US20100194040A1

US20100194040A1 - Dissection puzzle

Info

Publication number: US20100194040A1
Application number: US12/733,040
Authority: US
Inventors: Mark Thornton Wood
Original assignee: Individual
Current assignee: Global on Puzzles Pty Ltd
Priority date: 2007-08-02
Filing date: 2008-08-01
Publication date: 2010-08-05
Also published as: AU2008281334A1; WO2009015442A1

Abstract

A dissection puzzle comprising a set of pieces, each piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is selected from a set areas of different rational ratios to the area of the assembled puzzle and characterised in that: the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that when arranged to form the assembled puzzle and viewed from one side, each edge of each piece adjacent another piece is adjacent a piece having a different marking.

Description

FIELD OF INVENTION

This invention relates to a dissection puzzle. The present invention is primarily directed to dissection puzzle using what has become known by various names including the “ostomachion” and “loculus of Archimedes”. The general area of the art is sometimes known as recreational mathematics. However, the invention is limited to neither the ostomachion as such nor to such field of use.

BACKGROUND ART

Teaching mathematics, particularly geometry, is sometimes perceived as being difficult on the presumption that the subject is essentially uninteresting by its nature. In short, the subject as normally taught is prosaic in both normal meanings of the word. Puzzles have been proposed to encourage students to learn arithmetic by turning learning time into playtime with little or no compromise on the learning component of the experience. Dissection puzzles have been proposed to assist with the education of students of mathematics. Such puzzles fall into the category of recreational mathematics which includes such well known puzzles as the Chinese tangram, the term “tangram” itself sometimes being used to refer to dissection puzzles generally. In this specification, unless the context requires otherwise, the term “dissection puzzle” refers to puzzles of the type where a number of straight-edged pieces may be placed in edge-to-edge abutting relationship to form a predetermined straight-sided geometric shape. Bearing the above in mind, it might be useful to extend the utility of dissection puzzles involving shaped pieces to include an arithmetic dimension.
In this specification, unless the context requires otherwise, the term loculus of Archimedes may be taken to mean a dissection puzzle wherein a square is divided into fourteen straight-sided pieces, each piece circumscribing an area selected from a set areas of different rational ratios to the area of the square, although it will be appreciated that it is not necessarily common general knowledge in the art to refer to the loculus of Archimedes as a dissection puzzle.
The present invention aims to provide a dissection puzzle which alleviates one or more of the aforementioned deficiencies of the prior art. Another aim is to provide a dissection puzzle which aid in enlivening interest in learning mathematics. Other aims and advantages of the invention may become apparent from the following description.

DISCLOSURE OF THE INVENTION

With the foregoing in view, the present invention in one aspect resides broadly in a dissection puzzle comprising a set of pieces, each piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is selected from a set areas of different rational ratios to the area of the assembled puzzle and characterised in that:
the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that when arranged to form the assembled puzzle and viewed from one side, each edge of each piece adjacent another piece is adjacent a piece having a different marking.
Preferably, the markings are also arranged such that when arranged to form the assembled puzzle, there is provided an alternative arrangement of pieces such that the assembled puzzle is comprised of a plurality of regions, one for each marking and of equal area to each of the other regions. The rational ratios may also be selected such that one or more of the areas are in rational ratio to one or more of the other areas.
In another aspect, the present invention resides broadly in a dissection puzzle comprising a set of pieces, each piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is selected from a set areas of different rational ratios to the area of the assembled puzzle and characterised in that:
the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that when arranged to form the assembled puzzle and viewed from one side, each edge of each piece adjacent another piece is adjacent a piece having a different marking, and
the markings are also arranged such that when arranged to form the assembled puzzle, there is provided an alternative arrangement of pieces such that the assembled puzzle is comprised of a plurality of regions, one for each marking and of equal area to each of the other regions.
In another aspect, the present invention in one aspect resides broadly in a dissection puzzle comprising a set of pieces, each playing piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle in the form of a straight-sided geometric shape by assembling the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is selected from a set areas of different rational ratios to the area of the assembled puzzle and characterised in that:
the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that the pieces may be assembled to form an assembled puzzle comprising contiguous areas for each marking in exact ratio for the number of different markings.
In another aspect, the present invention reside broadly in a dissection puzzle comprising a set of pieces, each playing piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle in the form of a straight-sided geometric shape by assembling the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is selected from a set areas of different rational ratios to the area of the assembled puzzle and characterised in that:
the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that said assembled puzzle may be formed from all of the pieces with all but one of the markings common to one side of the assembled puzzle.
Preferably, the assembled puzzle is square in shape and the set of pieces is comprised of fourteen pieces. In such form, the pieces preferably conform to the shapes comprising the loculus of Archimedes. It is preferred that the markings be provided in the form of distinctive colours. It is preferred that the predetermined set of marking comprises four different markings. For example, the three primary colours and black may be selected for the colour markings on the faces of the pieces.
In another aspect, the present invention resides broadly in a dissection puzzle comprising a set of pieces, each piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece being a rational ratio to the area of the assembled puzzle, and the area of at least some of the pieces being in rational ratio to the other pieces, and characterised in that:
the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that when arranged to form the assembled puzzle and viewed from one side, each edge of each piece adjacent another piece is adjacent a piece having a different marking.
In another aspect, the present invention resides broadly in a dissection puzzle comprising a set of pieces conforming to the loculus of Archimedes, each piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship and characterised in that:
the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that when arranged to form the assembled puzzle and viewed from one side, each edge of each piece adjacent another piece is adjacent a piece having a different marking.

BRIEF DESCRIPTION OF THE DRAWINGS

In order that the invention may be more readily understood and put into practical effect, one preferred embodiment of the present invention will described with reference to the following drawings, and wherein:

FIG. 1 is a diagrammatic representation of the dissection puzzle in the form of a solved loculus of Archimedes known in the art;

FIG. 2 shows the set of pieces comprising a preferred embodiment of the dissection puzzle according to the invention;

FIG. 3 shows the obverse faces of the pieces of the set of FIG. 2;

FIG. 4 shows the reverse faces of the pieces of FIG. 3; and

FIGS. 5 and 6 each show one arrangement of pieces of the puzzle of FIGS. 1 to 4 into a square in accordance with the invention.

DETAILED DESCRIPTION OF THE DRAWINGS

The loculus of Archimedes 20 shown in FIG. 1 comprises the fourteen pieces numbered 1 to 14 against a twelve-by-twelve square grid to demonstrate the feature that each piece is in rational ratio in area to the square. The pieces all fit in to a large square 21, the dimensions of the large square being indicated by the 144 small squares indicated typically by reference numeral 22. In FIGS. 2 to 4, each piece of the puzzle is shown with the same piece number as is given in FIG. 1. However, the pieces are coloured on the obverse and reverse faces to provide the features of the invention herein defined and described. The allocation of the colours is set forth in Table 1.
The pieces 1 to 7 and 9 to 12 are all scalene triangles. Piece 8 and 14 are quadrilaterals and piece 13 is a pentagon. The pieces 1 and 7 are right triangles, and the quadrilateral 14 includes a right angle. The pentagon is an uneven sided pentagon, but has two parallel sides and two right angles. Although there are many solutions for arranging the pieces into a square, the obverse and reverse sides are selected for the solution as shown in FIG. 6 which has four equal and contiguous areas of each colour in the shape of a right triangle in which one of the sides opposite the hypotenuse is double the length of the other.
In an educational setting, the 14 pieces may be used as an introduction to conservation and calculation of area, transformations, shapes and forms, logical deductive thinking, geometry of angles, combinations and permutations, topology and possible many other areas of mathematics, particularly the mathematics involved in geometry. It may be observed that pieces 9 and 10 are identical in shape, but exhibit the property of chirality when one of the pieces is reversed. The puzzle may be solved so that each puzzle piece is adjacent a piece of a different colour, and such an arrangement of the pieces may admit that pieces 9 and 10 and pieces 4 and 5 which are the chiral pieces of the set arranged with the opposite handed shape of each pair made visible. It is believed by the inventor that such an arrangement, absent, of course, the colours selected to provide the present invention, may be the preferred solution proposed by Archimedes, many solutions proposed by Archimedes being lost in antiquity. One such solution in accordance with the present invention is shown in FIG. 5.

TABLE 1

Piece number	Obverse Face Colour	Reverse Face Colour

1	Black	Yellow
2	Blue	Black
3	Blue	Yellow
4	Blue	Red
5	Black	Blue
6	Black	Red
7	Yellow	Blue
8	Red	Black
9	Black	Yellow
10	Red	Black
11	Red	Blue
12	Red	Yellow
13	Blue	Red
14	Yellow	Red

It will be appreciated that the pieces forming the loculus of Archimedes have corners which overlie an intersection of the grid lines of the twelve-by-twelve grid in over which the pieces may be placed to solve the loculus of Archimedes puzzle. It has been suggested that there are 576 different arrangements of the puzzle pieces into the square, but it is to be appreciated that the loculus of Archimedes was not developed to provide a challenge to assemble the pieces into the square, nor to determine the number of solutions to do so, but as a result of the challenge to divide the square up into fourteen pieces each of rational ratio to the area of the square. In other words, the divisor selected for obtaining the number of pieces was a divisor in respect of which a simple even division into like pieces is not possible.
Insofar as use of the puzzle in the teaching of geometry of angles is concerned, it has been discovered that propositions set forth in Euclid's “Elements Book 1” may be demonstrated. Pieces 9 and 10 can be used to demonstrate proposition 6 from Euclid's Elements Book 1. If in a triangle to angles be equal to one another, the sides which subtend the equal angles will also be equal to one another. In similar fashion, two or more pieces of the puzzle of the present invention may be laid out on a flat surface to demonstrate at least propositions 8, 10, 11, 17, 18, 19, 20, 21, 32, 33, 34, 37, 38, 41 and 46 from Euclid's “Elements Book 1”. Propositions 11 to 14 from Book 1 by Archimedes on the “Equilibrium of Planes” can also be demonstrated. His proposition 14 says that it follows at once from the previous proposition that the centre of gravity of any triangle is the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively. This can be demonstrated by pieces 10, 11 and 12, the longest side of piece 12 placed against the second longest side of piece 11 whereupon the shortest sides of each triangle in alignment. Piece 10 may then be placed so that one of its sides matches the sum of the lengths of the two shortest sides of pieces 11 and 12, whereupon one of the other sides of the triangle of piece 10 is in alignment with the second longest side of piece 12.
Different pieces may also be laid against one another or in combination to produce parallelograms. In order to provide a challenge, users may be requested to construct a parallelogram of, say, 12 square units, 24 square units, 48 square units or 72 square units using selected pieces from the puzzle of the present invention, the units of the square being those depicted in FIG. 1. In similar fashion, shapes such as hexagon and parallelogram may also be set as challenges for solution to uses of the pieces of the puzzle of the present invention.
Although it is one of the features of the present invention to provide puzzles the solutions of which comprise 3 of the 4 different colours, such a feature can be presented to uses as a challenge. It has been found that there are several solutions providing such a feature using the pieces of the puzzle of the present invention.
Additionally, if it is so desired, the colours can be used to create aesthetic or artistic patterns from the pieces of the set. In addition to the above, it is believed by the inventor that the use of the puzzle pieces of the present invention enhances the learning experience of those who would otherwise find the learning of mathematics in general and geometry in particular uninteresting. It is believed by the inventors that the use of the invention and the teaching of principles involved may stimulate creativity in individuals. Because of the provision of a tactile experience in respect of the demonstration of such geometric principles as those found for example in Euclid's Elements Book 1, both retention and understanding of geometric principles may be enhanced by users of the puzzle of the present invention.
Although the invention has been described with reference to one specific example, it will be appreciated by those skilled in the art that the invention may be embodied in other forms within the broad scope and ambit of the invention as herein set forth and defined by the following claims.

Claims

1. A dissection puzzle comprising a set of fourteen pieces, each piece in the set having an obverse face and a reverse face substantially parallel to the obverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is an integer fraction of the area of the assembled puzzle, said integer fractions being selected from a set of integer fractions, each different one from the other, at least one of said pieces having an area different from the area of the other pieces and characterised in that:

the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that when arranged to form the assembled puzzle and viewed from one side, each edge of each piece adjacent another piece is adjacent a piece having a different marking.

2. A dissection puzzle according to claim 1, wherein the markings are arranged such that when the pieces are arranged to form the assembled puzzle, there is provided an alternative arrangement of pieces such that the assembled puzzle is comprised of a plurality of regions, one for each marking and of equal area to each of the other regions for each other marking.

3. A dissection puzzle comprising a set of fourteen pieces, each piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is an integer fraction of the area of the assembled puzzle, said integer fractions being selected from a set of integer fractions each different one from the other, at least one of said pieces having an area different from the area of the other pieces and characterised in that:

the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that when arranged to form the assembled puzzle and viewed from one side, each edge of each piece adjacent another piece is adjacent a piece having a different marking, and

the markings are also arranged such that when arranged to form the assembled puzzle, there is provided an alternative arrangement of pieces such that the assembled puzzle is comprised of a plurality of regions, one for each marking and of equal area to each of the other regions for each other marking.

4. A dissection puzzle comprising a set of fourteen pieces, each playing piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is an integer fraction of the area of the assembled puzzle, said integer fractions being selected from a set of integer fractions each different one from the other, at least one of said pieces having an area different from the area of the other pieces and characterised in that:

the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that the pieces may be assembled to form an assembled puzzle comprising contiguous areas for each marking in exact ratio for the number of different markings.

5. A dissection puzzle comprising a set of fourteen pieces, each playing piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is an integer fraction of the area of the assembled puzzle, said integer fractions being selected from a set of integer fractions each different one from the other, at least one of said pieces having an area different from the area of the other pieces and characterised in that:

the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that said assembled puzzle may be formed from all of the pieces with all but one of the markings common to one side of the assembled puzzle.

6. A dissection puzzle comprising a set of fourteen pieces conforming to the loculus of Archimedes, each piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship and characterised in that: