Computational Diversions: The Game of HullGrams

Abstract

This installment of the computational diversions column introduces a new game (at least I think it’s new, and original–I haven’t seen it anywhere before). The game is called HullGrams, which is intended to suggest a blend of the classic mathematical pastime of tangrams with the geometric notion of a convex hull. By way of preface–before we get to the rules of HullGrams–let’s begin with its inspiration in the puzzle of tangrams. Many readers will be familiar with tangrams; but for those who have never seen the puzzle, books such as (Crawford, 2002) and (Read, 1965) are recommended. Martin Gardner, in his Scientific American column, discussed the pastime and researched its history (Gardner (1988), chapters 3–4); that history, by the way, is resolutely less romantic than the fable originally spun by the larger-than-life American ‘‘puzzle king’’ Sam Loyd, who did not invent tangrams but popularized it early in the twentieth century. Briefly, the basic idea of the tangram puzzle is that we are provided with a set of seven geometric pieces: five of these are isosceles right triangles (two large, two small, and one medium-sized), one is a square, and one a parallelogram, and all angles within the shapes are multiples of 45 degrees. By placing the seven shapes flat on a plane in different arrangements, we can create an astonishing range of composite shapes. The essential idea of the tangram puzzle is conveyed in Fig. 1, which shows at left a photograph of my own set of plastic pieces, arranged to form a square. A typical tangram puzzle will begin with a solid black silhouette (think of this as the ‘‘goal shape’’), and the job of the player is to arrange the pieces so that their overall silhouette matches the goal shape. In Fig. 1, then, the simple black square shown at right would thus be the goal shape for the arrangement at left. I could spend much more time on tangrams, but there is already an extensive literature; and my hope is that by now you’ve gotten the idea, even if this is your first encounter with the puzzle. For those who have spent time with tangrams, you will probably have observed that in fact most goal shapes are pretty easy to match. The truly tough puzzles are the ones with relatively simple silhouettes, like the square in Fig. 1; but most goal shapes are more challenging in their original composition than they are in their solution. That is, it’s hard to