Pentominoes ''Puzzle shapes to make you think''

PENTOMI NOES ";Puzzle shapes to make you think"

PENTOMINOES If you take five squares of the same size and join them in every possible way, you get 12 different shapes. These shapes are called 'Pentominoes' and they form the basis of an enormous range of interesting puzzles and investigations.

PENTOMINOES JON MILLINGTON "Puzzle shapes to make you think"

TARQUIN PUBLICATIONS

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RECOGNISING THE PIECES

At first it may seem difficult to recognise the twelve pieces, as each can appear in any position. The easiest way is to connect them with letters of the alphabet. Of course, you need a little imagination and several could be called by different letters. For instance, the U could be called C and the W could be called M. But the letters used here mean that there are pieces to correspond with the last seven letters of the alphabet.

THE 12 PENTOMINOES AND THEIR LETTERS I

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© 1987 I.S.B.N: DESIGN: PRINTING:

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TARQUIN PUBLICATIONS STRADBROKE DISS NORFOLK IP21 5JP

Jon Millington 0906212 57X Paul Chilvers THE FIVE CASTLES PRESS All rights reserved

If you would like an up-to-date catalogue of other Tarquin books, please write to the publisher at the address above.

Ifyou take five squares of the same size and join them edge to edge, the shape you get is called a pentomino. You can put five squares together in only twelve different ways and so there are just twelve different pentominoes. As shown opposite, they are usually named after the letters of the alphabet F, 1,L, N, P, T, U, V,W, X, Y,Z. Sets c)f pentominoes to cut out Why pentominoes fit together in so many interesting ways, no-one knows, but the fact is that they do. This collection of 94 puzzles is only a sm Ii seecio Iro wnat is possie. nere Iare pient mor iua toJ-a small selection trom what Is possiDle. I here are plenty more i~des tO try and discoveries to be made. Perhaps some problems may look hard at first, but as you become familiar with the pieces, you will soon find that all you need is a little patience. At the end of the book you will find three pages with solutions to all of the problems. Some of the solutions given are the only ones possible, while others might well be different from yours. Fortunately, with pentomino puzzles, it is usually easy to see if you are right, even if An e nvelope to keep them in getting there may sometimes be tricky! Several books have one or more chapters on pentominoes, particularly Golomb also discussed them in 'Polyminoes'. Some of the problems were first suggested in one or other of these books, but many are new. I hope they will inspire you to create your own pentomino designs, and I would like to thank my wife for all her help with this book. J.M.M. Bristol.

94 puzzles to solve, and their solutions.

LET'S GET ' One of the amazing things about pentominoes is that you can fit them together in all kinds of unexpected ways. Some people find it hard to imagine that the twelve pentominoes can be arranged in a rectangle, yet there are several differently-shaped ones that the pieces will fit into. This rectangle measures six squares wide and ten squares long. It has an area of sixty squares because each of the twelve pentominoes has an area of five squares. Since there are over two thousands different ways of filling in the 6 by 10 rectangle, the puzzle can be made more interesting by saying how many pentominos should not touch the border. In the rectangle above, W and F are the only pentominoes not touching the border. Also shown is a 'crossroads' where the corners of four pieces (Z, P, T and F) meet. Avoid crossroads if possible, as solutions look better without them.

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Now here are some problems to try: Problem 1. Make a 6 x 10 rectangle with four pieces not touching the border. (It might be possible to do this with five pieces inside, but I have not managed it.) Problem 2. Make 6 x 10 rectangle with all the pieces touching the border. You probably used the long side of the I as part of the border when trying to make these rectangles, and this is where you will find it in the answers at the end, but here I touches the border by one of its shorter edges.

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Ifyou put I upright and near the middle, then you get these two 6 x 5 rectangles which together make one measuring 6 x 10. Problem 3. Try filling in this outline with your pentominoes. Although you might think it would be impossible for I not to touch the border at all, there are actually several ways of doing it. See if you can find one for problem 4. Any outline to be filled completely by the pentominoes must have an area of sixty squares, and rectangles measuring 5 x 12, 4 x 15 and 3 x 20 can all be made. The 5 x 12 rectangle, like the 6 x 10, can be solved with up to four pieces inside. Problem 5. Make a 5 x 12 rectangle with all pieces touching the border. Problem 6. Make a 5 x 12 rectangle with four pieces inside. Problem 7. Make a 5 x 12 rectangle with I inside. It looks as if no more than one piece can be inside a 4 x 15 rectangle. You might like to see if you can put two pieces inside after trying these: Problem 8. Make a 4 x 15 rectangle with all the pieces touching the border. Problem 9-12. Make four 4 x 15 rectangles, each with one of the pieces L, N, P, and Y inside. Problem 13. Make a 3 x 20 rectangle.

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X and Y are both inside.

There are only two ways of making the 3 x 20 rectangle, and you can change one solution into the other by turning round a group of seven pieces in the middle. .

HOW DOES SYMMETRY HELP? When

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to tit in all the pieces except A and find that the shape you needed was P? Ifyou can resist temptation and keep P to the end, you might be more likely to complete your rectangle. Why is P so useful? Maybe it is because it has more compact shape than most of the other pentominoes? Also, it is the only one with a perimeter of ten. The rest all have perimeters of twelve units. In use, these six pentominoes need not to turned over because they are symmetrical:

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Three of the pentominoes, 1,X, and Z, have another kind of symmetry

because they can be rotated through part of a turn and yet look the same as they did before being moved. This is called turning symmetry. Since the remaining pentominoes, F, L, N, P, and Y, have neither line nor turning symmetry, you can put them in a

rectangle in eight different ways. For example, here are the positions in which F might appear: w

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You will often find that it is best to leave the unsymmetrical pentominoes till later. When the time comes to insert them, there are more different ways in which they may fit.

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It is possible to buy attractive sets of pentominoes in wood or plastic, but it is also satisfying to make you own and this raises some interesting problems. To cut round a right-angle you would need to use a coping saw or a fret saw which do not easily produce straight edges. So you would almost certainly get a better result with an ordinary tenon saw. You might think that each piece would then have to be made from a separate block of wood. Luckily this is not so and the 8 x 10 rectangle here can be cut into the 12 pieces using a saw which only makes straight cuts. Only the U Diece has to be finished off afterwards with other tools. When you have marked out the rectangle, two cuts will release 1,two more will free U and then L can be parted from the rest with two cuts. Keep going like this until all the pieces have been cut out. Naturally the shaded areas are wasted wood. They amount to 20 squares. Now we can wonder if it is possible to reduce the wastage. n l~--R kIhq ln IU7R +n An {i- 1 -;-I,+ liM V-l I %JLAI iiii iI ILI MICfl LU LIAy %J 1 IAV II rL 3.LIAUI P.

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(77 squares), 4 x 19 (76 squares), and 5 x 15 (75 squares). For the 72 square rectangle, 4 x 18 and 8 x 9 might be possible but I have managed to do only the 6 x 12 arranged as two separate 6 x 6 squares. Problem 14. Lay out a 6 x 12 rectangle, so that each L1I pentomino could De cut out witn a tenon saw, leaving only the U to be completed with other tools. __ -

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It is not very likely that it is possible to waste fewer than 12 squares. The next smaller rectangle is 7 x 10 so that is the one to investigate.

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A CHESS BOARD WITH HOLES An ordinary chess board is an 8 x 8 grid, which makes 64 squares. A set of pentominoes takes 60 squares. This raises the question. Can the 4 spare squares be arranged in an interesting pattern?

Arrange the pentominoes to leave a 2 x 2 hole in the centre for problem 15.

Putting the four squares together in the centre is only one way of placing them in an 8 x 8 square. You can try to put them anywhere, either together or separately. For problem 16, make the pattern above.

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Ifthe four holes lie along a diagonal, you have an 8 x 8 square that can be made into a 7 x 9 rectangle by moving the lower half of the square to the right and then up. See if you can solve these for Problem 17. 8x8

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There are several ways of cutting the 8 x 8 square so that the two halves can be re-formed to make a 6 x 10 rectangle. The snag is that two of the holes in the 8 x 8 square will have to be on the edge. You can also have two identical shapes of six pieces each which can be arranged into either 8 x 8, 7 x 9, or 6 x 10 outlines. Make the 6 x 10 rectangle from the 8 x 8 square simply by moving the right-hand half of the square down and to the left. One half has to be turned over to obtain the 7 x 9 rectangle. Problem 18. Arrange the pentominoes to make two identical halves that can be fitted together to make any of these three:

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ABSENT FRIENDS A 5 x 13 rectangle has 65 squares, which is five more than those occupied by the set of pentominoes. Let's try to arrange the five gaps to be in the shape of a pentomino, an absent friend.

Here, the outline of the F is placed as centrally as possible (although it could be elsewhere), with the black dot right in the middle of the whole rectangle. Problems 19-30. Make a 5 x 13 rectangle so that the gaps form each of the pentominoes in turn. In each case the black dot marks the centre of the rectangle.

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You can make enlargements of each pentomino that will be three times the height and width of the pieces itself. Each will occupy 45 squares, the equivalent of nine pieces.

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Here are the outlines for F and X, each three times larger than the basic pentominoes and each using nine pieces. If three are not to be used, which three? Notice that the F uses the shape F, but X does not use the shape X. Problems 31-41. Make enlargements of F, I, L, N, P, T, U, V,W, Y,and Z using nine pentominoes for each, but leaving out the shape in question.

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A pentomino shape of twice the height and twice the width of the piece itself will have four times its area and will therefore need four basic pieces.

These double enlargements are possible for all pentomino shapes except V and X, and there will be numerous ways of selecting the four pieces needed. One of the many solutions of a double size P is illustrated here. A more challenging double enlargement is given below.

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Take two pieces, put them together and then make the same outline with two more pieces. Eight pieces then remain. Is it possible to use these eight pieces to make a double sized version of the same shape? The outline above shows that it is. Problem 42. Starting with the two pairs above, find another way of making the enlarged outline.

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Other enlargements can be made with different sets of two pairs. See if you can make the enlarged versions of these pairs, as problems 43-49.

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These animals look rather strange but each is made from a complete set of pentominoes. Problems 50-53. Find out how.

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Can you make any more for yourself?

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Problems 54-58. Make each of these outlines from a complete set of pentominoes.

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Are there any other animals you can find down on the pentafarm?

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SHAPES WITH JAGGED EDGES This attractive shape contains 61 squares, so you might think that it could be covered by a set of pentominoes, leaving the centre square free. However much you try, you will not succeed because it is truly impossible. Problem 59. Make this shape by removing a corner square rather than a middle one.

This outline also contains 60 squares, but is impossible because even if F and X are used as corner pieces the pentominoes can only provide a maximum of 21 edge squares and 22 are needed here.

Now for some outlines which you can solve! Problems 60-67. cl bU

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Ifyou solve problem 65 to have two separable parts as shown, you could move the smaller part over to the right and make a 6 x 10 rectangle.

This shape is about as round as you can get using pentominoes.

Here is an identical pair of shapes which can be put together in several ways, including both the 5 x 12 and 6 x 10 rectangles. 17

REPETITIONS TO MAKE TESSELLATIONS =-~'

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Instead of using a complete set of pentominoes, we can experiment with several copies of each shape. The diagrams below show how eight of the shapes will tessellate, or cover the surface without any gaps. Of course I can be added to the list, making nine altogether.

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The eight pentominoes on the left, and of course 1,can be used to tile a surface without being turned round and over. So, in every Y the single square is in exactly the same position pointing upwards and to the left. The final three pentominoes will also tile a surface, although half the pieces have to be turned upside down.

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With tilings, it is always interesting to experiment with different combinations of colouring.

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PENTACUBES

v If you replace the five squares of the pentominoes with five cubes you have a three-dimensional set of pieces which can be used to make some very interesting puzzles. You can make a set for yourself using the plan on page 7, or there is a very well-made mahogany set called 'Pentacubes' available from Pentangle, Tarquin or other suppliers. Each set of Pentacubes occupies 60 cubes and the first shape to try is the 5 x 4 x 3 rectangular solid illustrated on the left. It is the one which has the largest number of different solutions.

Once you move into three dimensions it becomes important to find a wav to --- ' -*- -^ -_-_ - -- - ....- -_ write down the solutions. To show the method, here is one solution for the 5 x 4 x 3 cuboid. Figure 1 shows 1,T and a partly hidden W in the bottom layer with a vertical X on the right. In figure 2 you can see that U and F have been added. Figure .5 snows thie completed figure. ___ - snow tne layers one - .is to __-_ ___way . - easier An after another, writing letters for any pieces which are not complete. All the solutions in the back are written this wav the tonr-~vand bottom*8v1 Swanninn -_1rr-.Zl --layers gives you a mirror image of the original.

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Two pieces, P and U can be entirely hidden inside a 5 x 4 x 3 solid. See ifyou can hide P for problem 68 and U for problem 69. For problem 70, try making 5 x 1 x 3 and 5 x 3 x 3 solids that can be fitted together to make a 5 x 4 x 3. Did you try to make the 5 x 4 x 3 solid as three separate 5 x 4 layers? In fact it is easy to prove that this is impossible. With X placed like this, either two l's or two U's are required to complete the rectangle.

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make it so that it does or does not separate into two layers. Problem 71 is to make it so that it does not separate. Only the pieces L, N, P, U, Y can connect the two layers and problem 72 is to make it so that all of them do connect the layers. There is only one other rectangular solid, the 10 x 3 x 2, which will not divide into two layers. Problem 73 is to make it with only L and Y connecting the layers. Problem 74 is to make it so that all the pieces L, N, P, U, Y. connect them. The 10 x 3 x 2 solid that you may just have tried to solve is a two-fold enlargement of the I shaped pentomino, but with 3 layers not two. Except for W and X, similar models can be made for the other pieces. Here is what Z looks like. Problem 75-83. Make similar models of F, L, N, P, U, T, V,Y,and Z.

Looking at the right-hand side of the steps of problem 84, you can see that the nimer

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+ 4 + 5, and it is because the total of 15 divides exactly into 60 that the steps are possible. But the total of 1 + 2 + 3 + 4 and of 1 + 2 + 3 each go into 60 as well, which suggests two further sets of steps for you to make as problems 85 and 86.

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SOLID PUZZLES I

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Problems 87-90. Use a set of solid pentominoes to form each of these shapes. 88

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Problems 91-94 are to solve these four shapes with holes in them.

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Will any other pentominoes, apart from T, form the hole in a 2 x 5 x 7 solid?

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Because there are so many pentomino designs, it is impossible in a book of this size to mention them all. Here are two more ideas for you to try out. This 8 x 10 rectangle is filled with all the pentominoes so as to leave a gap in the middle with the shape of an enlarged F. You could move this one or two squares to the right, or turn it sideways for three more positions, making six altogether for F. In the same way, there are 14 distinct positions where the shape of the gaps is P, 11 when it is U, 5 for T, 4 for Z and 3 each for L, N, V,W, and Y, and 2 for X. This makes 57 positions altogether and at least 18 can be solved; 6 are clearly impossible.

There are dozens of solutions to the next puzzle where you divide the pentominoes into three groups of four and then make the same outline with each group. Like this, with no gaps

or this, with one gap

And so the ideas go on, and on, and on ................... . i

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NUMBER OF SIDES PERIMETER

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OPEN BOX WHEN FOLDED NUMBER OF DIFFERENT POSITIONS PAGE 6 NUMBER OF LINES OF SYMMETRY PAGE 6 TURNING SYMMETRY PAGE 6

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ISBN 0-906212-57-X

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