# .平面座標上長方形沙發旋轉問題之解的存在性

2005年

## 摘要或動機

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’).

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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