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Tangrams 330 Puzzles
By Ronald C. Read
Dover Publications, Inc.Copyright © 1965 Dover Publications, Inc.
All rights reserved.
Letters and Numbers
The seven tangram pieces can be fitted together in such a vast number of ways that in preparing a selection of them it is very difficult to know where to start. Probably as good a way as any is to go right back to our early school days, back to the days when we first learned the alphabet. This time we shall make the letters out of the tangram pieces instead of with pencil and paper.
In the following pages you will find tangram outlines for all the letters of the alphabet. See if you can make them all yourself before looking at the solutions which begin on page 94.
Reconstructing tangram outlines that have been obtained by others is only half the fun that can be derived from this puzzle; as much entertainment, if not more, can be obtained by inventing new outlines. The reader can try his hand at this right away by trying to improve some of the outlines given for the letters. Certainly several could stand improvement; the N, Q and X, in particular, are nothing like as satisfactory as one would wish. The outlines given are the best that I have been able to concoct, but the reader may well be more successful.
For good measure, the digits 1, 2, up to 8 have also been given. Since zero is the same as the letter o, and 9 is simply a 6 upside down, these two digits are not new outlines, and have not been included.CHAPTER 2
The Chinese tangram books, and the later Western books all contain large numbers of outlines representing animals. This book will be no exception, and our contribution to the tangram menagerie lies in the next few pages. Some of the outlines which follow are taken, or adapted, from the Chinese books, and some from Sam Loyd's book. As far as I know, the others are entirely new.
It might be though that the tangram pieces, with their sharp angles and straight sides, could not possibly portray the complicated curved shapes of living creatures, but it is truly surprising how a well-designed tangram can suggest curves where none exist, and complexity where there is only simplicity. See, for instance, how a single piece suffices to suggest the back-curving horns of the mountain goat (No. 35); notice the stratagem by which the thin legs of the stork (No. 50) have been portrayed by three tangram pieces which, individually, are much thicker; note how the careful arrangement of the pieces unerringly conveys the fright of the startled cat (No. 41), the haughtiness of the camel (No. 47) and the graceful lines of the shark (No. 78). We shall see many examples throughout the book of outlines which, by their careful construction, convey much more to the mind's eye than is actually there.
In addition to the animals that have already been mentioned we have:
36. A horned animal of some kind. (Let's not inquire too closely into its precise genus and species!)
37. A polar bear.
38. A giraffe.
39 to 44. Assorted cats, closely followed (naturally) by
45 and 46. Two dogs.
48. A squirrel.
49 to 61. Birds of various shapes and sizes.
62 to 65. Four sinister-looking vultures.
66 to 68. Three horses. Also No. 74.
69 to 72. Four different kinds of bats.
73. A kangaroo.
75. A crocodile.
There follow several sea creatures, mainly fish of various kinds, but also
79. A lobster.
81. A turtle.
82. A seal.
83. A shrimp.CHAPTER 3
While the evolutionist will not deny the propriety of dealing first with animals, we should delay no longer in looking at those tangrams which depict man and his activities. These are among the most interesting of tangrams: for since the human shape is more familiar to us than that of any other object in the universe, we can readily appreciate a tangram which succeeds in portraying the essence of some human form or posture; and just as readily detect the fault in those that do not manage to suggest what they are meant to represent.
Our human parade starts with a shoeshine boy and a customer (Nos. 90 and 91), two men very excited about something, and three acrobats (94, 95, 96), one of whom has just had a fall (or perhaps he is just taking a rest). On the next page are five men who are obviously in a great hurry and a sixth, strolling along with his hands in his pockets, who is clearly in no rush to get anywhere (97 to 102). Then we have four stately medieval ladies (103 to 106) accompanied by a servant (107) and a lady's maid (108).
Next we have "Sam Loyd's Portrait Gallery," twenty-six cleverly constructed heads taken from Sam Loyd's book, plus two more given by Dudeney. Sam Loyd gave names to some of these outlines, as follows:
120. Old Scotch Piper.
121. French Grenadier.
122. Colonial General.
123. A Turk.
124. Aunt Betsy.
125. Uncle Rhube.
126. Mary Smith.
128. John Knox.
129. Tom Sharkey.
130. The Professor.
131. Buffalo Bill.
132. "The Easy Boss."
The Indian chief and his squaw (137 and 138) are also from Sam Loyd's book.
Finally we have four horsemen (139 to 142), a runner (143), and a lady and a gentleman drinking a toast (144 and 145). (Or is he a gentleman ? He seems to be seated while the lady is standing—and seated on the floor at that!)CHAPTER 4
Around the House
It is only fitting that after looking at man and his activities we should dwell for a few moments on the things that he uses every day. Starting with his pipe (No. 146) we go through a motley collection of shoes, chairs and general odds and ends, ending up with a watering can and a pistol (Nos. 161 and 162). We then have three different kinds of baby carriages—all the latest models and very comfortable, despite the fact that they have square wheels (to which tangram babies have long grown accustomed!). Number 165 is a very superior model which comes complete with a nursemaid (No. 166) to push it. Finally we have Number 167, which is a—now what on earth was that thing meant to be? A saw? A nutmeg-grater? Well, no matter; it is an interesting shape to construct.CHAPTER 5
Boats and Bridges
The Chinese books which served to introduce the tangram to the Western world were published in the Treaty Ports of South China, and their pages are full of representations of ships, boats, junks and many other things connected with the sea. It is a curious fact that sailing boats of various kinds seem to be comparatively easy to depict using the seven tangram pieces, notwithstanding the rather complicated silhouettes which they present; whereas steamships, which with their more angular shapes might seem much more suitable for representation by tangrams, are in fact very difficult to concoct. I have been quite unable to produce a convincing tangram of any sort of mechanically propelled craft, with the exception of the launch given in Number 186 (and I won't argue with the reader if he says that this tangram, too, is not very convincing).
Here, then, is something on which the reader can exercise his creative ability; to produce a reasonably good outline of some kind of steamboat or other modern sea-going craft. It will not be easy!
Along with the ships and boats is given an assortment of bridges, mostly taken from the Chinese books. The last two tangrams of this chapter show a lighthouse and the lighthouse keeper being rowed out to it. (Or maybe it is just a sailor taking his maiden aunt for a row around the bay.)CHAPTER 6
Stories and Pictures
Two innovations introduced by Sam Loyd and H. E. Dudeney to extend the scope and interest of the tangram were to construct a series of outlines to illustrate a story, and to bring together several tangram outlines to make a more detailed picture. In this chapter we have two illustrated stories by Sam Loyd, and two pictures by Dudeney.
The first set of tangrams illustrates the well-known nursery rhyme of "The House That Jack Built." In outlines 189 to 199 we see
... the farmer sowing his corn,
That kept the cock that crowed in the morn,
That waked the priest all shaven and shorn,
That married the man all tattered and torn,
That kissed the maiden all forlorn,
That milked the cow with the crumpled horn,
That tossed the dog,
That worried the cat,
That killed the rat,
That ate the malt,
That lay in the house that Jack built.
To fill out the page, three outlines are given, representing animals that would very likely be found in the vicinity of "The House That Jack Built."
The second story (adapted from Sam Loyd's version) is that of Cinderella. We see Cinderella crying in front of the fireplace (204 and 203); the two ugly sisters (205 and 206), and the fairy godmother with the pumpkin that became a coach and two of the rats that became coachmen (207 to 21 o). Cinderella dances with the prince (211 and 212), heedless of the clock (213) which is about to strike twelve. The episode of the slipper (214) is too well known to need repetition; everyone knows the story ends with wedding bells (215 and 216) for Cinderella and her prince.
One outline that clearly ought to have been included in this sequence is Cinderella's coach (when it wasn't being a pumpkin). Sam Loyd does not give one in his book, and I have been unable to devise a satisfactory tangram coach; they all turn out looking like baby carriages! Here again the reader may be more successful. By way of compensation I have included Cinderella's coach among the double tangrams later on in this book.
On the next page we have a picture by H. E. Dudeney entitled "A Game of Billiards." Dudeney says, "The players are considering a very delicate stroke at the top of the table."
Dudeney's second picture consists of nine outlines (221 to 229) and will not fit on a single page of this book, so has been spread over two. The description of it is best left to Dudeney himself.
My second picture is named "The Orchestra", and was designed for the decoration of a large hall of music. Here we have the conductor, the pianist, the fat little cornet-player, the left-handed player of the double-bass, whose attitude is lifelike, though he does stand at an unusual distance from his instrument, and the drummer-boy, with his imposing music-stand. The dog at the back of the pianoforte is not howling; he is an appreciative listener.CHAPTER 7
A Little Mathematics
(Don't be alarmed! You will need very little knowledge of mathematics to follow this chapter.)
As we have already remarked more than once, the number of tangrams is very large. This is really an understatement, for the number is infinite! This we can easily see by looking at Number 229, for example. The bottom corner of the square piece that represents the head of the drummer-boy can touch the rest of the tangram at an unlimited number of points along the line that represents the shoulder and the outstretched arm. It is true that the different outlines that one would get by putting this piece in the different positions would not be very different each from the other, but in the strictest sense they would have to be counted as distinct.
It is a little quixotic, however, to accord the same weight to minute variations in a tangram outline as to the differences between two totally distinct outlines, and one naturally wonders whether, by ignoring trivial variations, one could ask the question, "How many tangrams are there?" with some hope of getting a clear-cut answer. Alternatively one might ask the question, "How many tangrams are there of such-and-such a particular kind."
In 1942, two Chinese mathematicians, Fu Tsiang Wang and Chuan-Chih Hsiung, asked, and answered, the question, "How many convex tangrams are there?" Now, before we go further we must be sure what is meant by "convex" in this connection. Roughly speaking, we can say that a convex figure is one that does not have any recesses in its outline. For angular figures (like tangram outlines) this means that all the angles are less than 1800; in other words that the corners all stick out instead of in. But the simplest way of seeing the difference between a convex figure and one that is not convex is to imagine a piece of string or an elastic band pulled tight around the figure, as in Nos. 230 and 231, below. If this causes the string to make contact with the figure all the way round its edge, then the figure is convex; but if there are gaps between the string and the edge of the figure, as at A in Number 231, then the figure is not convex, but concave or recessed, where these gaps occur.
Now that we know exactly what "convex" means in this context we can look again at the question asked by the Chinese mathematicians: "How many convex tangrams are there?" One might well imagine that there would be quite a large number of them, but it turns out that there are only thirteen! In the paper in which this result is proved the thirteen convex tangrams are not drawn, but are merely listed as follows:
Four-sided figures 6
Five-sided figures 3
Six-sided figures 3
In this chapter we give all thirteen convex tangrams. Some of them are very easy to construct, but a few are rather tricky. The reader may want to try to construct them all without first looking at the outlines given below (Nos. 232 to 243). Or he can examine the outlines and then try to construct the tangrams. The total of thirteen is completed with Number 230, but since this tangram was necessarily given in the introduction, it doesn't really count.
An Unsolved Problem
It is clear that convex tangrams are very special indeed; there are so few of them. Can we think of any other special kinds of tangrams that would be more numerous than the convex ones, and yet not infinite in number ? As far as I know, no one has ever done so, but I am going to propose a problem of this kind for the benefit of anyone who may feel inclined to tackle it. First we must specify the sort of tangram that we are going to talk about.
Let us imagine a set of tangram pieces of such a size that the equal sides of the small triangles are 1 inch in length. Then the third side of these triangles will be of approximately 1.414 inches (the square root of 2, to be precise). Now any side of any of the pieces of this set will be one of these lengths, or twice one of these lengths, and we can therefore imagine every side of each of the pieces to be made up of "sections" whose lengths are either 1 inch or 1.414 inches. There will be either one or two sections to each side. In Figure 3 (page 58), which shows the tangram pieces, the ends of the sections are indicated by blobs.
Imagine now a tangram that has been constructed in such a way that wherever two pieces are in contact at all, they are in contact along a whole section of each, so that the ends of these sections coincide. In other words, when two pieces are in contact, the blobs on the two edges will match. This is illustrated by Figure 4. This stipulation on the way in which the pieces are to be placed together considerably restricts the sort of outline that can be produced; on the other hand, large numbers of very interesting tangrams fall into this category. We shall apply one further restriction: namely, that the tangram should be all in one piece. Tangrams which conform to the above restrictions I call "snug" tangrams, because of the close way in which the pieces fit together. All the convex tangrams are snug; so is Number 231, and so are very many other tangrams in this book. Snug tangrams tend to be rather more difficult to reconstruct from their outlines than tangrams that are not snug, since the close fitting reveals less of the way in which the tangrams have been formed.
It makes sense to ask the question, "How many snug tangrams are there?" for it can be shown that snug tangrams, unlike tangrams in general, are limited in number—there is only a finite number of them. But how many, exactly ? At the moment, nobody knows, and I recommend the problem of calculating this number to anyone who finds it interesting and who has access to a large electronic computer. It is unlikely that the "snug tangram number" will be found without the use of a computer, for it is almost certainly very large, probably well up into the millions, if not considerably larger.
(A preliminary investigation of the problem indicates that it is too complex for the computer to which I have access, but that the problem of programming a computer to find the number of snug tangrams would be a fascinating one—a puzzle that out-tangrams the tangram! Unfortunately, it is a puzzle not available to everyone.)
Excerpted from Tangrams 330 Puzzles by Ronald C. Read. Copyright © 1965 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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