Spatial puzzles: a guide for researchers

Butler, Brian E

Abstract

Spatial puzzles consist of objects or pieces that must be fitted into a specified configuration. The puzzles are often complex and offer some unique opportunities for examining spatial skills. These puzzles have not been used to advantage by psychologists because most researchers are not aware of the rich variety that exist and the extensive literature on them. The present article attempts to provide a guide to the most common puzzles and some of the literature on them.

Man has never shown so much imagination as in the variety of games he has invented. (G. W. Leibnitz, as quoted by Odier, 1975a, p. 12)

In 1926, Henry Ernest Dudeney, an English puzzle expert, made an interesting observation: People are fascinated with puzzles and, when this curiosity is coupled with modern communications media, it periodically produces puzzle crazes that sweep society. To prove his point, Dudeney described some puzzles popular since medieval times and documented a series of crazes beginning with tangrams and wire – puzzles in the early 1800’s, progressing to Sam Loyd’s “14 – 15” puzzle in the 1880’s (see also Hordern, 1986), and continuing to cross – words in the 1920’s.

Events since 1926 support Dudeney’s claim. Jigsaw puzzles were a popular fad in the 1950’s while Piet Hein’s Soma Cube (Hein, 1969) and Instant Insanity captured the public’s imagination briefly in the late 1960’s. These and all other crazes pale in comparison to the popularity of Rubik’s Cube (see Hofstadter, 1982; Hordern, 1986) which was probably the greatest puzzle craze of all time. Since 1980, video games have become popular but most video games are not puzzles in the truest sense. Most video games involve a great deal of dexterity but entail no aspects of problem – solving (Loftus & Loftus, 1983); however, there are two exceptions and both have spawned puzzle crazes. In the early 1980’s, college students and computer buffs spent hours playing ROGUE (or HACK), a three – dimensional maze exploration game. That fad has passed but today’s students seem equally enthralled with TETRIS, a Russian game using interlocking pieces (Pajitnov, 1988).

All the puzzles described by Dudeney, and all the puzzles mentioned above, are spatial puzzles, that is, puzzles that consist of pieces or objects that must be manipulated into a specific spatial configuration. Perhaps an important corollary to Dudeney’s thesis is that,while people are fascinated with puzzles, they are especially fascinated with spatial puzzles.

Despite the popularity of spatial puzzles, psychologists know relatively little about the processes involved in solving them. There has been a considerable amount of research on problem – solving (e.g., Newell & Simon, 1972; Hayes, 1989; Wickelgren, 1976) but most of the research has focussed on conceptual puzzles, that is, problems that can be represented symbolically with little or no reference to spatial parameters. This lack of interest in spatial puzzles may be due to several factors. For example, with psychometrics, mechanical skills have been linked to less desirable blue – collar occupations such as “bomb – fuse assembler” (Cronbach, 1960, p. 280). Another reason is that the cognitive renaissance during the 1960’s was motivated in part by the inability to account for language within a Behaviourist framework so that much of the subsequent research has focussed on language – related problems, such as verbal memory, rather than non – verbal processes. The most likely reason that spatial puzzles have been ignored, however, is that most psychologists simply don’t know anything about them.

Using Puzzles for Research and Teaching: A Rich Environment

Spatial puzzles offer a rich environment for exploring problem – solving and spatial abilities in that many puzzles have a large assortment of problems of different and known degrees of complexity with variations that allow the assessment of transfer across different dimensions. Pentominoes, for example, include both 2 – dimensional and 3 – dimensional problems so we can assess the extent to which schema for a plane surface transfer to the larger space, or vice – versa. Some puzzles also allow the same problem to be generated in different forms, that is, problem isomorphs can be specified easily. MacMahon’s Coloured Tiles (see below), for example, offers problems that can be stated in terms of coloured edges or in terms of edge contours with monochromatic tiles. The capacity to present the equivalent problem in different forms is necessary for identifying general strategies and schemata (Hayes & Simon, 1974; Kotovsky & Fallside, 1989).

Recently, Eysenck and Keane (1990, p. 387 – 388) have argued that research with well – defined problems, such as those used by Newell and Simon (1972), tells us how the subject uses general heuristics but tells us little about the use of domain – specific knowledge. Spatial puzzles, however, are exempt from this criticism in that they are well – defined yet they also require specific knowledge about objects and space. With these puzzles, we can isolate and identify the role of spatial knowledge, determine whether it is object – specific, and examine individual differences.

Spatial puzzles also simplify the problem of tracking a subject’s performance. As Chi and Glaser (1985, p. 231) have noted, “Puzzle problems … lend themselves particularly well to uncovering solution processes, by tracing the sequence of operations applied to transform the initial state to the goal state”. Because the pieces must be physically moved, a complete protocol can be recorded with a video camera.

Because these puzzles do not entail verbal processes, they may be useful tools to study problems such as anterograde amnesia and cerebral lateralization. Squire (1987) has noted that amnesic patients show a deficit in declarative memory but no deficit in procedural or implicit memory; this has been shown using tasks such as jigsaw puzzles (Brooks & Baddeley, 1976) and the Tower of Hanoi (Cohen, 1984). Similar tasks may be useful for testing right – hemisphere functions with commissurotomy patients; Corballis (1983, p. 49) has noted that “split – brained patients are much better able to carry out certain manipulo – spatial tasks with their left hands than with their right hands” (see also Bradshaw & Nettleton, 1983, p. 65).

Spatial puzzles also present some unique opportunities for teaching. Puzzles can be used to demonstrate a wide range of phenomena. Kohler’s (1925) concept of insight, for example, is quite abstract but can be illustrated vividly with the Heart – and – Arrow puzzle (see below). At first students assume it is impossible to remove the heart but, as they “catch on”, they will experience the perceptual reorganization that is the essence of insight (Zangwill, 1987). (The artist who illustrated this puzzle in Mayer’s (1977, p. 24) text did not achieve insight; his version is impossible!) Similarly, the concept of a depth – first search of a game tree using a difference – reduction heuristic is difficult to grasp in the context of the DONALD + GERALD = ROBERT problem (Newell & Simon, 1972) but is easily illustrated using a set of 24 coloured tiles, devised by MacMahon (1921), which can be assembled into a 4 X 6 matrix (see below). Within a few trials, a student can become quite efficient at solving the Tower of Hanoi so one can use this to illustrate a learning curve; the student will generally be unable to describe exactly what they have learned which demonstrates procedural tuning in the absence of declarative knowledge. These are only a few examples of how puzzles can be used; the possibilities seem limited only by the instructor’s imagination.

Using puzzles as teaching aids with a small group is straightforward (although retrieving them is often difficult). With a larger class, puzzles can often be placed directly on an overhead projector so that the class can see a shadow outline; this works well for wire – puzzles, for example. In some cases, versions made of tinted plastic can be used with an overhead projector. Again, imagination seems to set the boundaries.

Learning More about Puzzles

Spatial puzzles are potentially very useful but it is difficult to obtain enough information to use them effectively. The literature on puzzles is scattered over a wide range of sources, both academic and popular, and most of the articles do not provide adequate references. This is not a new situation; Dudeney (1926, p. 871) observed, “There is no very complete bibliography on these lines of research, and much time is frequently wasted by students who investigate subjects that have already been thoroughly worked out.”

There are, however, some general sources worth consulting. Hoffman (1893), Ball (1939), and Kew (1975) provide excellent reviews of some older puzzles popular around the turn of the century. Some interesting but incomplete collections are presented by Stubbs (1931), Filipiak (1942), Wylie (1957), Abraham (1964), and Schuh (1968). Two more recent, and much more complete, compendia are provided by van Delft and Botermans (1978) and Slocum and Botermans (1978) and Slocum and Botermans (1986); both volumes provide some historical perspectives. The mathematical properties of many spatial puzzles are discussed extensively by Ball (1939), by Berlkamp, Conway, and Guy (1982), and by Martin Gardner in some of the books based on his “Mathematical Games” column in the Scientific American (Gardner, 1959; 1961; 1966; 1969; 1970; 1971).

Any search of the literature will lead to two individuals who dominated the world of puzzles at the turn of the century: Sam Loyd (1841 – 1911) and Henry Ernest Dudeney (1847 – 1930). Loyd, an American, wrote a popular newspaper column that combined puzzles with folksy humour and had a keen sense of showmanship. He made a fortune by offering a $1000 prize to anyone who could solve his version of the “14 – 15” puzzle and devised a “horse and rider” puzzle sold by Barnum and Bailey’s circus (see Scheerer, 1963). Sam Loyd’s most enduring legacy may be the “Get Off The Earth” puzzle, a picture of the globe surrounded by either 12 or 13 figures depending on how the globe is tilted (van Delft & Botermans, 1978, p. 35). Loyd’s work appears in two collections (Loyd, 1914; 1959). Henry Ernest Dudeney, an Englishman, also wrote a newspaper column but, in contrast to Loyd, took a more serious, scholarly approach and devised a wider range of puzzles. Dudeney was honoured by the Royal Society for an elegant demonstration that an equilateral triangle can be transformed into a square. His best work appears in two collections (Dudeney, 1908, 1917).

The sources listed above will provide hours of fascinating reading but none offers a good overview. This article is an attempt to provide a more useful starting point by reviewing some of the puzzles available and by offering a brief bibliography for each. The readings were compiled by searching through journals, hobby magazines, and books so the lists are not exhaustive. Many of the references are long out – of – print and hard to locate but it has been my experience that university inter – library loan services are quite ingenious at ferreting out even the most obscure works.

The Puzzles

TANGRAM

The tangram has been a popular puzzle for over two hundred years; its devotees included Edgar Allen Poe, Lewis Carroll, and Napoleon while in exile (Bell, 1973). Sam Loyd (1903) claimed the tangram originated in China about 4000 years ago but this was probably a spoof to help sell the puzzle. Imported ivory sets did appear in Britain in the late 1700’s and the puzzle is still available in more exclusive toy stores.

The tangram, shown in Figure 1, consists of a square tile cut into seven pieces; two large triangles, a medium triangle, two small triangles, a rhombus, and a square. These pieces can be arranged into a wide variety of geometric shapes with three, four, five, or six sides (Wang & Hsiung, 1942; Lindgren, 1961; 1968) or into outline shapes resembling sailboats, people, etc. (Loyd, 1903; Hartswick, 1925; Read, 1965; Elffers, 1973; see also van Delft & Botermans, 1978). The puzzle is often accompanied by a booklet showing some of the many possible shapes. The two shapes presented in Figure 2 were devised by Dudeney (1908, p. 47) who used all seven pieces to build each character then asked the reader “Where does the second man get his foot from?”.

The specific shape of the tangram pieces is very important. There are many ways to dissect a square and toy manufacturers have tried other combinations to avoid copyright restrictions. Hoffman (1893) mentions five variations; van Delft and Botermans (1978) add two more using circles. None of these variations have proved as popular or enduring as the tangram but they could be used to determine whether the knowledge gained by practicing with tangrams is specific to the original set of shapes.

Tangram problems are difficult enough to be challenging without being frustrating. Tangrams have been used to teach mathematical equivalences and problem – solving strategies to children and may be an ideal test of spatial skills for children.

The tangram is a specific example of a dissection puzzle. The more general form of the problem is to take a given shape and transform it into another shape by cutting the original into a number of smaller pieces which are then rearranged. Thus, a square can be changed to a decagon or a Greek cross can be rearranged as a square or a pentagon (van Delft & Botermans, 1978). The goal is to achieve the specified transformation with as few pieces as possible. The general problem of dissections has been discussed extensively by Dudeney (1908) and Lindgren (1972). Dissections have produced some important geometric proofs but are far beyond the capability of most subjects.

There is no three – dimensional equivalent to the tangram but there are two close approximations. The first is Rubik’s Snake (Ideal Toy Co.), a set of 24, interconnected right – angle prisms that can be twisted into a variety of shapes including a ball and a 2 X 3 rectangle (Fiore, 1981). The second is Yoshi’s Cube (Ideal Toy Co.), a cube dissected into 12 identical prisms which are hinged to form a ring and can be contorted into a solid cube or turned inside – out to form a rhomboid dodecahedron with a hollow interior the size of the orginal cube. Both puzzles produce a myriad of shapes that intrigue children.

POL YOMINOES & POL YAMONDS

Polyominoes are two – dimensional dissection problems using pieces constructed from specific rules. The rules involve a progression based on the names. Dominoes are built of two squares which form a 1 X 2 tile; triominoes are built with three squares and have two shapes, a 1 X 3 rectangle and an L – shape; tetrominoes are built with four squares and assume five shapes if we exclude rotations and reflections. The most popular and most versatile set is the set of pentominoes, tweleve shapes based on five squares each. Collectively these sets, and all others in the progression, are referred to as polyominoes.

The same logic and almost the same nomenclature can be applied to other shapes. Thus, a diamond consists of two joined equilateral triangles and adding more triangles produces higher – order sets of polyamonds. The most popular puzzle set based on triangles is the set of twelve hexiamonds, each formed of six triangles. O’Beirne (1962a) has explored a set based on right – angle triangles but these do not seem as flexible or intriguing as the sets based on squares and equilateral triangles. The most popular sets, pentominoes and hexiamonds, are shown in Figure 3.

Pentominoes were devised by Dudeney (1917) who told the story of an angry chess player and a chess board broken into the twelve pentomino shapes plus a 2 X 2 square. The reader was invited to reassemble the board. This puzzle attracted very little attention until it was reintroduced by Solomon Golomb (1954, 1965) who coined the name and devised a theorem to prove that the twelve pieces can cover any chess board with any four designated squares left vacant. Since then a wide variety of problems have been posed and analysed exhaustively; Malachy (1969), for example, has found 2,339 ways to form a 6 X 10 rectangle, 1,010 for a 5 X 12, 368 for a 4 X 15, but only 2 ways to form a 3 X 20 rectangle. Other problems include constructing shapes such as staircases, subdividing the set to duplicate individual pieces, or building “farms” (Anderson, 1962a, 1962b, Bouwkamp, 1970; Feser, 1968; Haselgrove & Haselgrove, 1960; Klamkin & Liu, 1980; Kramer, 1983; Kramer & Gobel, 1983; Liu, 1982; Mayer, 1973; Miller, 1960; Parlett, 1973a, 1973b; Philpott, 1972a, 1972b, 1973). Pentominoes have also been used as a game (Parlett, 1973a) in which players alternate placing pieces on a chess board until the loser is unable to move.

The twelve hexiamonds also form a variety of shapes but have not been examined as extensively as pentominoes. Reeve and Tyrrell (1961) first demonstrated that theywill form a rhombus or a hexagon. Thomas O’Beirne (1961i, j, k) has a series of articles in the New Scientist in which he explores possible configurations and shows that hexiamonds will fill a hexagonal “honeycomb”, can be subdivided into sets to make hexagons and triangles, and can be assembled into a giant triangle with truncated corners. Other hexiamond problems are discussed by Torbijn, (1969), Meese (1973) and Odier (1975a).

Pentominoes offer a wide range of problems of graded difficulty for training and testing subjects. If pentominoes are constructed of cubes, rather than squares, three – dimensional problems can be devised (see box – packing problems below). Since tetrominoes are used for the popular video game, TETRIS (Pajinov, 1988), some subjects may have had considerable experience dealing with pentomino – like problems. Hexiamonds offer fewer problems but it would be important to note whether skills acquired with pentominoes will generalize to hexiamonds, or vice – versa.

SOMA CUBE & BOX – PACKING PUZZLES

The three – dimensional equivalent of polyominoes involves shapes produced by joining cubes instead of squares. The best known set is the Soma Cube devised by Piet Hein (1969), who noted that a cube can be dissected into seven pieces that represent all the irregular shapes produced by joining either three or four cubes face – to – face. Like the tangram, Soma can be used to build a number of shapes (Golomb, 1965; Hein, 1969; van Delft & Botermans, 1978; Whinihan & Trigg, 1973) but the most basic is the cube. Building the cube is relatively easy because there are 240 possible solutions (Odier, 1975a). The difficulty of building the cube can be manipulated by restricting the opening position (van Delft & Botermans, 1978) or by changing the shape of the pieces. Wells (1983) describes a cube with five pieces that is considerable easier than Soma; on the other hand, J. G. Mikusinski (see Steinhaus, 1950) has devised a fiendishly difficult version that permits only two solutions. The Soma Cube and Mikusinski’s cube are shown in Figure 4.

Generically, these puzzles are box – packing puzzles and there are a number of remarkably interesting ways to pack a designated box (DeBruijn, 1969; O’Beirne, 1961c). Pentominoes built of cubes can be used to construct a wide range of shapes (Golomb, 1965; Harary & Weisbach, 1982; van Delft & Botermans, 1978; Verbakel, 1972), although they cannot form a cube. Bouwkamp (1969, 1971) found that there are 7935 ways to build a rectangular box with pentominoes including 3940 ways to pack a 3 X 4 X 5 box. Pentominoes built from cubes are still defined in terms of a plane surface; occasionally these are referred to as pentacubes, although technically this term ought to be reserved for the 29 distinct shapes produced by joining five cubes in all three dimensions. The 29 true pentacubes cannot pack any box unless the 1 X 1 X 5 piece is discarded (Wagner, 1972, 1973).

Meeus (1973) describes a set of tetracubes which includes all the four – cube pieces from Soma plus a 1 X 1 X 4 and a 1 X 2 X 2 piece. There are 1390 ways to pack a 2 X 4 X 4 box and 224 ways to pack a 2 X 2 X 8 box with these shapes. In addition, Meeus points out that boxes can be constructed using several copies of single tetracube shapes. The equivalent problem with pentacubes has been explored by Bouwkamp and Klarner (1970) and Klarner (1969, 1980).

J.H. Conway, a Cambridge mathematician, has devised some surprisingly difficult box – packing problems using rectangular solids. One problem (van Delft & Botermans, 1978, p. 82) involves packing a box that is 5 X 5 X 5 with six blocks that are 1 X 2 X 4, six that are 2 X 2 X 3, and five that are 1 X 1 X 1. Klarner (1973) describes a similar puzzle designed by two Dutch architects and another packing puzzle from Conway which involves packing three 1 X 1 X 3 pieces, one 2 X 2 X 2 block, one 1 X 2 X 2 block, and thirteen 1 X 2 X 4 blocks into a box 5 units on each side.

One final packing puzzle is worth mentioning because it appears simple but is very difficult to solve. The puzzle consists of a tetrahedron divided into two identical prisms by cutting across its square axis (Slocum & Botermans, 1986, Wyatt, 1946). There are only five ways to join the two pieces but most people cannot find the one that reassembles the tetrahedron.

COLOURED TILES & COLOURED CUBES

The standard set of European dominoes consists of 28 1 X 2 tiles with each half coded with 0 to 6 pips. The domino rule specifies that two tiles can touch if and only if they have the same number of pips. Dominoes can be used for a wide range of tiling problems, that is, problems in which a specified surface must be covered following particular rules (Philpott, 1971a, 1971b, 1971c, 1972a, 1972b; van Delft & Botermans, 1978, p. 57 – 65). Puzzles can also be constructed by applying the domino rule to other shapes with equivalent codes.

MacMahon (1915, 1921; Odier, 1975a, 1975b, 1975c) devised the set of 24 square tiles, shown in Figure 5, in which each tile is sub – divided into four triangles and each triangle can be painted one of three colours. Across the 24 tiles, every combination of triangle colour and position can be represented exhaustively if rotations are excluded (as they are with standard dominoes). These tiles can then be assembled into a number of shapes following the rule that adjoining edges must be the same colour. It is relatively simple to tile a 4 X 6 rectangle but is more difficult if the rectangle must have a solid – coloured border. A host of problems of progressive difficulty have been developed (MacMahon, 1921; O’Beirne, 1961b; Philpott, 1969, 1973). The tiles can even be used to cover the sides of four cubes and the same rule can then be applied for joining the cubes together (Philpott, 1974).

The principles underlying the coloured squares can lead to new variations either by using different shapes or by changing the contact rule. MacMahon has explored the equivalent notion using other shapes such as equilateral triangles with three colours (Philpott, 1972a, b, c) and hexagons with six colours. One version of the hexagon puzzle is still available in many toy stores. MacMahon also varied the contact system which allowed a switch from colour coding to edge contour. Normally the contact rule is that adjoining edges must match in colour but this can be varied to define some complimentary colours. For example, the pieces in Figure 4 can be rearranged with colour 1 meeting colour 1 but colour 2 meeting colour 3. If the pieces are recoded so that 1 is a straight edge while 2 is concave and 3 is convex, the new puzzle is isomorphic to the colour problem even though the pieces can be all the same colour. This puzzle was commercially available at one time.

Another variation uses cubes instead of tiles. MacMahon pointed out that 30 cubes can be painted with six colours so that each colour appears on each cube and yet the permutation of colours on each cube is unique. These cubes can then be used for construction problems in which the adjoining faces of two cubes must match in colour. Eight of these, for example, can be used to construct a 2 X 2 X 2 cube with solid faces that duplicate the colours of any other cube selected at random (Johnson, 1956; Ehrenfeucht, 1964; Farrell, 1969).

One coloured cube puzzle that was popular about 1968 was a puzzle called Instant Insanity (Wahl, 1968; Wells, 1983). This consisted of four coloured cubes, painted with four colours, that had to be stacked so that each colour was visible on each side of the column. This has been a recurring favourite since the late nineteenth – century, often being extended to five cubes (Filipiak, 1942; O’Beirne, 1961f, g). What appears to be an arbitrary set of colours follows a definite mathematical rule (Berlekamp, Conway, & Guy, 1982; Brown, 1968). Shader (1978; Lulli, 1980; Perisho, 1960) has produced a puzzle using coloured tetrahedrons that follow the same principle.

PEG SOLITAIRE

For puzzle aficionados, solitaire refers to a single player peg game using a board in which all but one hole is filled with pegs. The rules allow one peg to jump another and the jumped peg is then removed. Only vertical and horizontal moves are permitted and the goal is to remove all but one peg from the board. The modern version, which is available in most toy stores, uses the 33 – hole board shown in Figure 6 (Beasley, 1985; Maltby, 1974; Ramsay, 1974; Rohrbaugh, 1930). The older version, popular in 17th – century France, had 37 holes that added an extra hole in each corner of the cross (Slocum & Botermans, 1986; van Delft & Botermans, 1978). According to legend, Solitaire was invented by a prisoner in the Bastille who had no opponent for Fox and Geese (Bell, 1973) and so devised a new game on the same board. The legend is probably not true; O’Beirne (1961a,d) reports similar puzzles from ancient Crete and India.

Traditionally, the goal is to remove all but one peg which should end up in the centre hole. Bergholt (1920) demonstrated that this could be done in 18 moves; Beasley (1962) later showed it could not be done in fewer. Berlekamp, Conway, and Guy (1982) provide an extensive analysis of strategies for the game and regard it as “the hardest game of its kind to have gained substantial popularity” (p. 698). Other mathematical analyses are offered by Davis (1967), DeBruijn (1972), and McKerrell (1972). Many variations on the puzzle use different positions for the remaining peg or require the player to finish with a particular pattern (Dudeney, 1917) or to begin with a limited pattern of pegs (van Delft & Botermans, 1978).

There are several variations on the board design. The simplest is a line of seven holes with three pegs of one colour at one end and three pegs of an opposing colour at the other and the goal is to exchange the peg positions following the jumping rule without removing pegs (Hoffman, 1893; Filipiak, 1942). Other variations include overlapping squares (Hoffman, 1893; van Delft & Botermans, 1978), triangles, (Hentzel, 1973), squares, and even an extension to a cube (Cross, 1968). Most of these have been analysed extensively to determine for the minimum number of moves.

SLIDING PIECE PUZZLES

A sliding piece puzzle consists of a set of tiles, held in a frame, that are set to an initial configuration or are randomized and then must be moved, one at a time, to a new arrangement. Two of these are the most successful commercial ventures of all time: Sam Loyd’s “14 – 15” puzzle and Erno Rubik’s Magic Cube.

Loyd didn’t invent the sliding piece puzzle but did make it famous (Hordern, 1986). His “14 – 15” puzzle consisted of 15 square tiles, numbered 1 to 15, set in a 4 X 4 frame. The puzzle had a parity restriction (Gardner, 1959) that limits the combinations that can be produced from a given initial arrangement. Loyd used this to advantage by offering a $1000 prize to anyone who could begin with an ordered sequence with two digits reversed and manipulate all the tiles into the correct order. This puzzle captivated the public in the 1880’s, made a fortune for Sam Loyd, and intrigued mathematicians both then (Fraser, 1880; Johnson, 1879; Story, 1879; Tait, 1880) and later (Austin, 1979; Davies, 1970; Liebeck, 1971).

While Loyd’s puzzle is the best known, there are many variations on this theme, some of which appear in Figure 7. Hordern (1986) lists 23 British and 120 American patents for sliding puzzles, including three designed by a blind woman. Filipiak (1942) describes ten different puzzles and lists the minimum number of moves for each; Hordern presents a much larger variety, shows the minimum solution for each, and introduces a five star rating system to indicate the degree of difficulty. Other examples are shown by Hoffman (1893), Gardner (1966), and van Delft and Botermans (1978). Berlekamp, Conway, and Guy (1982) discuss general strategies for the puzzles. Puzzles with square tilesare the easiest to solve; adding rectangular or L – shaped pieces makes the puzzles progressively more difficult. According to Hordern, the most elegant and difficult puzzles were designed by a Japanese coffeeshop owner who used them to entice customers.

The most famous recent variation is Rubik’s Magic Cube, apparently invented almost simultaneously by Erno Rubik, a Hungarian architect and Terutoshi Ishige, a Japanese engineer. The standard version is a 3 X 3 X 3 cube with a different colour on each face (Ewing & Kosniowski, 1982; Frey & Singmaster, 1982; Rubik, Varga, Keri, Marx, & Vekerdy, 1987; Singmaster, 1981; Taylor, 1981). Each layer along any axis can be moved independently so that it can be scrambled into approximately 4 “Formula not transcribed” different combinations. The goal is to learn a general strategy for unscrambling the cube which takes about two weeks of concentrated effort (Singmaster, 1981).

There are three different ways to vary the difficulty of Rubik’s Cube. The first involves changing the number of tiles, producing 2 X 2 X 2 and 4 X 4 X 4 versions. With the smaller version (which Hofstadter (1982) dubbed “Twobik’s Cube”), all the moves are trivial except the last which is topologically equivalent to the last move with the 3 X 3 X 3 version and is extremely difficult to discover; the larger version, introduced as “Rubik’s Revenge” (Adams, 1982), requires a considerably more complex algorithm than the original and was not popular probably because the moves are too difficult to remember even if executed correctly.

The difficulty can also be varied by changing the shape. The Pyraminx Puzzle, a tetrahedron designed by Uwe Meffert in 1972 (Hofstadter, 1982), involves a simple algorithm that can be learned in just a few hours (Schwartz, 1982); Alexander’s Star, a starcovered dodecahedron (Alexander, 1982), appears more complex than the cube but also allows a simpler solution. Hofstadter (1982) lists some other variations on the shape of the cube, some of which are quite exotic and may be more difficult than the original.

The simpliest and most direct way to manipulate the difficulty of Rubik’s Cube is to change the number of pieces to be oriented. With the original, the edge and corner pieces have to be moved to their correct positions then turned, or oriented, in the appropriate direction but the centre pieces on each face are fixed and are solid colours so they need not be oriented. One version of the cube has truncated corner pieces so that the edges have to be oriented but the corners do not; this avoids the difficult final step with the original (Bossert, 1981; Bandelow, 1982; Nourse, 1981) and produces a version that is considerably easier to solve. Another variation, called Rubik’s World, is more difficult than the original cube because it is printed with a copy of the globe so that four of the six centre pieces must be oriented correctly.

The sliding puzzles appear to require strong visuo – spatial abilities but some reservations may be in order. Erno Rubik invented his cube to foster spatial abilities among architecture students but Hofstadter (1981, 1982) has suggested that there may be two different strategies for these problems. Hofstadter (1982, p. 31) contrasts a nonspatial “algebraic approach … that is efficient but risky” with a “geometric one … that is inefficient but reliable” and claims the two approaches can “serve as metaphors for styles of attacking problems”. If Hofstadter is correct, Rubik’s Cube may not always be linked to spatial skills which could, to some extent, account for its popularity.

MECHANICAL PUZZLES

The term mechanical puzzle is reserved for the largest and most diverse category. These involve pieces of wood, wire, or string that seem inextricably linked until one discovers the right trick. Hoffman (1893, p. 84) described them as “dependent on some secret or trick”. There is usually no element of learning or practice and no variation on the problem with the the same pieces. In theory, there could be an infinite number of such puzzles but in practice the same themes are repeated again and again (Collins, 1927; Leeming, 1946, 1947; Pearson, 1907; Wells, 1983). Most require a great deal of manipulation and are ideal for classroom demonstrations of set and insight.

The mechanical puzzle mentioned most often in psychology texts is the Tower of Hanoi, which was introduced in 1883 by Edouard Lucas, a French mathematician, as the Tower of Benares (Ball, 1939; Crowe, 1956: Roth, 1974). This puzzle, shown in Figure 8, consists of a set of disks, of decreasing size, stacked on one of three posts. The disks must be moved, one at a time, to the third post while following the rule that a large disk may never cover a smaller one. The puzzle is easy, even for children, but requires practice to do it in the minimum number of moves possible, which is “Formula not transcribed” where n is the number of disks. The secret as Berlekamp, Conway, and Guy (1982) reveal, is to imagine the disks alternately made of gold and silver and “never place a disk immediately above another made of the same metal” (p. 754). Filipiak (1942) and Brousseau (1975) explore a variation using four posts.

The Chinese Rings puzzle, also shown in Figure 8, is similar to the Tower puzzle but dates back to the 16th century (Ball, 1939; Bell, 1973; Dudeney, 1917; Hoffman, 1893; van Delft & Botermans, 1978). This puzzle consists of a number of rings, all joined to a central bar, covering a U – shaped rod that ends with a handle. The object is to remove the rings. As Berlekamp, Conway, and Guy (1982) point out, the task seems quite hopeless because of the bar but the actual solution is easier to visualize if the solver imagines that the U – shaped rod is a flexible loop of string. A variation using an actual loop of string is described by van Delft and Botermans (1978, p.152).

Some of the oldest mechanical puzzles involve a simple loop of string attached to pieces of wood, often with rings or balls attached, and the goal is to remove the string or move one of the objects. One example is shown at the top of Figure 9. The string forms a “lark’s head” knot, not a “clove hitch”, with the hole in the middle of the stick and the ends of the string are firmly attached to each end of the stick. The goal is to move the ring from the loop on side A to the loop on side B. Since the ring is too large to pass through the centre hole, the secret is to bring the knot through the hole in order to allow the ring to pass the middle. This general principle is used with a wide variety of “ring – string – ball” puzzles; some of the best compilations are presented by Hoffman (1893), Filipiak (1942), Slocum (1955), Kew (1975), van Delft and Botermans (1978), and Berlkamp, Guy, and Conway (1983). This range of variations introduces the possibility of examining the degree of transfer that occurs with a principle that was learned in one specific context, an idea first explored by Ruger (1910) but one that is more pertinent to today’s discussions of implicit memory and amnesia.

Wire puzzles are similar to the ring – string – ball puzzles in that two shapes seem inextricably linked and the object is to separate them and, again, success depends on discovering the trick. The shape of the objects is often the most important clue. The shapes for the heart – and – arrow problem, shown in Figure 9, are not arbitrary; the exaggerated bend in the heart is essential for solving the problem. This is also true for the double – bow in Figure 9 although that solution is less obvious (Kraitchik, 1953). A variety of wire puzzles were used by Ruger (1910) but the best sources include Hoffman (1893), Slocum (1955), and van Delft and Botermans (1978). [However, one of the puzzles mentioned by van Delft and Botermans (1978, p. 154), the “Loony Loop”, is quite impossible (see Wells, 1976, 1977).]

Mechanical puzzles also include the burr puzzles, which consist of interlocking pieces of wood forming a regular three – dimensional object, often a cube or cross. The most common consists of six pieces, each about 2 X 2 units square and about 6 units in length, which are notched in such a way that they can be assembled into a three – dimensional cross with a pair of cross – pieces in each direction. The method of assembly depends entirely on the way in which the notches are formed and a very wide variety is possible (Slocum, 1955). Wyatt (1946) reported 17 different six – piece burrs while Filipiak (1942) had earlier described 73. Cutler (1978), in what appears to be a definitive analysis, reports that there are 314 burrs that can be formed of notchable pieces, i.e. pieces in which the notches are complete saw cuts perpendicular to the axis of the rod, and a total of 119,979 if all manner of cuts are allowed. Many of these appear to have been explored by J.H. deBoer (see van Delft & Botermans, 1978) who apparently has made over 2,096 variations.

The six – piece burr is the most common but there are at least two types of three – piece burrs (Hoffman, 1893; Wyatt, 1946) and one, a cube, with only two pieces (O’Beirne, 1962b). The six – piece also has a counterpart in which the pieces are fitted diagonally and resemble a star (O’Beirne, 1961e; Wyatt, 1946). Wyatt (1946) has the most diverse compendium with burrs of 3, 6, 9, 12, 15, 19 and 30 pieces. The record, however, may be a burr with no fewer than 51 pieces (van Delft & Botermans, 1978, p. 77).

Burr puzzles, although intriguing, should not be used for research on spatial skills. The trick with these puzzles is to find the key piece that allows the puzzle to be disassembled but putting it back together again can a heruclean effort. Burr puzzles have an aesthetic appeal that transcends all other qualities. These are objets d’art, not puzzles waiting to be solved. Researchers be warned: Students will reduce a burr to a pile of sticks then completely lose all interest in it!

Conclusions

Spatial puzzles are more than mere toys but the fact that they are toys and are very popular toys, as Dudeney (1926) attests, is important. Do children develop good spatial skills by playing with such toys or does interest in the toys reflect latent skills? In order to answer such questions, psychologists need appropriate environments in which to test and train spatial abilities. The purpose of the present article is to demonstrate that there is a rich array of spatial puzzles that have been explored by puzzle enthusiasts and mathematicians and can be used to advantage by psychologists.

Preparation of the manuscript was supported by NSERC grant OGP – 9581 to the author. The author would like to thank Carol, Gregg, Sara, and Michael for patiently testing all these puzzles. If any readers are familiar with other puzzles or wish a reprint, they are invited to write to the author at: Department of Psychology, Queen’s University, Kingston, Ontario, Canada, K7L 3N6.

References

Abraham, R.M. (1964). Diversions and Pastimes. New York: Dover.

Adams, J. (1982). How to Solve Rubik’s Revenge. New York: Dial Press.

Alexander, A. (1982). The Official Solution to Alexander’s Star, New York: Ballentine.

Anderson, J.H. (1962a). Polyominoes — the twenty problem. Recreational Mathematics Magazine, 9, 25 – 30.

Anderson, J.H. (1962b). Polyominoes — the twenty problem and others. Recreational Mathematics Magazine, 10, 25 – 28.

Austin, A.K. (1979). The 14 – 15 puzzle. Mathematical Gazette, 63, 45 – 46.

Ball, W.W.R. (1939). Mathematical Recreations and Essays. New York: Macmillan.

Bandelow, C. (1982). Inside Rubik’s Cube and Beyond. Boston, MA: Birkhauser.

Beasley, J.D. (1962). Some notes on solitaire. Eureka, 25, 13 – 18.

Beasley, J.D. (1985). The Ins and Outs of Peg Solitaire. Oxford: Oxford University Press.

Bell, R.C. (1973). Fox & geese and solitaire. Games & Puzzles, issue #17, 4 – 5.

Bell, R.C. (1973). The wisdom puzzle: An introduction to tangrams. Games & Puzzles, issue #17, 8 – 9.

Bergholt, E. (1920). Complete Handbook to the Game of Solitaire on the English Board of Thirty – Three Holes. New York: E.P. Dutton.

Berlkamp, E.R., Conway, J.H., & Guy, R.K. (1982). Winning Ways for Your Mathematical Plays, Vol. 2. New York: Academic Press.

Bossert, P. (1981). You Can Do the Cube. New York: Puffin.

Bouwkamp, C.J. (1969). Packing a rectangular box with the 12 solid pentominoes. Journal of Combinatorial Theory, 278 – 280.

Bouwkamp, C.J. & Klarner, D.A. (1970). Packing a box with y – pentacubes. Journal of Recreational Mathematics, 3, 10 – 26.

Bouwkamp, C.J. (1970). Simultaneous 4 X 5 and 4 X 10 pentomino rectangles. Journal of Recreational Mathematics, 3, 125.

Bouwkamp, C.J. (1971). A new solid pentomino problem. Journal of Recreational Mathematics, 4, 179 – 186.

Bradshaw, J.L. & Nettleton, N.C. (1983). Human Cerebral Asymmetry. Englewood Cliffs, NJ: Prentice – Hall.

Brooks, D.N. & Baddeley, A. (1976). What can amnesic patients learn? Neuropsychologica, 14, 111 – 122.

Brousseau, A. (1975). Tower of Hanoi with more pegs. Journal of Recreational Mathematics, 8, 169 – 175.

Brown, T.A. (1968). A note on Instant Insanity. Mathematics Magazine, 41, 68.

Chi, M.T.H. & Glaser, R. (1985). Problem – solving ability. In R.J. Sternberg (Ed.), Human Abilities: An information – processing approach. San Francisco, CA: W. H. Freeman, 227 – 250.

Cohen, N.J. (1984). Preserved learning capacity in amnesia: Evidence for multiple memory systems. In L. R. Squires & N. Butters (Eds.), Neuropsychology of Memory. New York: Guilford Press, 83 – 103.

Collins, A.F. (1927). The Book of Puzzles. New York: D. Appleton & Co.

Corballis, M.C. (1983). Human Laterality. New York: Academic Press.

Cronbach, L.J. (1960). Essential of Psychological Testing, 2nd ed. New York: Harper & Row.

Cross, D.C. (1968). Square solitaire and variations. Journal of Recreational Mathematics, l, 121 – 123.

Crowe, D.W. (1956). The n – dimensional cube and the Tower of Hanoi, American Mathematical Monthly, 63, 29 – 30.

Cutler, W.H. (1978). The six – piece burr. Journal of Recreational Mathematics, 10, 241 – 250.

Davies, A.L. (1970) Rotating the fifteen puzzle. Mathematical Gazette, 54, 237 – 240.

Davis, H.D. (1967). 33 – solitaire, new limits, small and large. Mathematical Gazette, 51, 91 – 100.

DeBruijn, N.G. (1969). Filling boxes with bricks. American Mathematical Monthly, 76, 37 – 40.

DeBruijn, N.G. (1972). A solitaire game and its relation to a finite field. Journal of Recreational Mathematics, 5, 133 – 137.

Dudeney, H.E. (1908). The Canterbury Puzzles and Other Curious Problems. NewYork: E.P. Dutton & Co. (reprinted Dover, 1958).

Dudeney, H.E. (1917). Amusement in Mathematics. London: Thomas Nelson & Sons. (reprinted Dover, 1958).

Dudeney, H.E. (1926). The psychology of puzzle crazes. The Nineteenth Century Magazine, 100, 868 – 879.

Dudeney, H.E. (1919). The Canterbury Puzzles. New York: Dover. (Original 1917).

Ehrenfeucht, A. (1964). The Cube Made Interesting. Oxford: Pergamon.

Elffers, J. (1973). Tangrams. Cologne: M. DuMont Schauberg.

Ewing, J. & Kosniowski, C. (1982). Puzzle It Out: Cubes, Groups and Puzzles. Cambridge: The University Press.

Eysenck, M.W. & Keane, M.T. (1990). Cognitive Psychology: A student’s handbook. Hillsdale, NJ: Lawrence Erlbaum.

Farrell, M.A. (1969). The mayblox problem. Journal of Recreational Mathematics, 2, 51 – 56.

Feser, Fr. V. (1968). Pentomino farms. Journal of Recreational Mathematics, l, 675 – 682.

Filipiak, A.S. (1942). Mathematical Puzzles and Other Brain Teasers. New York: A.S. Barnes.

Fiore, A. (1981). Shaping Rubik’s Snake. New York: Penguin.

Fraser, P. (1880) Three methods and forty – eight solutions to the fifteen problem. Proceedings of the American Philosophical Society, 18, 505 – 510.

Frey, A.H. Jr. & Singmaster, D. (1982). Handbook of Cubik Math. New York: Enslow Publishers.

Gardner, M. (1959). The Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon & Schuster.

Gardner, M. (1961). The Second Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon & Schuster.

Gardner, M. (1966). New Mathematical Diversions from Scientific American. New York: Simon & Schuster.

Gardner, M. (1969). The Unexpected Hanging and Other Mathematical Diversions. New York: Simon & Schuster.

Gardner, M. (1970). Further Mathematical Diversions. London: George Allen and Unwin.

Gardner, M. (1971). The Sixth Book of Mathematical Games from Scientific American. New York: Charles Scribner’s.

Golomb, S. (1954). Checker boards and polyominoes. American Mathematical Monthly, 61, 675 – 682.

Golomb, S. (1965). Polyominoes. New York: Charles Scribner’s.

Harary, F. & Weisbach, M. (1982). Polycube achievement games. Journal of Recreational Mathematics, 15, 241 – 146.

Hartswick, F.G. (1925). The Tangram Book. New York: Simon & Schuster.

Haselgrove, C.B. & Haselgrove, J. (1960). A computer program for pentominoes. Eureka, 23, 16 – 18.

Hayes, J.R. (1989). The Complete Problem – Solver, 2nd. ed. Philadelphia, PA: Franklyn Institute Press.

Hayes, J.R. & Simon, H.A. (1974). Understanding written problem instructions. In L. Gregg (Ed.), Knowledge and Cognition. Hillsdale, NJ: Lawrence Erlbaum, 167 – 200.

Hein, P. (1969). Introducing Soma. Salem, MA: Parker Brothers.

Hentzel, I.R. (1973). Triangular puzzle peg. Journal of Recreational Mathematics, 6, 280 – 283.

Hoffman, Professor [pseudonym of Angelo Lewis] (1893). Puzzles Old and New. New York: Frederick Warne.

Hofstadter, D.R. (1981). Metamagical themas: The magic cube’s cubies are twiddled by cibists and solved by cubemeisters. Scientific American, 244, 20 – 39.

Hofstadter, D.R. (1982). Metamagical themas: Beyond Rubik’s Cube. Scientific American, 247, 16 – 31.

Hordern, E. (1986). Sliding Piece Puzzles. Oxford: Oxford University Press.

Johnson, P.B. (1956). Stacking colored cubes. American Mathematical Monthly, 63, 392 – 395.

Johnson, W.W. (1879). Notes on the ’15’ puzzle, I. American Journal of Mathematics, 2, 397 – 399.

Kew, B. (1975). Patently puzzling: Parts 1 & 2. Games & Puzzles, 36, 17 – 19; 37, 18 – 20.

Klamkin, M.S. & Liu, A. (1980). Polyominoes on the infinite checkerboard. Journal of Combinatorial Theory, 28a, 7 – 16.

Klarner, D.A. (1969). Packing a rectangle with congruent n – ominoes. Journal of Combinatorial Theory, 7, 107 – 115.

Klarner, D.A. (1973). Brick – packing puzzles. Journal of Recreational Mathematics, 6, 112 – 117.

Klarner, D.A. (1980). A search for n – pentacube prime boxes. Journal of Recreational Mathematics, 12, 252 – 257.

Kohler, W. (1925). The Mentality of Apes. London: Routledge & Kegan Paul.

Kotovsky, K. & Fallside, D. (1989). Representation and transfer in problem – solving. In D. Klahr & K. Kotovsky (Eds.), Complex Information – Processing: The impact of Herbert A. Simon. Hillsdale, NJ: Lawrence Erlbaum, 69 – 108.

Kraitchik, M. (1953). Mathematical Recreations. New York: Dover.

Kramer, E.S. (1983). Tiling rectangles with T and C pentominoes. Journal of Recreational Mathematics, 16, 102 – 113.

Kramer, E.S. & Gobel, F. (1983). Tiling rectangles with pairs of pentominoes. Journal of Recreational Mathematics, 16, 198 – 206.

Leeming, J. (1946). Fun with Puzzles. Philadelphia: J.B. Lippincott Co.

Leeming, J. (1947). More Fun with Puzzles. Philadelphia: J.B. Lippincott Co.

Liebeck, H. (1971) Some generalizations of the 14 – 15 puzzle. Mathematics Magazine, 44, 185 – 189.

Lindgren, H. (1961). Going one better in geometrical dissections. The Mathematical Gazette, 45, 94 – 97.

Lindgren, H. (1968). Tangrams. Journal of Recreational Mathematics, l, 1 – 9.

Lindgren, H. (1972). Recreational Problems in Geometric Dissections and How to Solve Them, Rev. ed. New York: Dover.

Liu, A. (1982). Pentomino problems. Journal of Recreational Mathematics, 15, 6 – 13.

Loftus, G.R. & Loftus, E. F. (1983). Mind at Play: The psychology of video games. New York: Basic Books.

Loyd, S. (1903). The Eighth Book of Tan: Seven Hundred Tangrams, Rev. ed. New York: Dover. Reprinted 1968.

Loyd, S. (1914). The 14 – 15 puzzle in puzzleland. In S. Loyd (Ed.), Cyclopedia of 5000 Puzzles. New York: Lamb Publishing Co.

Loyd, S. (1959). The Best Mathematical Puzzles of Sam Loyd. (Ed. by M. Gardner). New York: Dover.

Lulli, H. (1980). The icosahedron and the tangled tetrahedron. Journal of Recreational Mathematics, 12, 172 – 176.

MacMahon, P.A. (1915). Combinatory Analysis. Cambridge: The University Press.

MacMahon, P.A. (1921). New Mathematical Pastimes. Cambridge: The University Press.

Malachy, J.S. (1969). Pentominoes: Some solved and unsolved problems. Journal of Recreational Mathematics, 2, 181 – 188.

Maltby, J. (1974). The solitaire revival. Games & Puzzles, issue #30, 8 -9.

Mayer, J.A. (1973). Pentomino problem. Journal of Recreational Mathematics, 6, 105 – 108.

Mayer, R.E. (1977). Thinking and Problem Solving: An introduction to human cognition and learning. Glenview, IL: Scott, Foresman.

McKerrell, A. (1972). Solitaire: An application of the four – group, Mathematics Teaching, (60), 19 – 22.

Meeus, J. (1973). Some polyomino and polyamond problems, Journal of Recreational Mathematics, 6, 215 – 220.

Meeus, J. (1973). Tetracubes. Journal of Recreational Mathematics, 6, 257 -265.

Miller, J.C.P. (1960). Pentominoes. Eureka, 23, 13 – 16.

Newell, A. & Simon, H. (1972). Human Problem Solving. Englewood Cliffs, NJ: Prentice – Hall.

Nourse, J. (1981). The Simple Solution to Rubik’s Cube. New York: Bantam Books.

O’Beirne, T.H. (1961a). Puzzles and paradoxes: “Going straight” and the siege of London. The New Scientist, issue 218, 170 – 171.

O’Beirne, T.H. (1961b). Puzzles and Paradoxes: MacMahon’s three – colour set of squares. The New Scientist, issue #220, 288 – 289.

O’Beirne, T.H. (1961c). Puzzles and paradoxes: A six – block cycle for six step – cut pieces. The New Scientist, issue #224, 560 – 561.

O’Beirne, T.H. (1961d). Puzzles and Paradoxes: Pentalpha and pentomega. The New Scientist, issue #233, 264 – 265.

O’Beirne, T.H. (1961e). Puzzles and paradoxes: Stars mystic and more mystic. The New Scientist, issue #243, 110 – 111.

O’Beirne, T.H. (1961f). Puzzles and paradoxes: “Plus Ca change – ” or the great tantalizer. The New Scientist, issue #247, 358 – 359.

O’Beirne, T.H. (1961g). Puzzles and paradoxes: How to make chains with patterns on cubes. The New Scientist, issue #249, 486 – 487.

O’Beirne, T.H. (1961h). Puzzles and paradoxes: Impossible dovetails and ways of making them. The New Scientist, issue #252, 678 – 679.

O’Beirne, T. H., (1961i). Puzzles and paradoxes: Pell’s equation in two popular problems. The New Scientist, issue #258, 260 – 261.

O’Beirne, T.H. (1961j). Puzzles and paradoxes: Pentominoes and hexiamonds. The New Scientist, issue #259, 316 – 317.

O’Beirne, T.H. (1961k). Puzzles and paradoxes: Thirty – six triangles make six hexiamonds make one triangle. The New Scientist, issue #263, 706 – 707.

O’Beirne, T.H. (1962a). Puzzles and paradoxes: Some tetrabolical difficulties. The New Scientist, issue #270, 158 – 159.

O’Beirne, T.H. (1962b). Puzzles and paradoxes: Taking two bites at an impossible two – part cube. The New Scientist, issue #272, 276 – 277.

Odier, M. (1975a). Patterns in space: Part 1. Games & Puzzles, issue #37, 12 – 17.

Odier, M. (1975b). Patterns in space: Part 2. Games & Puzzles, issue #38, 16 – 20.

Odier, M. (1975c). Patterns in space: Part 3. Games & Puzzles, issue #39, 14 – 16.

Pajitnov, A. (1988). TETRIS. Alameda, CA: Spectrum Holobyte.

Parlett, D. (1973a). Pentominoes: The game that has everything and costs nothing. Games & Puzzles, issue #9, 4 – 6.

Parlett, D. (1973b). Pentominoes: Try it with trellises. Games & Puzzles, issue #9, 4 – 6.

Parlett, D. (1973c). Pentominotation. Games & Puzzles, issue #11, 24 – 25.

Pearson, A.C. (Ed.), (1907). The Twentieth — Century Standard Puzzle Book. London: G. Rougledg & Sons.

Perisho, C.R. (1960). Colored polyhedra: A permutation problem. Mathematics Teacher, 53, 253 – 255.

Philpott, W.E. (1969). MacMahon’s threecolor squares, Journal of Recreational Mathematics, 2, 67 – 78.

Philpott, W.E. (1971a). Domino and superdomino recreations: Part 1, Journal of Recreational Mathematics, 4, 2 – 18.

Philpott, W.E. (1971b). Domino and superdomino recreations: Part 2, Journal of Recreational Mathematics, 4, 79 – 87.

Philpott, W.E. (1971c). Domino and superdomino recreations: Part 3, Journal of Recreational Mathematics, 4, 229 – 243.

Philpott, W.E. (1972a). Domino and superdomino recreations: Part 4, Journal of Recreational Mathematics, 5, 102 – 122.

Philpott, W.E. (1972b). Domino and superdomino recreations: Part 5, Journal of Recreational Mathematics, 5, 177 – 196.

Philpott, W.E. (1972c). Solution to problem 167: A MacMahon triangle problem. Journal of Recreational Mathematics, 5, 72 – 73.

Philpott, W.E. (1973). Domino and superdomino recreations: Part 6, Journal of Recreational mathematics, 6, 10 – 34.

Philpott, W.E. (1974). Covering cubes with MacMahon’s three – color squares. Journal of Recreational Mathematics, 7, 266 – 275.

Ramsay, D. (1974). Solitaire. Games & Puzzles, issue #31, 6 – 13.

Read, R.C. (1965). Tangrams: 330 puzzles. New York: Dover.

Reeve, J.E. & Tyrrell, J.A. (1961). Maestro puzzles. The Mathematical Gazette, 45, 97 – 99.

Rohrbaugh, L. (1930). Puzzle craft. Delaware, Ohio: Co – operative Recreation Service.

Roth, T. (1974). The Tower of Brahma revisited. Journal of Recreational Mathematics, 7, 116 – 119.

Rubik, E., Varga, T., Keri, G., Marx, G., & Vekerdy, T. (1987). Rubik’s Cubeic Compendium. Oxford: Oxford University Press.

Ruger, H. (1910). The psychology of efficiency. Archives of Psychology, No. 5.

Schuh, F. (1968). The Master Book of Mathematical Puzzles and Recreations. New York: Dover.

Scheerer, M. (1963). Problem solving, Scientific American, 208(4), 118 – 128.

Schwartz, B.L. (1982). Pyraminx – an improved solution. Journal of Recreational Mathematics, 15, 31 – 38.

Shader, L. (1978). Cleopatra’s Pyramid. Mathematics Magazine, 51, 57 – 58.

Singmaster, D. (1981). Notes on Rubik’s Magic Cube. New York: Enslow.

Slocum, J. (1955). Making and solving puzzles. Science & Mechanics, October issue, 121 – 126.

Slocum, J. & Botermans, J. (1986). Puzzles Old and New: How to Make and Solve Them., Seattle, WA: University of Washington Press.

Squire, L.R. (1987). Memory and Brain. New York: Oxford University Press.

Steinhaus, H. (1950). Mathematical snapshots, New York: Oxford United Press.

Story, W.E. (1879). Notes on the ’15’ puzzle, II. American Journal of Mathematics, 2, 399 – 404.

Stubbs, A.D. (1931). Miscellaneous Puzzles. New York: Frederick Warne.

Tait, P.G. (1880) Note on the theory of the ’15’ puzzle. Royal Society of Edinburgh Proceedings, 10, 664 – 665.

Taylor, D. (1981). Mastering Rubik’s Cube. New York: Holt, Rinehart & Winston.

Torbijn, I.P.J. (1969). Polyiamonds. Journal of Recreational Mathematics, 2, 216 – 227.

van Delft, P. & Botermans, J. (1978). Creative Puzzles of the World. New York: Harry N. Abrams.

Verbakel, J.M.M. (1972). Pentacube problem. Journal of Recreational Mathematics, 5, 20 – 21.

Wahl, P. (1968). Crazy cubes, Popular Science, 193, issue #5, 132 – 133.

Wagner, N.R. (1972). Constructions with pentacubes. Journal of Recreational Mathematics, 5, 266 – 268.

Wagner, N.R. (1973). Constructions with pentacubes – 2. Journal of Recreational Mathematics, 6, 211 – 214.

Wang, F.T. & Hsiung, C.C. (1942). A theorem on the tangram. American Mathematical Monthly, 49, 596 – 599.

Wells, D. (1976). Solid Gold. Games & Puzzles Magazine, issue #52.

Wells, D. (1977). Solid Gold. Games & Puzzles Magazine, issue #59.

Wells, K. (1983). Wooden Puzzles and Games. New York: Sterling Publishing Co.

Whinihan, M.J. & Trigg, C.W. (1973). Parity and centeress applied to the soma cube. Journal of Recreational Mathematics, 6, 61 – 66.

Wickelgren, W.A. (1976). How to Solve Problems. San Francisco, CA: W. H. Freeman.

Wyatt, E. M. (1946). Wonders in Wood. Milwaukee, WI: Bruce Publishing, (reissued in 1956 as Puzzles in Wood).

Wylie, C.R. (1957). 101 Puzzles in Thought and Logic. New York: Dover.

Zangwill, O.L. (1987). Wolfgang Kohler. In R.L. Gregory (Ed.), The Oxford Companion to the Mind. Oxford: Oxford University Press, 412 – 413.

Resume

Les casse – tete en trois dimensions

Les casse – tete en trois dimensions, tels que le cube Rubik ou la tour de Hanoi, sont des pieces ou un ensemble d’objets que l’on doit manipuler jusqu’a ce qu’on leur donne une configuration precise. A premiere vue, ils semblent n’etre que des jouets pour les enfants et de simples passetemps, mais la plupart de ces casse – tete sont, en realite, tres complexes, d’un grand interet pour les adultes (Dudeney, 1926), ainsi que l’objet d’analyses approfondies par les hobbyistes et les mathematiciens.

Les casse – tete en trois dimensions representent un objet d’etude unique pour les chercheurs qui se penchent sur les techniques de resolution de problemes et les habiletes spatiales. Plusieurs de ces cassetete constituent un environnement tres riche pour les chercheurs car ils leur proposent une grande variete de problemes de degres de complexite variables. Les chercheurs peuvent egalement modifier les scenarios de problemes a partir de ces casse – tete, ce qui leur permet de determiner si des strategies developpees sous un ensemble specifique de regles se reproduiront pour resoudre des problemes qui seraient definis par un nouvel ensemble de regles. Le probleme que l’on doit resoudre lorsque l’on manipule les tuiles de couleur de MacMahon (1921), par exemple, est de parvenir a les assembler en joignant les tuiles qui ont la meme couleur sur les bordures exterieures de la figure que l’on creera. La resolution de ce probleme, toutefois, peuvent etre abordee sous l’angle de la forme des bords exterieurs, plutot que sous l’angle de la couleur des tuiles a assembler, produisant ce qui est en apparence un nouveau probleme, mais qui demeurerait en fait identique au probleme initial. D’autres casse – tete requierent que l’on assemble des pieces pour former des figures a deux ou trois dimensions, nous permettant ainsi d’etudier combien un certain entrainement a construire des figures a deux dimensions se reproduit lorsqu’il s’agit de former celles a trois dimensions. Ces casse – tete peuvent aider a evaluer les habiletes spatiales mais, ce qui est plus important encore, a enseigner des habiletes spatiales aux participants, dont on enregistre les progres sur videocassette.

Sept groupes de casse – tete sont decrit dans le present article. Le jeu de tangram (figure 1) est un carre compose de pieces a trois et quatre cotes, que l’on doit assembler afin de reproduire differentes formes precises; deux exemples y sont representes a la figure 2. Les Polyominos et les polyamonds (figure 3) sont un ensemble de douze pieces distinctes, chacune etant une combinaison distincte de carres pour le premier et de triangles pour le second, que l’on peut manipuler pour construire de figures plus grandes. Les cubes soma et les box – packing puzzles, ou casse – tete en trois dimensions dont les pieces s’emboitent l’une dans l’autre (figure 4), ressemblent aux polyominos mais sont composes de cubes et s’assemblent pour former des figures plus grandes et en trois dimensions. Les tuiles de couleur (figure 5), ainsi que les cubes de couleur sont egalement composes de pieces a assembler en des figures de plus grande dimension;les pieces ont toutes la meme forme mais une distribution differente de couleurs, et doivent etre assemblees de sorte que les cotes ou les faces qui se touchent aient la meme couleur. Le solitaire (figure 6) ainsi que les sliding block puzzles (figure 7), ne sont pas des casse – tete d’assemblage comme les precedents, mais ils impliquent quand meme un deplacement systematique des elements qui le composent afin de creer une nouvelle figure. La categorie la plus generale, les mechanical puzzles, dont on montre quelques exemples dans les figures 8 et 9, sont tous des casse – tete constitues de deux ou plusieurs morceaux que l’on doit desengager ou separer des autres. La tour de Hanoi pourrait nous paraitre differente des casse – tete appartenant a cette categorie, puisqu’elle ne semble pas comporter de >, mais le casse – tete montre en – dessous dans la figure 8, que l’on nomme > en France, et qui comprend quand meme des morceaux que l’on doit a separer, est l’equivalent mathematique de la tour de Hanoi.

Les chercheurs n’ont pas utilise a profit ces casse – tete parce que la plupart d’entre eux ne connaissent ni la variete de casse – tete qui existent ni la quantite d’ecrits sur le sujet. Pour ceux qui sont desireux d’en apprendre davantage, les recherches en bibliotheque peuvent parfois s’averer longues et difficiles. Il n’existe que peu de livres qui se consacrent exclusivement aux casse – tete et ceux qui existent n’offrent pas au lecteur de bibliographies etoffees. Des articles sur les casse – tete paraissent dans une certain nombre de revues mais, souvent, ne comportent pas, eux non plus, une liste exhaustive de references. En 1926, H.E. Dudeney, l’expert anglais des casse – tete, se plaignait qu’> (traduction Dudeney, 1926, p. 871), et cela demeure vrai, meme de nos jours.

Le but de cet article est de familiariser le lecteur avec les differents types de cassetete, ainsi qu’avec quelques exemples de recherches poussees deja entreprises sur le sujet. Neanmoins, les exemples de recherches sur les casse – tete sont tires exclusivement de sources anglophones. Les lecteurs francophones devraient savoir qu’il existe de nombreuses recherches sur les casse – tete redigees en franCais, qui remontent aux Problemes plaisants et delectables, de C.G. Bachet, publie en 1612 et aux Recreations mathematiques et physiques, de Jacques Ozanam, publie en 1694, et repris dans l’edition de 1960 du classique d’Edouard Luca, Recreations mathematiques.

Copyright Canadian Psychological Association Jan 1994

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