政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/32598

政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/32598

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政大典藏 > College of Science > Department of Mathematical Sciences > Theses > Item 140.119/32598

Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/32598

Title:	迪菲方塊 Diffy Box
Authors:	蔡秀芬
Contributors:	李陽明蔡秀芬
Keywords:	四重數
Date:	2009
Issue Date:	2009-09-17 13:49:16 (UTC+8)
Abstract:	本篇論文的主題是在討論一個名為迪菲方塊(DIFFY BOX)的數學遊戲。內容是：在一個正方形的四個頂點處各寫下一個非負整數，然後算出相鄰兩角數字差的絕對值，寫在四條邊線的中點，再算出相鄰中點的數字差的絕對值，寫在四個頂點處，繼續重覆這個程序，直到出現四個為零的數。三年前，我用這個主題指導學生參加科展，當時也深信任給四個非負整數，最後一定出現四個為零的數，但整個研究到最後卻未證明出結論。於是，再次把這主題拿出來，試圖解決當初的問題。這篇論文一開始我將之前的研究與問題先提出來，然後從第四章開始，使用數學歸納法及遞減數列的概念來證明非負整數的迪菲方塊(DIFFY BOX)在有限步驟內確實會出現四個為零的數。接著，我將非負整數的迪菲方塊(DIFFY BOX) 在有限步驟內會出現四個為零的數之結論推廣至有理數的迪菲方塊(DIFFY BOX)也會成立；並利用【方法4.9】(p.26)，將在第三章討論的非負整數迪菲方塊(DIFFY BOX) 且，轉變成的樣子，目的是將第三章用四個未知數在討論時的複雜情形轉成只剩下兩個未知數來討論。而此時的迪菲方塊(DIFFY BOX) 滿足，然後，在這個範圍內發現大部分的迪菲方塊(DIFFY BOX)最多經過七個步驟就可以得到四個為零的數，如左下圖；但是在某個越來越小範圍的會需要經過很多的步驟才能出現四個為零的數。論文最後，提出一個猜想：當迪菲方塊(DIFFY BOX) 為 (此為的實數解) 時，。本篇論文至此，雖未繼續研究此奇妙現象，但也開闢了另一個新的研究方向。
Reference:	[1] Antonio Behn,Christopher Kribs-Zaleta,and Vadim Ponomarenko ,The Convergence of Difference Boxes. American Math. Monthly, volume 112,426-438, (1995) [2] Peter Trapa, DIFFY BOXES(ITERATIONS OF THE DUCCI FOUR NUMBER GAME). (September 27,2006) http://www.math.utah.edu/mathcircle/notes/diffybox.pdf [3] G.Sch¨offl,Ducci-Processes of 4-Tuples. The Fibonacci Quarterly 35.3,269-276, (1997) [4] Florian Breuer, Ducci sequences over abelian groups. Comm. Algebra 27, no. 12, 5999-6013, (1999) [5] B. Freedman, The four number game. Scripta Math. 14, 35-47,(1948) [6] M. Chamberland, Unbounded Ducci sequences. J. Difference Equ. Appl 9 , no. 10, 887-895, (2003) [7] P.Zvengrowski,Iterated Absolute Differences. Math. Mag, 52,36-37, (1979) [8] L.Meyers,Ducci’s four-number problem: a short bibliography.Crux Math, 8,262-26, (1982) [9] W.Webb,The length of the four-number game.Fibonacci Quart. 20,33-35,(1982) [10] R. Brown and J. Merzel,The length of Ducci’s four-number game. Rocky Mtn. Math J. 37,45-65,(2007) [11] S. Bezuszka and L. D`Angelo, An application of tribonacci numbers. Fibonacci Quart, 15,no.2,140-144,(1977) [12]蔡淑英,數字方塊.科學研究第43卷第五期 ,26,93年9月
Description:	碩士國立政治大學應用數學研究所 95972003 98
Source URI:	http://thesis.lib.nccu.edu.tw/record/#G0095972003
Data Type:	thesis
Appears in Collections:	[Department of Mathematical Sciences] Theses

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