Exploring Liu Hui’s Cube Puzzle - Introduction

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.