Liu Hui was a third century Chinese mathematician widely known for his ingenious commentaries on the Nine Chapters on the Mathematical Art, a summary of ancient Chinese mathematics that became the foundation of traditional Chinese mathematics. His approach to cube dissection served as the basis of a classic puzzle referred to as "Liu Hui's Cube Puzzle" in this article. The cube is dissected into three solids that are one half, one third, and one sixth of its original volume. The distinctive structure of the cube puzzle allows students to explore its appealing mathematics in pedagogically flexible ways. From paper folding to 3-D modeling, design, and printing, Liu Hui's Cube Puzzle sustains students' mathematical curiosity as they come to appreciate the convergence of a classic puzzle and modern modeling technologies in supporting mathematical thinking.
The cube, formally known as a regular hexahedron, is one of the classic mathematical objects and has both mathematical and pedagogical significance. It is one of the five Platonic solids, and is also fundamental for the representation of the three-dimensional (3-D) Cartesian coordinate system. The unit cube is further the defining element for the mathematical idea of volume. Aesthetically, the cube is the foundation for a host of geometric puzzles that have been popular at science centers, math clubs, and toy stores. As 3-D design and 3-D printing become accessible to school and college students, the cube has naturally become the universal starting point for 3-D modeling. The cube serves numerous roles in software packages such as Autodesk 123D Design® and 3D Slash®, where it is not only a constituent building block but is also used as a reference for sketching and orientation in a 3-D design environment. The cube truly stands out in both informal and formal mathematics, as well as emerging 3-D technologies, for its historical, mathematical, and technological appeal (Banchoff, 1990; Bolt, 1993; Cundy & Rollett, 1961; Gardner, 2001; Steinhaus, 1969), which will be further demonstrated in this article.
In mathematics education and recreational mathematics, the cube has long served as the foundation for a multitude of puzzles and cube construction tasks. For example, in the elementary and middle grades, nets or templates are often used for students to explore the process of cube construction. A cube can be built from up to 11 distinct nets (Weisstein, n.d.). Furthermore, the cube is often sliced or dissected in ingenious ways for construction puzzles or problem posing. For example, a cube can be dissected into two, three, or four identical pieces in numerous ways, each of which may serve as a puzzle for a certain audience (e.g., Banchoff, 1990; Bolt, 1993; Steinhaus, 1969). As another example, a cube can be dissected using four triangular planes, each defined by three face diagonals around a vertex. When the resulting four corners (triangular pyramids) of the cube are removed, there is a regular tetrahedron left at the core (see Figure 1, below). It is easy to see that each of the four corners is one sixth of the cube and the core is one third of the cube in terms of volume.
Figure 1. When four corners of a cube are sliced away by four planes, each containing three face diagonals, a tetrahedron is left in the middle. Click and drag the image below to see this tetrahedron from different perspectives.
Among the numerous cube dissections, one of them stands out for its aesthetic appeal, pedagogical richness, and historical significance—Liu Hui's Cube Puzzle, which decomposes the cube into three solids that are one half, one third, and one sixth of the cube volume (Figure 2). Liu Hui's cube dissection appears in a classic puzzle, although it is less well documented in educational literature or recreational mathematics. The lack of popularity of Liu Hui's cube puzzle is perhaps due to the foreign terms involved in its description, as is discussed below. In recent years, however, this mathematical curiosity has been re-discovered by mathematics educators, who appreciate its mathematical ingenuity, simplicity, and pedagogical richness. In Hong Kong, for example, Guan and Ke (2009) revisited Liu Hui's cube dissection, using both paper-folding and modeling technologies, and made a convincing case for the use of its history in middle grades mathematics education. Liu Hui's approach not only helps students understand the textbook formulas for solid volumes, but also provides new opportunities for mathematical exploration using emerging technologies. In the United States, Silverman (2013) "stumbled upon" Liu Hui and his ancient ideas at a mathematics conference and later discovered diagrams illustrating Liu Hui's cube dissection. She first modeled it with various materials, then designed templates using GeoGebra, and further created colorful paper solids after Liu Hui's Cube Puzzle. In Japan, mathematics educators have made Liu Hui's cube models out of wood, paper, and even radishes (四角錐, n.d.).
Figure 2. A cube is dissected, using Liu Hui's approach, into three solids that are \({\frac{1}{2},} {\frac{1}{3},} \) and \({\frac{1}{6}}\) of its volume.
In my work with children and mathematics teachers at Southern Illinois University Carbondale and local schools, I have tried to revisit classic modeling activities with emerging technologies such as GeoGebra and Autodesk 123D Design®. Liu Hui's cube dissection has proved to be an engaging activity that appeals to both young children and their teachers. Initially, I designed a dynamic GeoGebra worksheet to generate templates for the three solids in Figure 2 and printed them on cardstock paper for classroom use. Later on, as 3-D design and 3-D printing became accessible to my audience, the cube puzzle evolved into a simple yet captivating 3-D design task and an opportunity for mathematical re-discovery. The convergence of all these activities, from paper-folding to 3-D design and printing, provides a unique experience for teachers to embrace classic mathematics and engage in deep reflection on mathematics teaching and learning in today's classrooms, where traditional mathematical ideas and teaching tools blend naturally with advanced modeling technologies. In this article, I first review the history of Liu Hui's approach to cube dissection and then discuss its educational implications with the overarching goal of enhancing student engagement and sense-making in mathematics teaching and learning.
Figure 2. A cube is dissected, using Liu Hui's approach, into three solids that are \({\frac{1}{2}}, {\frac{1}{3}},\) and \({\frac{1}{6}}\) of its volume. To see the dissection from different perspectives, click then drag the image below.
Liu Hui (刘徽, ca. 220 AD – ca. 280 AD) was a third-century Chinese mathematician, well known as the “Euclid of China” for his dominant contributions to Chinese mathematics (Dauben, 2007). Although little is known today about his personal life, Liu Hui is widely recognized for his original commentaries on the ancient Chinese mathematical text, Jiu Zhang Suan Shu or the Nine Chapters on the Mathematical Art, which laid the foundation for the development of Chinese mathematics (Li & Du, 1987). Over time, Liu Hui's commentaries were further edited and annotated by successive mathematicians such as Li Chunfeng (李淳风, 602 – 672 AD) and Li Huang (李潢, ca.1746 – 1812). Thus, there are in existence various editions of this classic of Chinese mathematics (e.g, Liu & Li, n.d.; Li, 1820). In his commentaries, Liu Hui elaborated on 246 problems in nine chapters, covering a wide variety of mathematical ideas from arithmetic to matrix algebra, along with geometry in various real-world applications characteristic of his era, such as farming and earthworks.
In Chapter Five of Nine Chapters on the Mathematical Art, entitled Shang Gong (商功, construction consultations), Liu Hui commented on a series of 28 problems concerning the volumes of solids in the context of civil engineering and grain measurement. Here, Liu Hui may not have been as much interested in designing a puzzle as finding algorithms to calculate the volumes of various polyhedral solids. Problem 15 of Chapter Five is of special significance in our present case. The original problem, answer, and the corresponding rule in the Nine Chapters on the Mathematical Art are as follows (trans. Shen, Crossley & Lun, 1999, p. 269):
Now given a yangma, with a breadth of 5 chi, a length of 7 chi and an altitude of 8 chi. Tell: what is the volume? Answer: \(93{\frac{1}{3}}\) [cubic] chi. The Rule [for a yangma]: Multiply the breadth by the length, then multiply by the altitude, divide by 3.
The term yangma (阳马), which makes little sense nowadays in modern Chinese, is an ancient Chinese term for a special kind of rectangular pyramid. The term chi is a Chinese linear unit, still in limited use today with a modern definition. In ancient Chinese, chi was also used as a square unit and a cubic unit (as in the problem above), subject to the specific context. In his commentary on the problem above, Liu Hui detailed his method of cube dissection for finding the volume of a yangma. He started with a cube, which has three equal dimensions, and dissected it along two face diagonals, as described below (trans. Shen, Crossley, & Lun, 1999, p. 269):
Take a cube 1 chi in each of breadth, length and altitude. The product is 1 [cubic] chi. Cutting the cube on a diagonal gives two right triangular prisms (qiandu). Dissecting a right triangular prism (qiandu) on a diagonal gives a yangma and a bie'nao. The yangma occupies 2 parts and the bie'nao 1 part. These are fixed rates. Two bie'nao constitute 1 yangma, and 3 yangma constitute 1 cube.
Liu Hui's approach is clear once we get hold of a cube and perform the dissecting operations (see Figure 2, above). His use of terms, however, calls for some explanation.
All three terms were essential in Liu Hui's efforts to find the volumes of polyhedral solids, although they do not make much sense in modern Chinese without further explanation. Therefore, they ought to be used with respect to Liu Hui's specific geometric dissections. The context will help readers understand the underlying mathematical processes and the resulting objects. About the terminology, Martzloff (1987/1997) suggested, "Whatever the exact etymologies, there is no doubt that here these terms have no more to do with architecture than terms like 'pyramid' used in a mathematical context" (p. 283). Indeed, Liu Hui may have just used these pieces as qi (棋, chess pieces) to make sense of other solids or to play games, hence the puzzle nature of Liu Hui's dissections (Martzloff, 1987/1999). As a further linguistic note, Chinese nouns do not have plural forms. Accordingly, in literal translations, their plural forms are the same as their singular forms, such as in the case of “3 yangma constitute 1 cube.”
It is noteworthy that Liu Hui's approach and his use of terms such as qiandu, yangma, and bie'nao are not restricted to cubes only but are applicable to any rectangular prism or rectangular parallelepiped. In fact, Liu Hui used lifang (立方, a rectangular prism) in the Chinese text. When he needed to specify a cube, he would give the lifang three equal dimensions. Therefore, it is important that we make sense of Liu Hui's approach in the context of arbitrary rectangular prisms and treat a cube as a special case (Wagner, 1979; Ying, 2011; Li, 2002). In English translations, an arbitrary rectangular prism (or rectangular parallelepiped) is sometimes called a (rectangular) box or cuboid. For a mathematical puzzle, a cube is more appropriate in light of its rich symmetries. When a cube is used, Liu Hui's approach leads to one type of qiandu, one type of yangma, and two types of bie'nao, depending on the choice of the second diagonal cut. The two types of bie'nao are mirror images of each other. It is visually evident that two congruent qiandu make a cube (see Figure 3) and three congruent yangma also make a cube (see Figure 4). In the case of a cube, therefore, it is fairly easy to see that a yangma is one third of the cube volume. Given six bie’nao, if three of them are mirror images of the rest, one can also make a cube (see Figure 5). Thus, a bie’nao is one sixth of the cube volume. These relationships can also be explored visually using dynamic models.
Figure 3. Two qiandu (triangular prisms) make a cube. To see the dissection from different perspectives, click then drag the image below.
Figure 4. Three yangma (triangular prisms) make a cube. To see the dissection from different perspectives, click then drag the image below.
Figure 5. Six bie'nao (triangular pyramids) make a cube after mirroring three of them. To see the dissection from different perspectives, click then drag the image below.
By contrast, an arbitrary rectangular prism is more complicated because it does not have all the symmetries of a cube. Consequently, Liu Hui's dissection leads to three types of qiandu, six types of yangma, and twelve types of bie'nao (with four right triangular faces), depending on the choice of both diagonals during the dissecting process. Three of the yangma are mirror images of the rest; six of the twelve bie'nao are mirror images of the rest (Shen, Crossley, & Lun, 1999, pp. 272-273). Figure 6 shows Liu Hui's dissection of an arbitrary rectangular prism, where the green piece is a qiandu (1/2 of the rectangular prism), the red piece is a yangma (1/3 of the rectangular prism), and the purple piece is a bie'nao (1/6 of the rectangular prism). Figure 7 further shows three ways to dissect the same rectangular prism, each way yielding a distinct qiandu, yangma, and bie'nao. The three yangma in Figure 7 have distinct bases, and they can be assembled to form the original rectangular prism (Figure 8). Their mirror images also form the original rectangular prism in a different orientation. As can be seen, Liu Hui's approach seems to apply to all arbitrary rectangular prisms as well as to a cube.
Figure 6. Liu Hui's dissection of an arbitrary rectangular prism into a quiandu (green piece), a yangma (red piece), and a bie'nao (purple piece).
Figure 7. Three different dissections of an arbitrary rectangular prism that yield three distinct quiandu, yangma, and bie'nao. The blue pieces are quiandu, the red pieces yangma, and the green pieces bie'nao.
Figure 8. Three distinct yangma can be assembled into the original rectangular prism.
But, in the general case, does each yangma (rectangular pyramid) have the same volume? If so, what is that volume? Liu Hui did not answer these questions directly using the visual method in Figure 8, but instead set out to show that there is a 2-to-1 ratio between the volumes of a yangma and a bie'nao. Given that the qiandu has half the volume of the rectangular prism, this 2-to-1 ratio of the volume of the yangma to that of the bie'nao would then guarantee volumes of \({\frac{1}{2},} {\frac{1}{3},} \) and \({\frac{1}{6}}\) of the volume of the rectangular prism for, respectively, the qiandu, yangma, and bie'nao.
Through the illustrations in Figures 2-8 (preceding webpage), we have come to understand Liu Hui's ideas about cube (and rectangular prism) dissection and the three important Chinese terms used in his description, quiandu, yangma, and bie'nao. But what is the volume of a yangma (rectangular pyramid) that results from such a dissection? And how does the bie'nao (triangular pyramid) compare with the yangma in terms of volume? Although it may be evident in the case of a cube that the yangma volume is one third of the cube volume, the yangma volume is not as clear in the general case of a rectangular prism. Liu Hui, in his commentary, wanted to give a more rigorous justification for the volumes of a yangma and a bie'nao. Toward this goal, he started with a cube and divided it into two congruent qiandu, then divided one of the qiandu into a yangma and a bie’nao. He then made his crucial argument that a yangma is twice as much as a bie'nao in volume, and, once he had established that relationship, concluded that each yangma is one third of the cube volume and each bie'nao is one sixth of the cube volume. Finally, he extended these relationships to a general rectangular prism, but without detailed discussion. How he did so has been an area of research in the history of Chinese mathematics (Wagner, 1979; Shen, Crossley, & Lun, 1999). Another source of confusion is the modern interpretation of ancient Chinese. Ancient Chinese texts can be potentially interpreted in multiple ways. However, there is general agreement today among researchers that, with technical emendations, Liu Hui's approach presented in the case of a cube is still valid in the case of a rectangular prism.
Figure 9. A yangma and a bie'nao are bisected in all three dimensions for a comparison of their volumes. On a smaller scale, the yangma (at right) consists of a cube, two qiandu, and two yangma; the bie'nao (at left) has two qiandu and two bie'nao.
As noted above, Liu Hui sought a general solution on the basis of the special case of a cube. He started with a cube that is two chi in all three dimensions. From such a cube, he took a yangma and a bie'nao. Then, in order to show that the volume of the yangma is twice that of the bie'nao, he dissected the yangma in half along all three dimensions into, on the smaller 1-chi scale, a cube, two qiandu, and two yangma, as illustrated in Figure 9 (above). The bie'nao was dissected in the same way, producing, on the 1-chi scale, two qiandu and two bie'nao. After the bisections, it is clear that the original yangma consists of a smaller cube (equivalent to two qiandu), two smaller qiandu, and two smaller yangma; and the original bie'nao has two smaller qiandu and two smaller bie'nao. Now, because two smaller qiandu make a smaller cube on a 1-chi scale, the original yangma is equivalent to two smaller cubes plus two smaller yangma, and the original bie'nao is equivalent to one smaller cube and two smaller bie'nao. The same bisection can be applied to the smaller yangma and bie'nao iteratively.
Modern mathematicians have tried hard to make sense of Liu Hui's ideas. The following is a synthesis of modern interpretative efforts in the literature (Guo, 2009; Quan & Ke, 2009; Shen, Crossley, & Lun, 1999; Li, 2002; Wagner, 1979). To show the progression of Liu Hui's iterative approach, let us use \(Y_n\) and \(B_n\) for the volumes of the yangma and the bie'nao, respectively, after the \(n\)th iteration, with \(n=0\) for the initial volumes so that \(Y_0\) is the volume of the original yangma and \(B_0\) is the volume of the original bie'nao. We further use \(V\) for the volume of a unit cube, substituting \(V=1\) only at the end of the process. Then, we have the following expressions for the volumes of the original yangma and bie'nao:
When \(n=1,\) we have \[Y_0 = 2V + 2Y_1,\,\,\,\, B_0 = V + 2B_1.\]
When \(n=2,\) we have $$Y_0= 2V + 2\left(2\cdot{\frac{1}{8}}V + 2Y_2\right) = 2V\left(1+\frac{2}{8}\right) + 2^2Y_2$$ and \[B_0 = V + 2\left(\frac{1}{8}V + 2B_2\right) = V\left(1+\frac{2}{8}\right) + 2^2B_2.\]
When \(n=3,\) we have $$Y_0= 2V\left(1+\frac{2}{8}\right) + 2^2\left(2\cdot\frac{1}{8^2}V + 2Y_3\right) =2V\left(1+\frac{2}{8}+\frac{2^2}{8^2}\right) + 2^3Y_3$$ and \[B_0 = V\left(1+\frac{2}{8}\right) + 2^2\left(\frac{1}{8^2}V + 2B_3\right) =V\left(1+\frac{2}{8}+\frac{2^2}{8^2}\right) + 2^3B_3.\]
For the general case, after the \(n\)th iteration, with \(V=1,\) we have \[Y_0 = 2\,{\sum_{i=0}^{n-1} \frac{1}{4^i}} + 2^nY_n,\] and \[B_0 = \sum_{i=0}^{n-1} \frac{1}{4^i} + 2^nB_n.\]
In the equations above, the last two components, \(2^nY_n\) and \(2^nB_n,\) are particularly interesting. Recall that at each step of Liu Hui's dissection of the yangma and the bie'nao, there are two smaller yangma and two smaller bie'nao left over, which together form a smaller cube with one eighth of the previous cube's volume (see Figures 10 and 11). Therefore, remembering that \(V=1,\)
\[2^nY_n+2^nB_n=2^{n-1}(2Y_n+2B_n)= 2^{n-1}\cdot\frac{1}{8^{n-1}}V=\frac{1}{4^{n-1}},\]
which means the sum \(2^nY_n + 2^nB_n\) goes to zero as the process is iterated indefinitely. Indeed, Liu Hui seems to have argued that this “remainder,” the leftover yangma and bie'nao at each step of the iteration, disappears (Shen, Crossley, & Lun, 1999, p. 270):
To exhaust the calculation, halve the remaining breadth, length and altitude respectively, … . The smaller the halves, the finer the remainder. Extreme fineness means infinitesimal, which is formless. In that case, how can one have a remainder?
Since, with each iteration, dissection of the yangma produces two smaller cubes while dissection of the bie’nao produces just one smaller cube, Liu Hui concluded that, in a qiandu formed from a yangma and a bie'nao, there is a 2:1 ratio between the volumes of the yangma and the bie'nao. Thus, the original yangma is twice as much as the original bie'nao in terms of volume. In other words, two bie'nao make one yangma, which is often called “Liu Hui's Principle” (Shen, Crossley, & Lun, 1999, p. 275). Since the qiandu has half the volume of the original cube, it follows that a yangma is one third and a bie'nao is one sixth of the original cube volume. A little algebraic notation can sometimes help convince prospective teachers (and others) of these conclusions. Thus, let us assume that the bie'nao has a volume of B. Then, the yangma has a volume of 2B, and the qiandu has a volume of 3B. The whole cube then has a volume of 6B. Therefore, the yangma is one third and the bie'nao is one sixth of the cube volume.
Although Liu Hui presented his approach in the case of a cube, the iterative process can be extended to the general case of a rectangular prism, using the same analysis shown above. Now, since an arbitrary rectangular prism consists of two qiandu and a qiandu can be dissected into a yangma and a bie'nao in a 2:1 ratio, then a yangma is one third and a bie'nao is one sixth of the original rectangular prism in volume. That is essentially Liu Hui's argument for the volume of a yangma, which involves an iterative process of bisection of edges and, at least in its modern interpretation, the idea of a limit, instead of using an experimental approach that allows three yangma to fit in a cube or a rectangular prism. It seems that Liu Hui had the general case in mind when he was using a cube to illustrate his dissecting strategy.
Liu Hui's original comment, however, is slightly different than the modern interpretation given above and has proven “very difficult to understand” (Shen, Crossley, & Lun, 1999, p. 277). After bisecting the yangma and bie'nao along all three dimensions, Liu Hui proposed that the two blocks be colored black and red, respectively, and combined into a qiandu, as shown in Figures 10 and 11 (below). Then, he halved the breadth and the height and focused on the two black-and-red qiandu (Figure 11), each of which is made of a smaller yangma and a smaller bie'nao (Wagner, 1979, pp. 179-182). Liu Hui thus stated that (Shen, Crossley, & Lun, 1999, p. 270):
[T]he red and black right triangular prisms constitute, in such a case, a cube with altitude of 1 chi and a square base of 1 [square] chi. Every two bie'nao make one yangma.
Liu Hui further indicated an iterative process and the applicability of his approach to an arbitrary rectangular prism without giving detailed justification. Wagner's (1979) interpretation of Liu Hui's original comment has been generally accepted as appropriate since the 1980s by Chinese mathematicians and historians of mathematics (Shen, Crossley, & Lun, 1999). According to Wagner, Liu Hui may have just used the colors to explore the relationships among the various pieces. In the case of a cube, the two red qiandu in Figure 10 make a cube, and the two black qiandu also make a cube. By contrast, in the general case, the two red qiandu in Figure 10 do not fit in a rectangular prism; nor do the two black qiandu. However, two qiandu in the same color have the same volume as a corresponding black-and-red rectangular prism (Guo, 2009). To justify this claim, note that in the general case a red qiandu and a corresponding black qiandu make a black-and-red rectangular prism. Thus, a red qiandu is one half of the black-and-red rectangular prism. Therefore, the two red qiandu in the bie'nao have the same volume as the black-and-red rectangular prism, although they do not fit in such a rectangular prism. The two black qiandu in the yangma, by the same argument, also have the same volume as the black-and-red rectangular prism. In short, the analytic process used for the cube applies to the general case of a rectangular prism. Liu Hui's ingenious dissection techniques and his use of coloring are of pedagogical significance today, especially so when we consider the use of 3-D modeling technologies.
Figure 10. A yangma and a bie'nao are bisected in all three dimensions and colored black and red, respectively.
Figure 11. A yangma and a bie'nao, after their bisections and coloring, are recombined to form a qiandu. The smaller black-and-red qiandu, each made of a yangma and bie'nao, constitute a smaller cube. Note that some of the colors from Figure 10 are removed to highlight the two black-and-red qiandu.
In summary, Liu Hui elaborated on a systematic approach to the dissection of various polyhedral solids in his commentaries on the Nine Chapters on the Mathematical Art. His purpose was obviously to calculate the volume of such solids for practical needs in farming and earthworks. An arbitrary rectangular prism, using his approach, can be decomposed into a qiandu (triangular prism), a yangma (rectangular pyramid), and a bie'nao (triangular pyramid). While a qiandu is one-half of the original volume of the rectangular prism, a yangma is one third, and a bie'nao is one-sixth. Every qiandu can be further dissected into a yangma and a bie'nao, with the latter two pieces in a 2-to-1 ratio in volume. Toward the end of his comment on Problem Fifteen of Chapter Five, Liu Hui summarized his systematic method of cube dissection—his overarching goal and the interconnections among the constituent solids (Shen, Crossley, & Lun, 1999, p. 270):
Exhaustive calculation flows from the situation without involving counting rods. The form of the bie'nao is different from a practical object, and that of a yangma also shows variations in dimensions. Nevertheless, without the definite volume of the bie'nao, there is no way to survey the yangma, and without the definite volume of the yangma, there is no way to know the volume of solids such as cones [zhui, 锥] and frusta [ting, 亭], which are fundamental to the calculations of the tasks of labourers and volumes.
Although Liu Hui used a cube in his original text to illustrate the process of dissection, his method applies to arbitrary rectangular prisms and the various relationships remain constant. In the following sections, however, we focus on the case of cube dissection for its rich symmetry and pedagogical implications. For the purpose of this article, we refer to Liu Hui's cube dissection as "Liu Hui's Cube Puzzle," to suggest how it might be approached in the classroom. Indeed, a puzzle designation may well be appropriate for Liu Hui's conception of cube dissection in light of the way he sought to understand the ratio between a yangma and a bie'nao in the context of a cube (Martzloff, 1987/1999, p. 283). With a working knowledge of Liu Hui's approach to cube dissection, we now move on to discuss its classroom applications, starting with paper folding and GeoGebra-based modeling.
GeoGebra is a free and open-source software environment for mathematical modeling, available at www.geogebra.org for all major computing platforms. Initially, I planned to design some flexible templates for prospective teachers to build the three solids in Liu Hui's Cube Puzzle and further use them as motivation to discuss multiple modes of student engagement in mathematics teaching. GeoGebra proves adequate for this purpose. After defining an independent variable for the length of a cube edge, one can construct the three templates using the line and circle tools in GeoGebra, paying attention to the various right triangles that form the faces of the three solids (Figure 12). To facilitate assembly and subsequent discussions, each template should be printed on a letter-sized cardstock sheet of a distinct color. For example, in Figure 13, the qiandu (triangular prism) is in green, the yangma(rectangular pyramid) is in red, and the bie'nao (triangular pyramid) is in purple. In the beginning, students have little knowledge of the fractional relationships among the three solids. Therefore, it is pedagogically advisable to refer to them as the green, red, or purple pieces.
Figure 12. Templates for the three component solids of Liu Hui's Cube Puzzle designed with GeoGebra. Download all three templates here.
Figure 13. The three solids constructed by students from cardstock paper templates—from left to right: a yangma, a bie'nao, and a qiandu.
In light of its rich mathematical structure, Liu Hui's Cube Puzzle allows for various forms of student interaction. To begin with, each student will cut out and assemble (using adhesive tape) all three solids. Then students should be directed to work individually to try to put the three pieces together to make a shape with which they are familiar. Some students may make something other than a cube, which is perfectly acceptable. Some students will eventually produce a cube to their own amazement. Gradually, the whole class will catch up as students help each other or watch others solve the puzzle.
The game is not over yet. Next, the instructor could ask students to work in groups of two, using only the green (qiandu) solids, and see if they can make a cube (Figure 14). A question about the volume of the green piece (qiandu) naturally follows. If two green pieces (qiandu) make a whole cube, its volume is one half of the cube. Subsequently, students will work in groups of three, using three red pieces (yangma) and try to put them together in a cube (Figure 15). They may have to try a few times before settling on a cube, which is part of the instructional plan. If three red pieces (yangma) make a cube, then one piece (yangma) must be one third of the cube in terms of its volume.
The last task is more challenging and is also more interesting. Students will now work in groups of six, using six purple pieces (bie'nao). They may initially have trouble putting the blocks together in a cube because they don't seem to fit in with each other. Some students will try to make a red (yangma) piece using two purple ones (bie'nao), but cannot get the orientation correct. Depending on the instructional objectives, the instructor could make a suggestion at this moment that the purple one (bie'nao) can be folded backward, thus making a mirror image of the original bie’nao. The original and mirrored ones will fit nicely in a red piece (yangma). The rest will be straightforward, because three yangma make a cube, as is already known. After mirroring three of the six purple pieces (bie'nao), each group will be able to make three red pieces, which form a cube (Figure 16).
Figure 14. Two qiandu (triangular prisms) make a cube.
Figure 15. Three yangma (triangular prisms) make a cube.
Figure 16. Six bie'nao (triangular pyramids) make a cube after mirroring three of them.
Liu Hui's Cube Puzzle provides multiple opportunities for students to work with each other in collaborative groups while they reason repeatedly about the mathematical relationships among the three polyhedral solids. A few beautiful observations will surface after the group work: \[\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1,\,\,\, \frac{1}{2}=\frac{1}{3}+\frac{1}{6},\,\,\,{\rm{and}}\,\,\, \frac{1}{3}=\frac{1}{6}+\frac{1}{6}.\] It is eye-opening for the students to experience these amazing connections between geometric objects and between geometry and arithmetic. The cube puzzle construction task can be used across the grades to facilitate students' mathematical exploration and sense-making for various instructional goals. Students may initially have difficulty using the various terms, whether they are English translations, numeric descriptions, or original Chinese names. Pedagogically, colors are useful for smooth communication at the beginning of the activities. Even Liu Hui had to use red and black colors to explore the relationships in his cube dissection. As students get to know the processes involved, other names may naturally surface in the classroom discussions. Otherwise, these terms should be introduced at an appropriate moment. Eventually, students will realize that, for example, the red piece, whose volume is one-third of the cube, is called a rectangular pyramid in English and it is called a yangma in ancient Chinese. The struggle with linguistic references is a healthy one in an educational context.
Pedagogically, the paper-based activities described above may take up to two hours. During the first hour, students will make the three solids from paper templates and explore their relationships in an open manner. During the second hour, students will conjecture, articulate, and justify the specific relationships among these solids with respect to each other and the whole cube. Alternatively, the construction task can be assigned as homework. When students bring ready-made solids to class for discussion, the whole activity may be implemented in one hour. Further extensions, as described on the next two webpages, can be considered at the discretion of the instructor in accordance with student need and curricular goals.
GeoGebra also comes with a powerful 3-D panel, where Liu Hui's Cube Puzzle can be modeled on the basis of a cube by constructing three face diagonals (FH, BD, BE) and one internal diagonal (BH) as shown in Figure 17. To define the three solids in Liu Hui's dissection, one can use the prism tool for the qiandu (green triangular prism) piece and the pyramid tool for the yangma (red rectangular pyramid) and the bie'nao (purple triangular pyramid). By rotating the 3-D view, one can obtain a better picture of all three solids. GeoGebra further has a net tool that flattens a chosen polyhedron to a template on a plane. Modeling Liu Hui's Cube Puzzle in GeoGebra 3-D allows students to appreciate the relationships between 3-D geometric operations and 2-D templates and further develop increasingly complex spatial reasoning skills, which are strikingly lacking in traditional school mathematics.
Figure 17a. Liu Hui's Cube Puzzle can be modeled using construction tools in the 3-D panel of GeoGebra. Click then drag the image above to see the dissection from different perspectives.
Figure 17b. Use the slider to fold up the 2-D nets into the 3-D pieces of Liu Hui's Cube Puzzle; namely, a red qiandu, green yangma, and blue bie'nao (GeoGebra applet by Lee Stemkoski).
Figure 17c. Use the slider to "solve" Liu Hui's Cube Puzzle; that is, to fit the red qiandu, green yangma, and blue bie'nao together to form a cube. You may also move the three pieces into position by dragging them (GeoGebra applet by Lee Stemkoski).
GeoGebra 3-D modeling is pedagogically powerful, allowing students to develop imaginative skills for sense-making in a 3-D environment. However, GeoGebra did not yet support 3-D printing as of early 2017. Although paper folding is practically adequate for classroom use, it is lacking in accuracy. Meanwhile, 3-D design software has become freely accessible, inviting mathematics educators to engage students with mathematical applications and design-based learning. Therefore, when 3-D printing service became available at my university library, I decided to take one step further along this cube excursion and explore Liu Hui's Cube Puzzle as a 3-D design task. As a matter of fact, the cube is a fundamental starting point for all 3-D design software. It is beneficial for prospective mathematics teachers to see the connections between basic 3-D design and the mathematics they are learning and will be teaching to school children.
I chose to use Autodesk 123D Design®, which is free for educational purposes. In addition, 123D Design has a simple and friendly user interface, which can be quickly introduced to middle and secondary students as well as prospective and classroom mathematics teachers, using specific design tasks. It turned out that Liu Hui's Cube Puzzle can be readily built out of a cube in 123D Design, using two splitting operations, both along a face diagonal. These splitting operations are accessible to middle and secondary students as well as to prospective teachers when using 123D Design. Figures 18a-f, below, show how to apply the two splitting actions in 123D Design in order to perform Liu Hui's dissection of the cube. (Download a 123D Design file for Liu Hui's dissection of the cube into a quiandu, yangma, and bie'nao. Download a 123D Design file for Liu Hui's dissection of an arbitrary rectangular prism into a quiandu, yangma, and bie'nao.)
Figure 18. Liu Hui's Cube Puzzle can be readily built in 123D Design® using two splitting actions. Perform the following steps, illustrated in Figures 18a-f, below, to carry out Liu Hui's cube dissection using 123D Design.
Figure 18a. Sketch a diagonal polyline on one of the faces in preparation to split the cube.
Figure 18b. Use the split solid tool to cut the cube in half into two quiandu (triangular prisms).
Figure 18c. The cube is now cut into two quiandu.
Figure 18d. Sketch a diagonal polyline on one of the square faces of a qiandu.
Figure 18e. Use the split solid tool to dissect the chosen quiandu into a yangma (rectangular pyramid) and a bie'nao (triangular pyramid).
Figure 18f. The cube is now split into Liu Hui's cube puzzle consisting of a quiandu, a yangma, and a bie'nao.
Once the cube has been split into its qiandu (triangular prism), yangma (rectangular pyramid), and bie'nao (triangular pyramid) pieces, students should be allowed to play with them within 123D Design and tweak their properties, such as colors and material, in order to develop a better understanding and appreciation of the puzzle and the geometric processes involved. This is a pleasant experience for the majority of students, who have rarely manipulated a virtual cube in such a manner. Once the three solids are tweaked to the students' satisfaction, they can be exported to a STereoLithography (STL) file and sent to a 3-D printing station, where the STL files will be processed (sliced) and printed. About half way through the printing task, small magnets can be dropped inside each solid for easy assembly, if so desired. An example is shown in Figure 19. (Download an STL file, suitable for direct 3-D printing of the three pieces of the cube puzzle.)
Figure 19. An example of 3-D printed solids for the cube puzzle—a qiandu, a yangma, and a bie'nao, from left to right.
If enough sets of qiandu, yangma, and bie'nao are printed, students could revisit the group work discussed previously. To see how six bie'nao come together in a cube, three of them should be mirrored either in 123D Design or in the printing software. (Download a 123D Design file for the case of three yangma making a rectangular prism.) By examining the paper models and 3-D printouts side by side, students will have an additional opportunity to reflect on the diverse connections between hands-on paper folding, computer-based 3-D design, and 3-D printing, all converging to a pleasant mathematical experience. Furthermore, 3-D printouts can be used to reconstruct Liu Hui's original dissection of yangma and bie'nao for an enhanced understanding of Liu Hui's argument for the 2-to-1 ratio between the volumes of a yangma and a bie'nao, as shown in Figure 20. (Download a 123D Design file for Liu Hui's further dissection of yangma and bie'nao.)
Figure 20. Liu Hui's original dissection of yangma and bie'nao can be modeled using 3-D printouts.
Liu Hui's Cube Puzzle has been successfully used with more than 12 classes of prospective elementary teachers in the teacher education program at Southern Illinois University Carbondale as well as several groups of inservice teachers at in-school professional development workshops in southern Illinois, from fall 2013 to spring 2016. Initially, paper folding was used as the primary pedagogical tool. GeoGebra modeling and 3-D design and printing were later incorporated when the technologies became accessible to the targeted audience. In teaching practice, either one or all of these activities can be used, depending on student need and the accessibility of technologies. The hands-on nature of these activities, their mathematical and aesthetic appeal, and their pedagogical flexibility resonated well with both prospective and classroom teachers. In spring 2016, a group of 18 prospective elementary teachers responded to a short questionnaire on their feelings about the instructional approach. The results are presented in Table 1, where one can see that almost all the prospective teachers reported a positive experience with the cube puzzle sequence, including a field trip to the university 3-D printing service.
Question |
Frequency |
The cube puzzle has worthwhile math ideas. | 18 |
The cube puzzle provides rich learning experiences. | 18 |
The paper-cutting activity is useful. | 18 |
The cube puzzle allows meaningful group work. | 15 |
The cube puzzle has an artistic appeal. | 17 |
I will consider using the cube puzzle with K-8 students. | 17 |
The cube puzzle is just confusing. | 1 |
There are many other questions I wanted to explore using the cube puzzle. | 3 |
3D modeling or printing is a useful teaching tool for teachers to know. | 17 |
The cube puzzle provides diverse learning experiences. | 17 |
I am interested in finding more information about similar puzzle activities for my future teaching. | 16 |
The cube puzzle offers useful problem solving opportunities. | 16 |
I learned useful things during the 3D trip to the library. | 15 |
Field trips are a waste of our time for learning. | 0 |
Table 1. Prospective teachers' perception of the cube puzzle activities (n = 18, AG = Agree, SAG = Strongly Agree).
Thirteen of the 18 prospective teachers also added their own comments about the cube puzzle and their field trip to the university library 3-D printing service. They were all very excited to learn about the process of 3-D printing and its intrinsic connections to mathematics. About the whole cube puzzle experience, their comments are essentially consistent with the survey statistics. One prospective teacher wrote:
I thought the cube puzzle was fun and a great way to get my brain cranking and to develop ideas on how to form the cube. It was not very difficult for me but I could see it being a challenge for children and a great learning experience when learning about 3D objects and geometry. I also loved the 3D printer field trip to the library. I had no idea we even could use the printer or how to use it. It will be super useful in the future as I continue to work with young children. I am very happy I had that experience (A prospective teacher, February 18, 2016).
Another prospective teacher expressed a similar view and further addressed the pedagogical benefits of the cube puzzle:
I have never seen a 3D printer or even thought that it would be available to me as a teacher, but now that I know I can think of many useful ways to incorporate it that will engage students. I also loved the cube puzzle because it was unlike any math I have done before. Not only was I problem solving, but I also got to be hands on which was very engaging. When multiple people worked together it also brought new ideas and really encouraged collaboration. This is something I will definitely use in my future classroom (A prospective teacher, February 16, 2016).
One of the prospective teachers, Rita (a pseudonym), was doing her practicum at a local elementary school in spring 2016. She decided to take the cube lesson to a first grade class and explore children's responses. She won the support of her cooperating teacher and eventually invited the university librarian in charge of 3-D printing to demonstrate 3-D printing to the children. After her lesson, Rita made a presentation to her college class about the children's work and her reflection on it. In her summary, Rita wrote about the children,
The students were very engaged from the start. They never lost focus and I never had any behavioral issues. Also some of the more advanced students were able to help the less advanced students.
About her cooperating teacher, Rita said,
She thought it was a great lesson and that it went very well. She was surprised to see every student so involved and engaged.
Reflecting on her own role as a prospective teacher, Rita reflected,
[I]n our practicums we have the choice to teach what we want. I do not think many people choose math lessons unless it is their area of interest. I was pretty insecure about my math abilities, but this definitely helped boost my confidence and teaching abilities.
In summary, there is consistent evidence that the cube puzzle and related activities provided an inviting experience for the prospective teachers to think deeply about mathematics, mathematic teaching, and their emerging ideas of technology use and student engagement. The cube puzzle served multiple purposes for the instructor to reach out to these novice teaching professionals, many of whom, unfortunately, did not experience school mathematics as relevant, meaningful, and beautiful.
Liu Hui's Cube Puzzle has proven a worthwhile case for teaching and learning of mathematics using historically significant tasks. It is well suited for K-12 and teacher education math classes as an art project, a geometric construction project, or a pedagogical task for teachers to reflect on the art of mathematics teaching and learning. In the lower grades, the focus can be on mathematical literacy in order for children to develop the vocabulary and visual experience for spatial reasoning. In the middle grades and above, the focus can be on geometric construction, geometric proof, 3-D design, and even Liu Hui's infinitesimal argument. Perhaps, the goal is not really making or solving the cube puzzle; rather, it is a way of thinking about the mathematical relevance of all kinds of cultural tools or designs in the history of mathematics, science, and culture. With the daily advancement of digital technologies, mathematical treasures of the past, whether recreational or professional, have acquired a new status as objects of mathematical teaching and pedagogical innovation. Initially, I had a hard time finding the historical source of the puzzle, due in part to the foreignness of the original Chinese terms. Once I located the historical origin of the cube puzzle, I found an excitingly rich world of mathematics, much of which I had heard about before, but for which I now have fresh connotations from a contemporary perspective, particularly in terms of K-12 mathematics teacher education. During my historical excursion, I encountered Friedrich Froebel, who invented similar puzzles and blocks in Europe more than 150 years ago (Kriege, 1876). I have also read Martin Gardner (2001), Brian Bolt (1993), H. Steinhaus (1969), Ian Stewart (2009), and H. Martyn Cundy and A. P. Rollett (1961), much of whose recreational mathematics is serious mathematics under a playful cover and invites fresh visits, with either traditional or contemporary technologies.
In the case of cube dissection, Liu Hui may not have intended his method to become a puzzle, as can be seen from his clear focus on practical matters of his own time. Nevertheless, it has over the centuries evolved into one of the most engaging mathematical inventions, frequently covered in texts on the history of mathematics, in China and internationally. From a modern perspective, Liu Hui's Cube Puzzle is also symbolic of a way of thinking about the mathematics education of school students and their teachers, an invitation to reconsider the design metaphor for mathematics teaching and learning, for, after all, our knowledge is a kind of design (Perkins, 1986). It is equally interesting as a case that supports the integration of mathematical history in mathematics education, an area that has attracted both mathematicians and mathematics educators in recent years. There is a wealth of similar projects, including much of Liu Hui's 1800-year-old commentary, that lend themselves to school students and mathematics teachers. When classic problems converge with modern technologies, mathematics educators have a fresh opportunity to engage children and adults in exploring and appreciating the nature and power of mathematical thinking.
The author is grateful to the anonymous reviewers and the MAA Convergence Editor, Dr. Janet Beery, for their constructive suggestions and thought-provoking questions, without which the article would not have come to its present shape. In addition to her kind encouragement, Dr. Beery took the trouble of locating some key references, which led the author to a rewarding journey of reading and research in ancient Chinese texts, modern Chinese interpretations, and English translations in the context of the problem. Special thanks are due to all the prospective and classroom teachers for their enthusiastic participation in the author's mathematics teaching experiments. The author is solely responsible for any mistakes or misinterpretations.
The article "Leonardo Da Vinci's Geometric Sketches" includes images of the tetrahedron and cube, along with the other Platonic solids, created for Luca Pacioli's De divina proportione (1509).
Liu Hui’s Commentary in Chinese on Problem Fifteen of Chapter Five in the Nine Chapters on the Mathematical Art
Jiu Zhang Suan Shu (Nine Chapters on the Mathematical Art) is an ancient Chinese mathematical text which was probably complied and annotated over time by many mathematicians. Liu Hui’s commentaries around 263 AD comprise the best known version and are widely considered the foundation of traditional Chinese mathematics. These commentaries were further annotated by successive mathematicians in the ensuing centuries (Lui & Li, n.d.; Sheng, Crossley & Lun, 1999). Thus, there exist various versions of the famous mathematical text even in the Chinese language. The following excerpts are screenshots of Liu Hui’s Chinese comments on Problem 15 of Chapter 5 from an official version of the Nine Chapters, appearing in the Masters’ Branch of Si Ku Quan Shu (Emperor’s Complete Library of the Four Branches of Literature) available at https://archive.org/details/06057482.cn. As is typical of ancient Chinese text, it should be read from the top to the bottom and from the right to the left.
Figure 21a. Part I of Liu Hui’s comment on Problem 15 of Chapter 5 about the volume of a yangma (rectangular pyramid). Screenshot from https://archive.org/details/06057482.cn
Figure 21b. Part II of Liu Hui’s comment on Problem 15 of Chapter 5 about the volume of a yangma (rectangular pyramid). Screenshot from https://archive.org/details/06057482.cn