Science Focus (issue 26)

19 What is the best way to lace your shoes? This isn’t a math question, so let’s change it slightly: What is the shortest way, and what is the strongest way to lace your shoes? My guess is you’ve never had to think about these questions, but maybe your parents remember cutting off lengths of shoelace to use or testing the strength of their own shoelaces. You can see how relevant those questions might be. Nowadays we don’t consider them because the shoe manufacturers do it for us; their costs and profits depend on it, and they may even use the same mathematics as this article describes to come up with answers. Lacing a Shoe Mathematically, a shoe has two columns of n eyelets each, which we can label A1, … , An, B1, … , Bn in columns A and B (Figure 1). Without loss of generality, let the gap between columns A and B be g units, and the height between successive eyelets in a column be h units. A segment is formed every time the lace go through an eyelet from the previous one. A lacing is a way to pass a shoelace through every eyelet once each, starting at An and ending at Bn. By counting, we know that a lacing consists of 2n – 1 segments. The Shoelace Problem– What’s the Best Way to Lace Your Shoes? 怎樣穿鞋帶才是最佳方法? 這不算是個數學問題,所以讓我們換個問法: 怎樣能用最少的鞋帶,以及怎樣能最穩妥地把鞋帶 穿好? 相信你從來都不用擔心這些問題,但也許你的 父母還記得以前剪下鞋帶來穿,然後確定鞋帶是否 穿得穩妥的日子,在這種情況下就顯出這些問題的 重要性了;現在我們之所以不用操心是因為製造商 在銷售前就已經穿好鞋帶,而他們的成本和利潤全 都取決於這些問題,他們或許也使用與本文相同的 數學角度作出選擇。 關於穿鞋帶 從數學角度看,一隻鞋有兩行孔眼,每行各有 n 個孔眼,我們可以把 A行和 B 行的孔眼以 A1 到 An,及B1 到Bn 表示(見圖一)。在不失一般性的情 況下,設 A、B 行之間的間隙為 g 單位,同一行上下 孔眼之間的高度則為h單位。由一個孔眼到另一 個孔眼之間的一段鞋帶被稱為一個線段;而一個綁 法是指把鞋帶由An 穿起,中間穿過每個孔眼一次, 最後穿過Bn 作結。透過簡單數算,我們知道一個 完整綁法裡會有2n – 1個線段。 Figure 1 A shoe in a lacing which contains 2n – 1 segments. 圖一 以由 2n – 1個線段組成的綁法穿好的鞋 By Peace Foo 胡適之