Now, all the other eleven solutions are non-symmetrical, and therefore each of them may be presented in eight ways by these reversals and reflections. It will thus be seen why the twelve fundamentally different solutions produce only ninety-two arrangements, as I have said, and not ninety-six, as would happen if all twelve were non-symmetrical. It is well to have a clear understanding on the matter of reversals and reflections when dealing with puzzles on the chessboard.

Can the reader place the eight queens on the board so that no queen shall attack another and so that no three queens shall be in a straight line in any oblique direction? Another glance at the diagram will show that this arrangement will not answer the conditions, for in the two directions indicated by the dotted lines there are three queens in a straight line. There is only one of the twelve fundamental ways that will solve the puzzle. Can you find it?

301.--THE EIGHT STARS.

+---+---+---+---+---+---+---+---+ |///| | | | | | |///| +---+---+---+---+---+---+---+---+ | |///| | | | |///| * | +---+---+---+---+---+---+---+---+ | | |///| | |///| | | +---+---+---+---+---+---+---+---+ | | | |///|///| | | | +---+---+---+---+---+---+---+---+ | | | |///|///| | | | +---+---+---+---+---+---+---+---+ | | |///| | |///| | | +---+---+---+---+---+---+---+---+ | |///| | | | |///| | +---+---+---+---+---+---+---+---+ |///| | | | | | |///| +---+---+---+---+---+---+---+---+

The puzzle in this case is to place eight stars in the diagram so that no star shall be in line with another star horizontally, vertically, or diagonally. One star is already placed, and that must not be moved, so there are only seven for the reader now to place. But you must not place a star on any one of the shaded squares. There is only one way of solving this little puzzle.

302.--A PROBLEM IN MOSAICS.

The art of producing pictures or designs by means of joining together pieces of hard substances, either naturally or artificially coloured, is of very great antiquity. It was certainly known in the time of the Pharaohs, and we find a reference in the Book of Esther to "a pavement of red, and blue, and white, and black marble." Some of this ancient work that has come down to us, especially some of the Roman mosaics, would seem to show clearly, even where design is not at first evident, that much thought was bestowed upon apparently disorderly arrangements. Where, for example, the work has been produced with a very limited number of colours, there are evidences of great ingenuity in preventing the same tints coming in close proximity. Lady readers who are familiar with the construction of patchwork quilts will know how desirable it is sometimes, when they are limited in the choice of material, to prevent pieces of the same stuff coming too near together. Now, this puzzle will apply equally to patchwork quilts or tesselated pavements.

It will be seen from the diagram how a square piece of flooring may be paved with sixty-two square tiles of the eight colours violet, red, yellow, green, orange, purple, white, and blue (indicated by the initial letters), so that no tile is in line with a similarly coloured tile, vertically, horizontally, or diagonally. Sixty-four such tiles could not possibly be placed under these conditions, but the two shaded squares happen to be occupied by iron ventilators.

+---+---+---+---+---+---+---+---+ | V | R | Y | G | O | P | W | B | +---+---+---+---+---+---+---+---+ | W | B | O | P | Y | G | V | R | +---+---*===*---+---*===*---+---+ | G | P H W H V | B H R H Y | O | +---+---*===*---+---*===*---+---+ | R | Y | B | O | G | V | P | W | +---+---+---+---+---+---+---+---+ | B | G | R | Y | P | W | O | V | +---+---+---+---+---+---+---+---+ | O | V | P | W | R | Y | B | G | +---+---+---+---+---+---+---+---+ | P | W | G | B | V | O | R | Y | +---+---+---+---+---+---+---+---+ |///| O | V | R | W | B | G |///| +---+---+---+---+---+---+---+---+

The puzzle is this. These two ventilators have to be removed to the positions indicated by the darkly bordered tiles, and two tiles placed in those bottom corner squares. Can you readjust the thirty-two tiles so that no two of the same colour shall still be in line?

303.--UNDER THE VEIL.

+---+---+---+---+---+---+---+---+ | | | V | E | I | L | | | +---+---+---+---+---+---+---+---+ | | | I | L | V | E | | | +---+---+---+---+---+---+---+---+ | I | V | | | | | L | E | +---+---+---+---+---+---+---+---+ | L | E | | | | | I | V | +---+---+---+---+---+---+---+---+ | V | I | | | | | E | L | +---+---+---+---+---+---+---+---+ | E | L | | | | | V | I | +---+---+---+---+---+---+---+---+ | | | E | V | L | I | | | +---+---+---+---+---+---+---+---+ | | | L | I | E | V | | | +---+---+---+---+---+---+---+---+

If the reader will examine the above diagram, he will see that I have so placed eight V's, eight E's, eight I's, and eight L's in the diagram that no letter is in line with a similar one horizontally, vertically, or diagonally. Thus, no V is in line with another V, no E with another E, and so on. There are a great many different ways of arranging the letters under this condition. The puzzle is to find an arrangement that produces the greatest possible number of four-letter words, reading upwards and downwards, backwards and forwards, or diagonally. All repetitions count as different words, and the five variations that may be used are: VEIL, VILE, LEVI, LIVE, and EVIL.

This will be made perfectly clear when I say that the above arrangement scores eight, because the top and bottom row both give VEIL; the second and seventh columns both give VEIL; and the two diagonals, starting from the L in the 5th row and E in the 8th row, both give LIVE and EVIL. There are therefore eight different readings of the words in all.

This difficult word puzzle is given as an example of the use of chessboard analysis in solving such things. Only a person who is familiar with the "Eight Queens" problem could hope to solve it.

304.--BACHET'S SQUARE.

One of the oldest card puzzles is by Claude Caspar Bachet de Meziriac, first published, I believe, in the 1624 edition of his work. Rearrange the sixteen court cards (including the aces) in a square so that in no row of four cards, horizontal, vertical, or diagonal, shall be found two cards of the same suit or the same value. This in itself is easy enough, but a point of the puzzle is to find in how many different ways this may be done. The eminent French mathematician A. Labosne, in his modern edition of Bachet, gives the answer incorrectly. And yet the puzzle is really quite easy. Any arrangement produces seven more by turning the square round and reflecting it in a mirror. These are counted as different by Bachet.

Note "row of four cards," so that the only diagonals we have here to consider are the two long ones.

305.--THE THIRTY-SIX LETTER-BLOCKS.