畢氏定理演繹的正三角形分割研究
畢氏定理(a²+b²=c²)歷經25世紀，發現了數百種的幾何論證法；而畢氏定理演繹出的正三角形 ( (/4) a²+(/4) b²=(/4) c² )幾何分割研究，卻一直沒有人研究。因此，承襲著之前處理幾何問題的經驗，決定挑戰畢氏定理演譯的正三角形分割研究。本文研究兩正三角形，經切割後拼成另一大正三角形；期間以GSP及AutoCAD繪製分析幾何圖形，並建立了4種分割模式，得到了3段式「最佳分割模式」及準「通用分割模式」，提供這方面問題一個可應用於所有條件之完善解決方案。本研究成果豐碩，補足了相關領域的空檔，且可製成益智又富挑戰性之拼圖系列，不管用做教具或遊戲，對建立意至己和相關資料有莫大貢獻！
Twenty five centuries after its discovery, hundreds of proofs have been given for the Pythagorean Theorem (a²+b²=c²). But, research about regular triangle dissection extending from Pythagorean Theorem has always been lacking. So, based on previous experience with geometric dissection problems, I have decided to do a research on regular triangle dissection extending from Pythagorean theorem. This research dissects two regular triangles and assembles them into a large regular triangle. Using GSP and AutoCAD to draw and analyze geometric shapes, four dissection models and nine dissection methods are constructed. The extreme values under all conditions are also discussed, as are the best and generic dissection models. There is a Three-section type “best dissection model” and a semi “generic dissection model.” offering a perfect solution to this kind of problem that can be used under all conditions. This study yields numerous results as well as filling in blanks in similar fields. It can also be made into challenging jigsaw puzzles for educational or entertainment purposes.
8x8 棋盤路徑解之一般化推廣
Abstract (一)、 In our study, we discuss a m×n chess and any beginning square p finding a directed path of chessman from p moving to an end square in which the chessman moves to adjacent squares including only three directions which are right move, up move and diagonal left down move. A m×n chess is ruled into m columns and n rows creating the number of (m×n) squares (二)、 A chess directed path moves from any beginning square to end square in a m×n chess and every other square is visited just once. In the view of the beginning squares, the chess paths are solvable paths in a mxn chess and the corresponding squares are solutions. (三)、 First, we find out that some beginning squares are located in a special area with no any solvable directed paths. We define the special area be no-solution area. (四)、 According the 3-color theorem, we determine more than two thirds of no-solution area. (五)、 Then, we derive properties of reversibility and symmetry in solvable paths. i.e. A solvable path exist another solvable path by reversibility and symmetry respectively. (六)、 Utilizing the generalization of no-solution area which is extended from the concept of no-solution area provides judgment for the next moves effectively. The judgment is defined as effective move principle. (七)、 Furthermore, using the other theorem called rules of shift Hamiltonian path gets augment solutions. (八)、 According to the effective move principle finding a number of solvable directed paths, use the reversibility and rules of shift Hamiltonian paths to get augment solutions. Finally, utilize symmetry to find out all solvable paths in the m×n chess. (一)、研究規則：在m×n 的格子中，任取一格A 當作「起點格」，在起點格上放一顆棋子，只能往「上」、往「右」、往「左下」的方向移動。(二)、定義：若棋子從「起點格」，按照上述規則能不重複的通過所有m×n 格子到達某一「終點格」，則對於「起點格」而言，此移動路徑稱為m×n 的「有解路徑」，其任4一「終點格」稱為「起點格」的「路徑解」。(三)、我們先研究出「基本無解區」。(四)、根據遊戲規則我們利用三種顏色將n × n 方格塗滿，並判斷出大部分的「無解起點格」。(五)、利用遊戲規則得到兩重要性質：(1)[可逆性性質] (2) [對稱性性質](六)、利用「廣義基本無解區」，當作我們[有效移動]的判斷，讓「有解路徑」快速的找出。(七)、利用本研究所稱的「平移哈式鏈」，得到[擴充解]。(八)、根據[有效移動]求出部分「路徑解」，再利用[可逆性性質]、 [擴充解] ，最後利用[對稱性性質]完成所有「路徑解」的尋找。
約瑟夫數列(Josephus Series)
所謂約瑟夫數列，就是有n 個數排成一環狀，從頭開始，殺1(個數)留1(個數)，求倒數第k 個留下的數會是多少？約瑟夫數列在台灣的全國中小學科學展覽出現多次（如下表）。全國科學展覽與本題類似的作品
資訊界演算法大師Donlad E. Knuth 在其著作The Art of Programing，CONCRETE MATHEMATICS，也針對該數列作詳細的說明。唯，不論是歷屆科學展覽或是大師的著作，對於該數列，都只是談及殺1 留β或是殺α留1。
筆者則在2005 年暑假，曾經提交於全國國小組比賽作品「老師無法解決的難題」討論到n 個人排成一圈經過殺α留β，最後留下來的情形。
本研究是將α、β、k 和n 作為變數，求：當有n 個數排成一環狀，從頭開始，殺α(個數) 留β(個數)，則倒數第k 個留下的數會是多少？
需符合α、β、k、n 皆∈N，且n≧k
1.直觀觀察:發現在每一個循環中，當n 等差α時，Aα,β,n,k 則等差α+β、n- Aα,β,n,k 則等差β。
2.分類:將其分類為cα,n，使當中有規律可求。
3.循環觀察:發現每個循環的尾數n- Aα,β,n,k 都小於β。
4.循環尾數:設計公式求出每個循環節的尾數n、留下數Aα,β,n,k 及n-Aα,β,n,k 。
5.倒推:由與循環節中有等差的性質，則可以由循環節的尾數，推論出循環節中的任意一數。
Joseph Sequence is the problem that discussed the situation of eliminating1 and retaining1 in the circle formed by n people. Joseph Sequence has appeared a number of times in National Elementary School and Middle School Science Fair in Taiwan (as shown in the table below). Past national science fairs and researches on Joseph Sequence
The publications，The Art of Programing，CONCRETE MATHEMATICS ，by the expert of mathematical calculation in the IT industry，Donlad E. Knuth，has provided detailed explanation on it. However, all of those only discussed eliminating 1 and retaining β or eliminating α and retaining 1.
The researcher proposed “Problems unsolved by teachers” in the national competition, and discussed the situation of eliminating α and retaining β in the circle formed by n people. This study continued the summer project of 2005, and conducted research on the question of when is the last kth person eliminated in a circle formed by n people. In the paper, α, β, n and k were independent variables and the research process was as follows:
1. Direct observation: the series shows equal difference in each cycle.
2. Classification: to search the pattern of the series based on cα,n classification.
3. Use the end number of each cycle to obtain the pattern.
4. Reverse induction: use the equal difference of each cycle to induce when the kth person would be eliminated.