臺灣國際科展

.平面座標上長方形沙發旋轉問題之解的存在性

科展類別
臺灣國際科展
屆次
2005年
科別
數學科
學校名稱
國立臺中第一高級中學
作者
周義超、林柏延

摘要或動機

這篇報告要探討下列的「轉沙發的問題」是否有解?有一個長方形的沙發,如圖一,若要求每次只能以「四個頂點逆時針或順時針連續旋轉90度」的方式轉動,請問當長寬具備何種關係時,沙發經數次轉動後,剛好可以「轉」到相鄰的位置,如圖一,而且沙發坐人的正面方向仍保持不變呢?

圖一

我們把原問題看成「平面座標上長方形旋轉的數學問題」,再利用「平面座標、三角函數、複數、複數的極式表示及向量」等數學工具,導出符合題目要求的方程式,最後證出當長與寬的比值為正實數時,有下列的結果:

1.當長與寬比值為無理數時,此問題無解。

2.當長與寬比值是最簡分數時,若分子為奇數,此問題無解。

3.當長與寬比值是最簡分數時,若分子為偶數,分母為奇數,此問題有解。


4.在有解的情況下,我們可以找出特定轉法的最小值。

5.當長與寬比值是最簡分數時,若分子為偶數,分母為奇數,沙發可轉至A點座標為(αp,0) 的位置,其中 α∈Z,且沙發坐人的正面方向保持不變。

6.當長與寬比值是最簡分數時,若分子為奇數,分母為偶數,沙發可轉至A點座標為(0,βq) 的位置,其中 β∈Z,且沙發坐人的正面方向保持不變。

7當的長與寬比值為正實數時,可將沙發轉至A點的座標為(2αp + 2βq,2γp + 2qω)的位置,其中 α,β,γ,ω∈Z,且沙發坐人的正面方向保持不變。

In this paper we discuss the solution of rotating sofa problem as follows : The
condition is : Merely allow to rotate the sofa several times by rotating 90 degrees
clockwise or counterclockwise around the vertex. (maybe A, B, C, or D in Fig. 1)
The question is : What’s the relationship between the length and the width of the
sofa, if we request the sofa translated next to the original position with direction
unchanged. (as shown in Fig. 1 with A’B’C’D’).

Fig. 1

We take this problem as a mathematical one of rotating a rectangle in plane coordinates.
Then we derive the desired equations by using the tools of plane coordinates, trigonometric
functions, complex number, polar form of complex number, and vector. Finally, we
prove that:


1. When the ratio of length and width is irrational, the problem has no solution.


2. When the length of sofa is odd in the ratio of length and width, the problem
has no solution.


3. When the ratio of length and width is even, the problem has solutions.


4. When the solutions exist , we can find the minimum of the number of rotations.


5. When the ratio of length and width is an irreducible fraction, which has the
even numerator and the odd denominator, the sofa can be rotated to the coordinate
(αp,0)(α∈Z)which is the new position of A and keep the original position with direction
unchanged.


6. When the ratio of length and width is an irreducible fraction, which has the
odd numerator and the even denominator, the sofa can be rotated to the coordinate
(0,βq)(β∈Z) which is the new position of A and keep the original position with direction
unchanged.



7. When the ratio of length and width is a real positive number, the sofa can be
rotated to the coordinate (2αp + 2βq,2γp + 2qω)(α,β,γ,ω∈Z)which is the new position
of A and keep the original position with direction unchanged.



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