Amusements in Mathematics Part 27

The general solution to the problem is, in fact, this: n C 2n ----- n + 1 where 2n equals the number of barrels. The symbol C, of course, implies that we have to find how many combinations, or selections, we can make of 2n things, taken n at a time.

280.--BUILDING THE TETRAHEDRON.

Take your constructed pyramid and hold it so that one stick only lies on the table. Now, four sticks must branch off from it in different directions--two at each end. Any one of the five sticks may be left out of this connection; therefore the four may be selected in 5 different ways. But these four matches may be placed in 24 different orders. And as any match may be joined at either of its ends, they may further be varied (after their situations are settled for any particular arrangement) in 16 different ways. In every arrangement the sixth stick may be added in 2 different ways. Now multiply these results together, and we get 5 24 16 2 = 3,840 as the exact number of ways in which the pyramid may be constructed. This method excludes all possibility of error.

A common cause of error is this. If you calculate your combinations by working upwards from a basic triangle lying on the table, you will get half the correct number of ways, because you overlook the fact that an equal number of pyramids may be built on that triangle downwards, so to speak, through the table. They are, in fact, reflections of the others, and examples from the two sets of pyramids cannot be set up to resemble one another--except under fourth dimensional conditions!

281.--PAINTING A PYRAMID.

It will be convenient to imagine that we are painting our pyramids on the flat cardboard, as in the diagrams, before folding up. Now, if we take any four colours (say red, blue, green, and yellow), they may be applied in only 2 distinctive ways, as shown in Figs, 1 and 2. Any other way will only result in one of these when the pyramids are folded up. If we take any three colours, they may be applied in the 3 ways shown in Figs. 3, 4, and 5. If we take any two colours, they may be applied in the 3 ways shown in Figs. 6, 7, and 8. If we take any single colour, it may obviously be applied in only 1 way. But four colours may be selected in 35 ways out of seven; three in 35 ways; two in 21 ways; and one colour in 7 ways. Therefore 35 applied in 2 ways = 70; 35 in 3 ways = 105; 21 in 3 ways = 63; and 7 in 1 way = 7. Consequently the pyramid may be painted in 245 different ways (70 + 105 + 63 + 7), using the seven colours of the solar spectrum in accordance with the conditions of the puzzle.

[Ill.u.s.tration: 1 2 +---------------+ +---------------+ R / B / B / R / / / / / / G / / G / -------/ -------/ / / Y / Y / / / ' '

3 4 5 +---------------+ +---------------+ +---------------+ R / R / R / G / Y / R / / / / / / / / G / / G / / G / -------/ -------/ -------/ / / / Y / Y / Y / / / / ' ' '

6 7 8 +---------------+ +---------------+ +---------------+ G / Y / Y / Y / G / G / / / / / / / / G / / G / / G / -------/ -------/ -------/ / / / Y / Y / Y / / / / ' ' '

282.--THE ANTIQUARY'S CHAIN.

[Ill.u.s.tration]

THE number of ways in which nine things may be arranged in a row without any restrictions is 1 2 3 4 5 6 7 8 9 = 362,880. But we are told that the two circular rings must never be together; therefore we must deduct the number of times that this would occur. The number is 1 2 3 4 5 6 7 8 = 40,320 2 = 80,640, because if we consider the two circular links to be inseparably joined together they become as one link, and eight links are capable of 40,320 arrangements; but as these two links may always be put on in the orders AB or BA, we have to double this number, it being a question of arrangement and not of design. The deduction required reduces our total to 282,240. Then one of our links is of a peculiar form, like an 8. We have therefore the option of joining on either one end or the other on every occasion, so we must double the last result. This brings up our total to 564,480.

We now come to the point to which I directed the reader's attention--that every link may be put on in one of two ways. If we join the first finger and thumb of our left hand horizontally, and then link the first finger and thumb of the right hand, we see that the right thumb may be either above or below. But in the case of our chain we must remember that although that 8-shaped link has two independent ends it is like every other link in having only two _sides_--that is, you cannot turn over one end without turning the other at the same time.

We will, for convenience, a.s.sume that each link has a black side and a side painted white. Now, if it were stipulated that (with the chain lying on the table, and every successive link falling over its predecessor in the same way, as in the diagram) only the white sides should be uppermost as in A, then the answer would be 564,480, as above--ignoring for the present all reversals of the completed chain. If, however, the first link were allowed to be placed either side up, then we could have either A or B, and the answer would be 2 564,480 = 1,128,960; if two links might be placed either way up, the answer would be 4 564,480; if three links, then 8 564,480, and so on. Since, therefore, every link may be placed either side up, the number will be 564,480 multiplied by 2^9, or by 512. This raises our total to 289,013,760.

But there is still one more point to be considered. We have not yet allowed for the fact that with any given arrangement three of the other arrangements may be obtained by simply turning the chain over through its entire length and by reversing the ends. Thus C is really the same as A, and if we turn this page upside down, then A and C give two other arrangements that are still really identical. Thus to get the correct answer to the puzzle we must divide our last total by 4, when we find that there are just 72,253,440 different ways in which the smith might have put those links together. In other words, if the nine links had originally formed a piece of chain, and it was known that the two circular links were separated, then it would be 72,253,439 chances to 1 that the smith would not have put the links together again precisely as they were arranged before!

283.--THE FIFTEEN DOMINOES.

The reader may have noticed that at each end of the line I give is a four, so that, if we like, we can form a ring instead of a line. It can easily be proved that this must always be so. Every line arrangement will make a circular arrangement if we like to join the ends. Now, curious as it may at first appear, the following diagram exactly represents the conditions when we leave the doubles out of the question and devote our attention to forming circular arrangements. Each number, or half domino, is in line with every other number, so that if we start at any one of the five numbers and go over all the lines of the pentagon once and once only we shall come back to the starting place, and the order of our route will give us one of the circular arrangements for the ten dominoes. Take your pencil and follow out the following route, starting at the 4: 41304210234. You have been over all the lines once only, and by repeating all these figures in this way, 41--13--30--04--42--21--10--02--23--34, you get an arrangement of the dominoes (without the doubles) which will be perfectly clear. Take other routes and you will get other arrangements. If, therefore, we can ascertain just how many of these circular routes are obtainable from the pentagon, then the rest is very easy.

Well, the number of different circular routes over the pentagon is 264. How I arrive at these figures I will not at present explain, because it would take a lot of s.p.a.ce. The dominoes may, therefore, be arranged in a circle in just 264 different ways, leaving out the doubles. Now, in any one of these circles the five doubles may be inserted in 2^5 = 32 different ways. Therefore when we include the doubles there are 264 32 = 8,448 different circular arrangements. But each of those circles may be broken (so as to form our straight line) in any one of 15 different places. Consequently, 8,448 15 gives 126,720 different ways as the correct answer to the puzzle.

[Ill.u.s.tration: ----- | | / | | / ----- / . . ----- . . ----- | | . . | o o | | o | -.--------.--- | | | | ... | o o | ----- .... ----- ... . / ----- .. ----- | o | . . |o | | | --------- | o | | o |. .| o| ----- ----- ]

I purposely refrained from asking the reader to discover in just how many different ways the full set of twenty-eight dominoes may be arranged in a straight line in accordance with the ordinary rules of the game, left to right and right to left of any arrangement counting as different ways. It is an exceedingly difficult problem, but the correct answer is 7,959,229,931,520 ways. The method of solving is very complex.

284.--THE CROSS TARGET.

[Ill.u.s.tration: -- -- (CD)( ) -- -- (AE)(A ) -- -- -- -- -- -- (CE)(E )(A )(AB)(C )(D ) -- -- -- -- -- -- (D )( )(B )(E )(EB)( ) -- -- -- -- -- -- (C )(B ) -- -- ( )(ED) -- -- ]

Twenty-one different squares may be selected. Of these nine will be of the size shown by the four A's in the diagram, four of the size shown by the B's, four of the size shown by the C's, two of the size shown by the D's, and two of the size indicated by the upper single A, the upper single E, the lower single C, and the EB. It is an interesting fact that you cannot form any one of these twenty-one squares without using at least one of the six circles marked E.

285.--THE FOUR POSTAGE STAMPS.

Referring to the original diagram, the four stamps may be given in the shape 1, 2, 3, 4, in three ways; in the shape 1, 2, 5, 6, in six ways; in the shape 1, 2, 3, 5, or 1, 2, 3, 7, or 1, 5, 6, 7, or 3, 5, 6, 7, in twenty-eight ways; in shape 1, 2, 3, 6, or 2, 5, 6, 7, in fourteen ways; in shape 1, 2, 6, 7, or 2, 3, 5, 6, or 1, 5, 6, 10, or 2, 5, 6, 9, in fourteen ways. Thus there are sixty-five ways in all.

286.--PAINTING THE DIE.

The 1 can be marked on any one of six different sides. For every side occupied by 1 we have a selection of four sides for the 2. For every situation of the 2 we have two places for the 3. (The 6, 5, and 4 need not be considered, as their positions are determined by the 1, 2, and 3.) Therefore 6, 4, and 2 multiplied together make 48 different ways--the correct answer.

287.--AN ACROSTIC PUZZLE.

There are twenty-six letters in the alphabet, giving 325 different pairs. Every one of these pairs may be reversed, making 650 ways. But every initial letter may be repeated as the final, producing 26 other ways. The total is therefore 676 different pairs. In other words, the answer is the square of the number of letters in the alphabet.

288.--CHEQUERED BOARD DIVISIONS.

There are 255 different ways of cutting the board into two pieces of exactly the same size and shape. Every way must involve one of the five cuts shown in Diagrams A, B, C, D, and E. To avoid repet.i.tions by reversal and reflection, we need only consider cuts that enter at the points a, b, and c. But the exit must always be at a point in a straight line from the entry through the centre. This is the most important condition to remember. In case B you cannot enter at a, or you will get the cut provided for in E. Similarly in C or D, you must not enter the key-line in the same direction as itself, or you will get A or B. If you are working on A or C and entering at a, you must consider joins at one end only of the key-line, or you will get repet.i.tions. In other cases you must consider joins at both ends of the key; but after leaving a in case D, turn always either to right or left--use one direction only. Figs. 1 and 2 are examples under A; 3 and 4 are examples under B; 5 and 6 come under C; [Ill.u.s.tration]

and 7 is a pretty example of D. Of course, E is a peculiar type, and obviously admits of only one way of cutting, for you clearly cannot enter at b or c.

Here is a table of the results:-- a b c Ways. A = 8 + 17 + 21 = 46 B = 0 + 17 + 21 = 38 C = 15 + 31 + 39 = 85 D = 17 + 29 + 39 = 85 E = 1 + 0 + 0 = 1 -- -- -- --- 41 94 120 255 I have not attempted the task of enumerating the ways of dividing a board 8 8--that is, an ordinary chessboard. Whatever the method adopted, the solution would entail considerable labour.

289.--LIONS AND CROWNS.

[Ill.u.s.tration]

Here is the solution. It will be seen that each of the four pieces (after making the cuts along the thick lines) is of exactly the same size and shape, and that each piece contains a lion and a crown. Two of the pieces are shaded so as to make the solution quite clear to the eye.

290.--BOARDS WITH AN ODD NUMBER OF SQUARES.

There are fifteen different ways of cutting the 5 5 board (with the central square removed) into two pieces of the same size and shape. Limitations of s.p.a.ce will not allow me to give diagrams of all these, but I will enable the reader to draw them all out for himself without the slightest difficulty. At whatever point on the edge your cut enters, it must always end at a point on the edge, exactly opposite in a line through the centre of the square. Thus, if you enter at point 1 (see Fig. 1) at the top, you must leave at point 1 at the bottom. Now, 1 and 2 are the only two really different points of entry; if we use any others they will simply produce similar solutions. The directions of the cuts in the following fifteen [Ill.u.s.tration: Fig. 1. Fig. 2.]

solutions are indicated by the numbers on the diagram. The duplication of the numbers can lead to no confusion, since every successive number is contiguous to the previous one. But whichever direction you take from the top downwards you must repeat from the bottom upwards, one direction being an exact reflection of the other.

1, 4, 8. 1, 4, 3, 7, 8. 1, 4, 3, 7, 10, 9. 1, 4, 3, 7, 10, 6, 5, 9. 1, 4, 5, 9. 1, 4, 5, 6, 10, 9. 1, 4, 5, 6, 10, 7, 8. 2, 3, 4, 8. 2, 3, 4, 5, 9. 2, 3, 4, 5, 6, 10, 9. 2, 3, 4, 5, 6, 10, 7, 8. 2, 3, 7, 8. 2, 3, 7, 10, 9. 2, 3, 7, 10, 6, 5, 9. 2, 3, 7, 10, 6, 5, 4, 8.

It will be seen that the fourth direction (1, 4, 3, 7, 10, 6, 5, 9) produces the solution shown in Fig. 2. The thirteenth produces the solution given in propounding the puzzle, where the cut entered at the side instead of at the top. The pieces, however, will be of the same shape if turned over, which, as it was stated in the conditions, would not const.i.tute a different solution.

291.--THE GRAND LAMA'S PROBLEM.

The method of dividing the chessboard so that each of the four parts shall be of exactly the same size and shape, and contain one of the gems, is shown in the diagram. The method of shading the squares is adopted to make the shape of the pieces clear to the eye. Two of the pieces are shaded and two left white.

The reader may find it interesting to compare this puzzle with that of the "Weaver" (No. 14, _Canterbury Puzzles_).

[Ill.u.s.tration: THE GRAND LAMA'S PROBLEM.

+===+===+===+===+===+===+===+===+ |:o:| : : : : : : : I...I...+===+===+===+===+===+===+ |:::| o |:::::::::::::::::::::::| I...I...I...+===+===+===+===+...I |:::| |:o:| : : : |:::| I...I...I...I...I===+===+...I...I |:::| |:::| o |:::::::| |:::| I...I...I...+===I===+...I...I...I |:::| |:::::::| |:::| |:::| I...I...+===+===+...+...I...I...I |:::| : : : |:::| |:::| I...+===+===+===+===I...I...I...I |:::::::::::::::::::::::| |:::| +===+===+===+===+===+===+...I...I | : : : : : : |:::| +===+===+===+===+===+===+===+===+ ]

292.--THE ABBOT'S WINDOW.

THE man who was "learned in strange mysteries" pointed out to Father John that the orders of the Lord Abbot of St. Edmondsbury might be easily carried out by blocking up twelve of the lights in the window as shown by the dark squares in the following sketch:-- [Ill.u.s.tration: +===+===+===+===+===+===+===+===+ | : : : : : : : | I...+===+...+...+...+...+===+...I | IIIII : : : IIIII | I...+===+===+...+...+===+===+...I | : IIIII : IIIII : | I...+...+===+===+===+===+...+...I | : : IIIIIIIII : : | I...+...+...+===+===+...+...+...I | : : IIIIIIIII : : | I...+...+===+===+===+===+...+...I | : IIIII : IIIII : | I...+===+===+...+...+===+===+...I | IIIII : : : IIIII | I...+===+...+...+...+...+===+...I | : : : : : : : | +===+===+===+===+===+===+===+===+ ]

Father John held that the four corners should also be darkened, but the sage explained that it was desired to obstruct no more light than was absolutely necessary, and he said, antic.i.p.ating Lord Dundreary, "A single pane can no more be in a line with itself than one bird can go into a corner and flock in solitude. The Abbot's condition was that no diagonal lines should contain an odd number of lights."

Now, when the holy man saw what had been done he was well pleased, and said, "Truly, Father John, thou art a man of deep wisdom, in that thou hast done that which seemed impossible, and yet withal adorned our window with a device of the cross of St. Andrew, whose name I received from my G.o.dfathers and G.o.dmothers." Thereafter he slept well and arose refreshed. The window might be seen intact to-day in the monastery of St. Edmondsbury, if it existed, which, alas! the window does not.

293.--THE CHINESE CHESSBOARD.

+===I===+===+===+===I===+===+===+ | |:::: 2 ::::| 3 |:::| 5 |:6:| I...+===+...+===+...I...I...+===I |:::: 1 |:::| ::::| 4 |:::| 7 | I...+===+===+...I===I...I===+===I | |:::: |:::| ::::| 9 |:::| I===I...I===============I...I...I |:::: 11|:::: ::::: 10|:::| 8 | I=======I===I===========I...I...I | ::::: 12|:::: 13::::| |:::| I=======+...I...+===+===|===+===I |:::: 14|:::| |:::| 16::::| 17| I...+...I===I===+...+...+===+...I | ::::| ::::: 15|:::| ::::| I=======+===========+===+=======I |:::: ::::: 18::::: ::::: | +===+===+===+===+===+===+===+===+ +===+===I===I===+===I===+===+===+ | ::::| |:::: |:::| ::::| I...+===I...I=======I...I===+...I |:::| |:::: |:::: |:::| | I...I===I===============I===I...I | |:::: ::::| ::::: |:::| I===I=======I=======I=======I===I |:::| ::::| ::::| ::::| | I...I===+...I...+...I...+===+...I | ::::| |:::: |:::| ::::| I...+===I...+===I===+...I===+...I |:::| |:::: |:::: |:::| | I===I...+=======I=======+...I===I | |:::: ::::| ::::: |:::| I...+=======+...I...+=======+...I |:::: ::::| |:::| ::::: | +===+===+===+===+===+===+===+===+ Eighteen is the maximum number of pieces. I give two solutions. The numbered diagram is so cut that the eighteenth piece has the largest area--eight squares--that is possible under the conditions. The second diagram was prepared under the added condition that no piece should contain more than five squares.

No. 74 in The Canterbury Puzzles shows how to cut the board into twelve pieces, all different, each containing five squares, with one square piece of four squares.

294.--THE CHESSBOARD SENTENCE.

+===I===I===I===I=======I=======+ | |:::| |:::| ::::| ::::| I===I...I===I...I...+===I...+===I |:::| ::::: |:::| ::::: | |...|...+===I...I...+===+...+===I | |:::| |:::| ::::| ::::| |...+===+...+===I===I===I=======I |:::: ::::: |:::| ::::: | I===========I===I...I===I===+...| | ::::: |:::| |:::| |:::| |...+===+...|...|...|...I===+...| |:::| |:::| |:::| |:::: | |...|...|...|...I===+...+===+...| | |:::| |:::| ::::: |:::| I===+...+===I...+=======I===+...| |:::: ::::| ::::: |:::: | +===========I===================+ The pieces may be fitted together, as shown in the ill.u.s.tration, to form a perfect chessboard.

295.--THE EIGHT ROOKS.

Obviously there must be a rook in every row and every column. Starting with the top row, it is clear that we may put our first rook on any one of eight different squares. Wherever it is placed, we have the option of seven squares for the second rook in the second row. Then we have six squares from which to select the third row, five in the fourth, and so on. Therefore the number of our different ways must be 8 7 6 5 4 3 2 1 = 40,320 (that is 8!), which is the correct answer.

How many ways there are if mere reversals and reflections are not counted as different has not yet been determined; it is a difficult problem. But this point, on a smaller square, is considered in the next puzzle.

296.--THE FOUR LIONS.

There are only seven different ways under the conditions. They are as follows: 1 2 3 4, 1 2 4 3, 1 3 2 4, 1 3 4 2, 1 4 3 2, 2 1 4 3, 2 4 1 3. Taking the last example, this notation means that we place a lion in the second square of first row, fourth square of second row, first square of third row, and third square of fourth row. The first example is, of course, the one we gave when setting the puzzle.

297.--BISHOPS--UNGUARDED.

+...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+...+...+ ::::: ::::: ::::: ::::: : +...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+...+...+ ::B:: B ::B:: B ::B:: B ::B:: B : +...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+...+...+ ::::: ::::: ::::: ::::: : +...+...+...+...+...+...+...+...+ ::::: ::::: ::::: ::::: : +...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+...+...+ This cannot be done with fewer bishops than eight, and the simplest solution is to place the bishops in line along the fourth or fifth row of the board (see diagram). But it will be noticed that no bishop is here guarded by another, so we consider that point in the next puzzle.

298.--BISHOPS--GUARDED.

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his puzzle is quite easy if you first of all give it a little thought. You need only consider squares of one colour, for whatever can be done in the case of the white squares can always be repeated on the black, and they are here quite independent of one another. This equality, of course, is in consequence of the fact that the number of squares on an ordinary chessboard, sixty-four, is an even number. If a square chequered board has an odd number of squares, then there will always be one more square of one colour than of the other.

Ten bishops are necessary in order that every square shall be attacked and every bishop guarded by another bishop. I give one way of arranging them in the diagram. It will be noticed that the two central bishops in the group of six on the left-hand side of the board serve no purpose, except to protect those bishops that are on adjoining squares. Another solution would therefore be obtained by simply raising the upper one of these one square and placing the other a square lower down.

299.--BISHOPS IN CONVOCATION.

The fourteen bishops may be placed in 256 different ways. But every bishop must always be placed on one of the sides of the board--that is, somewhere on a row or file on the extreme edge. The puzzle, therefore, consists in counting the number of different ways that we can arrange the fourteen round the edge of the board without attack. This is not a difficult matter. On a chessboard of n squares 2n - 2 bishops (the maximum number) may always be placed in 2^n ways without attacking. On an ordinary chessboard n would be 8; therefore 14 bishops may be placed in 256 different ways. It is rather curious that the general result should come out in so simple a form.

[Ill.u.s.tration]

300.--THE EIGHT QUEENS.

[Ill.u.s.tration]

The solution to this puzzle is shown in the diagram. It will be found that no queen attacks another, and also that no three queens are in a straight line in any oblique direction. This is the only arrangement out of the twelve fundamentally different ways of placing eight queens without attack that fulfils the last condition.

301.--THE EIGHT STARS.

The solution of this puzzle is shown in the first diagram. It is the only possible solution within the conditions stated. But if one of the eight stars had not already been placed as shown, there would then have been eight ways of arranging the stars according to this scheme, if we count reversals and reflections as different. If you turn this page round so that each side is in turn at the bottom, you will get the four reversals; and if you reflect each of these in a mirror, you will get the four reflections. These are, therefore, merely eight aspects of one "fundamental solution." But without that first star being so placed, there is another fundamental solution, as shown in the second diagram. But this arrangement being in a way symmetrical, only produces four different aspects by reversal and reflection.

[Ill.u.s.tration]

302.--A PROBLEM IN MOSAICS.

[Ill.u.s.tration]

The diagram shows how the tiles may be rearranged. As before, one yellow and one purple tile are dispensed with. I will here point out that in the previous arrangement the yellow and purple tiles in the seventh row might have changed places, but no other arrangement was possible.

303.--UNDER THE VEIL.

Some schemes give more diagonal readings of four letters than others, and we are at first tempted to favour these; but this is a false scent, because what you appear to gain in this direction you lose in others. Of course it immediately occurs to the solver that every LIVE or EVIL is worth twice as much as any other word, since it reads both ways and always counts as 2. This is an important consideration, though sometimes those arrangements that contain most readings of these two words are fruitless in other words, and we lose in the general count.

[Ill.u.s.tration: _ _ I V E L _ _ E V L _ _ I _ _ L _ _ I _ _ V E I _ V E _ _ _ L _ E _ _ L V _ I _ L I _ _ I _ E V /V _ E L _ _ I _ _ I _ _ V E L _ ]

The above diagram is in accordance with the conditions requiring no letter to be in line with another similar letter, and it gives twenty readings of the five words--six horizontally, six vertically, four in the diagonals indicated by the arrows on the left, and four in the diagonals indicated by the arrows on the right. This is the maximum.

Four sets of eight letters may be placed on the board of sixty-four squares in as many as 604 different ways, without any letter ever being in line with a similar one. This does not count reversals and reflections as different, and it does not take into consideration the actual permutations of the letters among themselves; that is, for example, making the L's change places with the E's. Now it is a singular fact that not only do the twenty word-readings that I have given prove to be the real maximum, but there is actually only that one arrangement from which this maximum may be obtained. But if you make the V's change places with the I's, and the L's with the E's, in the solution given, you still get twenty readings--the same number as before in every direction. Therefore there are two ways of getting the maximum from the same arrangement. The minimum number of readings is zero--that is, the letters can be so arranged that no word can be read in any of the directions.

304.--BACHET'S SQUARE.

[Ill.u.s.tration: 1]

[Ill.u.s.tration: 2]

[Ill.u.s.tration: 3]

[Ill.u.s.tration: 4]

Let us use the letters A, K, Q, J, to denote ace, king, queen, jack; and D, S, H, C, to denote diamonds, spades, hearts, clubs. In Diagrams 1 and 2 we have the two available ways of arranging either group of letters so that no two similar letters shall be in line--though a quarter-turn of 1 will give us the arrangement in 2. If we superimpose or combine these two squares, we get the arrangement of Diagram 3, which is one solution. But in each square we may put the letters in the top line in twenty-four different ways without altering the scheme of arrangement. Thus, in Diagram 4 the S's are similarly placed to the D's in 2, the H's to the S's, the C's to the H's, and the D's to the C's. It clearly follows that there must be 2424 = 576 ways of combining the two primitive arrangements. But the error that Labosne fell into was that of a.s.suming that the A, K, Q, J must be arranged in the form 1, and the D, S, H, C in the form 2. He thus included reflections and half-turns, but not quarter-turns. They may obviously be interchanged. So that the correct answer is 2 576 = 1,152, counting reflections and reversals as different. Put in another manner, the pairs in the top row may be written in 16 9 4 1 = 576 different ways, and the square then completed in 2 ways, making 1,152 ways in all.

305.--THE THIRTY-SIX LETTER BLOCKS.

I pointed out that it was impossible to get all the letters into the box under the conditions, but the puzzle was to place as many as possible.

This requires a little judgment and careful investigation, or we are liable to jump to the hasty conclusion that the proper way to solve the puzzle must be first to place all six of one letter, then all six of another letter, and so on. As there is only one scheme (with its reversals) for placing six similar letters so that no two shall be in a line in any direction, the reader will find that after he has placed four different kinds of letters, six times each, every place is occupied except those twelve that form the two long diagonals. He is, therefore, unable to place more than two each of his last two letters, and there are eight blanks left. I give such an arrangement in Diagram 1.

[Ill.u.s.tration: 1]

[Ill.u.s.tration: 2]

The secret, however, consists in not trying thus to place all six of each letter. It will be found that if we content ourselves with placing only five of each letter, this number (thirty in all) may be got into the box, and there will be only six blanks. But the correct solution is to place six of each of two letters and five of each of the remaining four. An examination of Diagram 2 will show that there are six each of C and D, and five each of A, B, E, and F. There are, therefore, only four blanks left, and no letter is in line with a similar letter in any direction.

306.--THE CROWDED CHESSBOARD.

[Ill.u.s.tration]

Here is the solution. Only 8 queens or 8 rooks can be placed on the board without attack, while the greatest number of bishops is 14, and of knights 32. But as all these knights must be placed on squares of the same colour, while the queens occupy four of each colour and the bishops 7 of each colour, it follows that only 21 knights can be placed on the same colour in this puzzle. More than 21 knights can be placed alone on the board if we use both colours, but I have not succeeded in placing more than 21 on the "crowded chessboard." I believe the above solution contains the maximum number of pieces, but possibly some ingenious reader may succeed in getting in another knight.

307.--THE COLOURED COUNTERS.

The counters may be arranged in this order:-- R1, B2, Y3, O4, GS. Y4, O5, G1, R2, B3. G2, R3, B4, Y5, O1. B5, Y1, O2, G3, R4. O3, G4, R5, B1, Y2.