节前安全大检查总结:中国古代“幻方”与现代“几何魔方”

幻方是中国古代的一种数学谜题，在正方形网格中填上整数，让每行、每列以及两条对角线上的数之和均相等。例如下面这个：        
它在任意方向之和都是”15“。这是我小时候很喜欢玩的游戏。
汉朝的数术记遗称幻方为九宫算或九宫图。现在趣味数学家Lee Sallows将数字幻方变成更复杂的几何幻方。他将新的结构称之为“geomagic square”，并在网上发布了一系列实例。伦敦大学玛丽女王学院的数学家Peter Cameron相信几何幻方中可能隐藏着更深层的结构。在Sallows的几何幻方中，网格中的数字被类似俄罗斯方块的几何形状取代，这种形状被称为polyomino。
Geomagic square: Inner cell missing 

In a geomagic square, each digit is replaced by a "polyomino" made up of different numbers of identical squares. There must be a way to combine the polyominos in each row, column and diagonal to build a single master shape, or target. 

This is a "normal" 3 × 3 geomagic square, meaning that the polyominos form the natural progression "1, 2, 3…" It is one of 4370 normal geomagic squares, not including rotations and reflections, that can be formed for which the target is a 4 × 4 square with a missing inner cell.

Geomagic square: Corner cell missing 

A second normal geomagic square, this time the target is a 4 × 4 square missing one corner cell. It is one of 27,110 normal squares with the same target. By comparison, there are 16,465 normal squares for which the 4 × 4 target is missing an edge cell.

Rare square 

In this geomagic square, all the polyominos have the same area. This type is much rarer than those formed using unequal polyominos.

Geomagic squares go 3D 

In this square, the target is a 3 × 3 × 3 cube. Note that the polyomino size forms the consecutive series of odd numbers "1, 3, 5, 7, 8, 13, 15, 17" while their shapes are derived from a formula for creating magic squares devised by the 19th-century French mathematician Édouard Lucas. 

Unlike the previous geomagic squares in this gallery, this one was not devised by Lee Sallows, who first came up with the concept. Instead, another recreational mathematician, Aad van de Wetering of Driebruggen, the Netherlands, submitted it to the Dutch mathematics periodical Pythagoras.

Impossible geomagic 

Some geomagic targets really are 3D. But in this one, the target is an "impossible figure", a two-dimensional object that the visual system interprets as a projection of a 3D object, even though such an object can't actually exist.
Smallest possible geomagic square 

Finding smaller geomagic squares is harder than larger ones, because the larger squares give you more options. Indeed, before launching his website of geomagic squares, Lee Sallows was unable to come up with an example of the smallest possible version – a 2 × 2. But shortly after his site went live, fellow square-hunter Frank Tinkelenbergsent him an example, as shown here.

更多有关“几何魔方”的资讯：http://www.geomagicsquares.com/index.php