Jigsaw puzzle

“ Widow Wilson has a smart son, who is reputed to have once won a prize for puzzle-solving. He asserts that as he cannot find any solution to the problem it must have something to do with the squaring of the circle, the duplication of the cube, or the trisection of an angle; at any rate, he has never before seen a puzzle on the principle, and he gives it up. ”

“ They may move in any order one step at a time, or jumping over one of the opposite colour to the place beyond, just as we play draughts to-day. The only other condition is that when all the frogs have changed sides, the 1 must be where the 12 now is and the 12 in the place now occupied by 1. The puzzle was to perform the feat in as few moves as possible. How many moves are necessary? ”

“ But it is generally understood that in dissection puzzles you are allowed to turn pieces over unless it is distinctly stated that you may not do so. And very often a puzzle is greatly improved by the added condition, "no piece may be turned over." I have often made puzzles, too, in which the diagram has a small repeated pattern, and the pieces have then so to be cut that not only is there no turning over, but the pattern has to be matched, which cannot be done if the pieces are turned round, even with the proper side uppermost. ”

“ But a sapient reader made all the people swim across without using the boat at all! Of course, some few puzzles are intended to be solved by some trick of this kind; and if there happens to be no solution without the trick it is perfectly legitimate. We have to use our best judgment as to whether a puzzle contains a catch or not; but we should never hastily assume it. To quibble over the conditions is the last resort of the defeated would-be solver. ”

“ It should be noted again that it would have been much easier to locate the cuts in the reverse puzzle of cutting the cross to form a half-square triangle.
Another ideal that the puzzle maker always keeps in mind is to contrive that there shall, if possible, be only one correct solution. Thus, in the case of the first puzzle, if we only require that a Greek cross shall be cut into four pieces to form a square, there is, as I have shown, an infinite number of different solutions. ”

“ Now we will try another card puzzle. Cut a piece of card or paper in the shape of a horse-shoe, and mark on it the places for the nails as represented in the subjoined sketch. [Pg 241]
The puzzle is with two cuts to divide it into six parts, each part containing one nail.
Of course you cannot do it; we could not do it ourselves, and had to get the white mice to show us the way.
Somehow or another we never can find out anything with half a dozen taper fingers fluttering before our eyes. They bewilder us terribly, getting between the feet of our ideas, so to speak, and tripping us up ”

“ I have already said that the ideal solution to a cutting-out puzzle is always that which requires the fewest possible pieces. We have just seen that two crosses of the same size may be cut out of a square in five pieces. The reader who succeeded in solving this perhaps asked himself: "Can it be done in fewer pieces?" ”

“ The commonest of all puzzles are coloured maps, pictures (“ jig-saw ”) or designs, dissected into numerous variously shaped pieces, to be fitted together to form the complete design. A great number of puzzles are based on mathematical principles, such as the “ fifteen puzzle,” the “ railway shunting puzzle,” and the like. ”

“ Here is an extension of the last game of solitaire. All you need is a chessboard and the thirty-two pieces, or the same number of draughts or counters. In the illustration numbered counters are used. The puzzle is to remove all the counters except two, and these two must have originally been on the same side of the board; that is, the two left must either belong to the group 1 to 16 or to the other group, 17 to 32. You remove a counter by jumping over it with another counter to the next square beyond, if that square is vacant, but you cannot make a leap in a diagonal direction. ”

“ And the puzzle which that boy accidentally discovered is now in all the old puzzle books.
Puzzles can be made out of almost anything, in the hands of the ingenious person with an idea. Coins, matches, cards, counters, bits of wire or string, all come in useful. An immense number of puzzles have been made out of the letters of the alphabet, and from those nine little digits and cipher, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0.
It should always be remembered that a very simple person may propound a problem that can only be solved by clever heads—if at all. ”

“ But in the meantime, as a temporary measure, I hold what I call the doctrine of the jig-saw puzzle. That is: this remarkable occurrence, and that, and the other may be, and usually are, of no significance. Coincidence and chance and unsearchable causes will now and again make clouds that are undeniable fiery dragons, and potatoes that resemble Eminent Statesmen exactly and minutely in every feature, and rocks that are like eagles and lions. All this is nothing; ”

“ Whether this was so or not, it is certain that all good dissection puzzles (for the nursery type of jig-saw puzzle, which merely consists in cutting up a picture into pieces to be put together again, is not worthy of serious consideration) are really based on geometrical laws. This statement need not, however, frighten off the novice, for it means little more than this, that geometry will give us the "reason why," if we are interested in knowing it, though the solutions may often be discovered by any intelligent person after the exercise of patience, ingenuity, and common sagacity. ”

“ The rule, in the case of my puzzle (where the moves resemble rook moves in chess, with the added condition that the rook may only move to an adjoining square) , is simply this. Where the number of squares on the same row, between the man or woman and the hog, is odd, the hog can never be captured; where the number of squares is even, a capture is possible. The number of squares between Hendrick and the black hog, and between Katrün and the white hog, is 1 (an odd number) , therefore these individuals cannot catch the animals they are facing. ”

“ Arrange the blocks so as to form the two multiplication sums 915 × 64 and 732 × 80, and the product in both cases will be the same: 58,560. [Pg 239]
94.—Foxes and Geese.
The smallest possible number of moves is twenty-two—that is, eleven for the foxes and eleven for the geese. Here is one way of solving the puzzle:
Of course, the reader will play the first move in the top line, then the first move in the second line, then the second move in the top line, and so on alternately. ”

“ The several pieces of wood were not only of different shapes and sizes, but they were as thin as the thinnest veneer; yet the box had been formed by simply joining them together. The man who made that box must have been possessed of ingenuity worthy of a better cause.
I perceived how the puzzle had been worked. ”

“ Geometrical Puzzle shown in Fig. 56. Each of these consists of a capital letter divided up by one or two straight lines into right-angled triangles and other geometrical shapes. While very simple to look at when completed, these little puzzles are by no means easy to solve when the odd pieces are given in a jumbled state. The capital letters should be drawn on a piece of cigar-box wood, and then carefully cut out with a fret saw, or, better still, with a tenon saw if you have one. If you cannot manage wood, then the puzzle can be done in stout cardboard and cut out with a sharp thin knife. ”

“ Place eight checker men upon the checker board as shown in the first row in the sketch. The puzzle is to get them in four piles of two men each without omitting to jump over two checker men every time a move is made. ”

“ The general solution in the case of this puzzle is 3n2 + 2n - 2 moves, where the number of frogs of each colour is n. The law governing the sequence of moves is easily discovered by an examination of the simpler cases, where n = 2, 3, and 4. ”

“ Now if you follow the same procedure with the left-hand reel you will find that the centre loop is released and can be pulled through the centre hole. Then will the two reels slide down side by side.
[pg 48] One thoroughly entertaining and, to a certain extent, bewildering puzzle is
The Three-hole Puzzle.—Really the puzzle consists of a piece of thin wood with three holes cut in it. These three holes are respectively circular, square, and triangular (Fig. 58) . The problem is to cut one block of wood which will pass through each hole and at the same time fit the hole exactly. ”

“ The practical usefulness of puzzles is a point that we are liable to overlook. Yet, as a matter of fact, I have from time to time received quite a large number of letters from individuals who have found that the mastering of some little principle upon which a puzzle was built has proved of considerable value to them in a most unexpected way. Indeed, it may be accepted as a good maxim that a puzzle is of little real value unless, as well as being amusing and perplexing, it conceals some instructive and possibly useful feature. ”

“ This variation of the last puzzle is also played by two persons. One puts a counter on No. 6, and the other puts one on No. 55, and they play alternately by removing the counter to any other number in a line. If your opponent moves at any time on to one of the lines you occupy, or even crosses one of your lines, you immediately capture him and win. ”

“ Pg 114The puzzle is to find the shortest way of performing the feat, which in itself is not difficult. Remember, a person cannot help himself by hanging on to the rope, the only way being to go down "with a bump," with the weight in the other basket as a counterpoise.
PROBLEMS CONCERNING GAMES. "The little pleasure of the game."
MATTHEW PRIOR.
Every game lends itself to the propounding of a variety of puzzles. They can be made, as we have seen, out of the chessboard and the peculiar moves of the chess pieces. ”

“ When this knight was called on to propound a puzzle, he said to the company, "This riddle a wight did ask of me when that I fought with the lord of Palatine against the heathen in Turkey. In thy hand take a piece of chalk and learn how many perfect squares thou canst make with one of the eighty-seven roses at each corner thereof." The reader may find it an interesting problem to count the number of squares that may be formed on the shield by uniting four roses. ”

“ The following is the simple solution of the three nines puzzle:—
To divide 18 by .9 (or nine-tenths) we, of course, multiply by 10 and divide by 9. The result is 20, as required. [Pg 248]
109.—Noughts and Crosses.
The solution is as follows: Between two players who thoroughly understand the play every game should be drawn. Neither player could ever win except through the blundering of his opponent. If Nought (the first player) takes the centre, Cross must take a corner, or Nought may beat him with certainty. ”

“ Then if you give a "crossing the river" puzzle, in which people have to be got over in a boat that will only hold a certain number or combination of persons, directly the would-be solver fails to master the difficulty he boldly introduces a rope to pull the boat across. ”

“ Take two pieces of string or tape, and round the wrists of two persons tie the string, as shown in Fig. 19. It adds to the amusement of the puzzle if one of the persons is a lady and the other a gentleman. The puzzle is for them to liberate themselves, or for any one else to release them without untying the string. To do this, B makes a loop of his string pass under either of A's manacles, slips it over A's hands, and both will be free. ”

“ The puzzle is to move the single rook over the whole board, so that it shall visit every square of the board once, and only once, and end its tour on the square from which it starts. You have to do this in as few moves as possible, and unless you are very careful you will take just one move too many. Of course, a square is regarded equally as "visited" whether you merely pass over it or make it a stopping-place, and we will not quibble over the point whether the original square is actually visited twice. We will assume that it is not. ”

“ The fewest possible moves, counting every move separately, are sixteen. But the puzzle may be solved in seven plays, as follows, if any number of successive moves by one frog count as a single play. All the moves contained within a bracket are a single play; the numbers refer to the toadstools: (1—5) , (3—7, 7—1) , (8—4, 4—3, 3—7) , (6—2, 2—8, 8—4, 4—3) , (5—6, 6—2, 2—8) , (1—5, 5—6) , (7—1) .
This is the familiar old puzzle by Guarini, propounded in 1512, and I give it here in order to explain my "buttons and string" method of solving this class of moving-counter problem. ”

“ As a matter of fact, the puzzle can be solved in seventeen moves if the men are to remain together; but if they separate at thePg 244 end, they may actually save a move and perform the feat in sixteen! The trick consists in placing the man in the centre on the back of one of the corner men, and then working the pair into the centre before their final separation.
Here are the moves for getting the men into one or other of the above two positions. The numbers are those of the men in the order in which they move into the cell that is for the time being vacant. ”

“ Disk Puzzle BALL PUZZLE
At first it is not evident why this is called a ball puzzle, but, when let into the secret, most people see at once a good reason for naming it so. The wooden ball or marble is hidden from sight inside of the wood and may be shifted in position from the middle to the upper piece of the puzzle and vice versa (Fig. 21) .
Fig. 21.
The problem is to slide the middle piece off of the pin that projects up from the lower piece and swing it around its pivot. This pivot is a 1-1/2" round-headed screw, fitting loosely in the upper two and fixed in the lower piece. ”