「环」以颜色从黑白棋到多色棋的循环探讨.PDF

「環」以顏色─從黑 白棋到多色棋的循環探討 新竹市國立高級 中學 張維哲 指導老師 ：吳智傑 Abstract Assume there are N black/white chess pieces placed around a circle. We perform the following process to change the placement of the N pieces. If two adjacent pieces are of the same color, we insert one black piece in between; otherwise, we insert a white piece. After that, we remove all the original pieces. In this way, we can create a new placement of N black/white pieces. We observe that after repeating the process, at some iteration, the new placement of the pieces will be the same as a placement in a earlier iteration. This is called a cycle. We use Pascal triangle and Euler theorem to derive formula for the cycle length. In addition to the model of black/ white pieces, we further investigate the multiple color cases and its corresponding cycle. We divide the multiple color case into two sub-cases. One has the number of colors to be a prime, while the other has the number of colors to be a composite number. We found that the prime color case can be solved similarly to the two color case. When the number of pieces is given, we are able to find the corresponding cycle of the prime color problem. For the composite number color case, we provide a new definition.((Is this necessary?)) By using the cycle calculation of the prime color problem, we expect the solution can be found by using matrices to represent the relationship between color changes. 中中中文文文摘摘摘要要要 假設有 N 顆棋子 （黑色或 白色 ）置於一圓周上 ，我們進行 以下的動作改變此 N 顆棋子的排 列 ：如果相鄰兩棋子為同色 ，則在兩棋所在圓弧 中點上放一顆黑棋 ，異色則放上 白棋 ，然後移除 原先的 N 顆棋子 。如此在圓周上可產生一組新的 N 顆棋子 。發現如果重複執行數次 以上的動 作