數學

在實驗中學2007 年校內科展，參展作品《三角形中的切圓》的研究中，研究三角形內的切圓時，發現連續切圓的圓心與拋物線的軌跡有關。於是去查資料，在偶然的情況下，翻閱《平面幾何中的小花》時，接觸了「六圓定理」。因為覺得這問題非常有趣，於是便著手證明（見報告內文）。 又發現，當移動六個圓中的起始圓時，總是在某種情況下，六個圓會重合成三個圓。繼續研究其重合的狀況，發現了馬爾法蒂問題（Malfatti's Problem）的一種代數解法。 當我試著推廣六圓定理至多邊形時，發現奇數邊的多邊形似乎也有如六圓定理般圓循環的狀況，於是著手證明，但目前尚未證明成功。而偶數邊的多邊形則無類似的結果。 ;In 2007 National Experimental High School Science Exhibition, one of the exhibit works, "Inscribed Circles in Triangles", shows that the centers of the consecutive inscribed circles has something to do with the parabola's trajectory. To learn more about inscribed circles and parabolas, I referred to literature. By accident, I am faced with the problem on six circles theorem, in the book The Small Flower of Plane Geometry(平面幾何中的小花). Out of my interest in this problem, I tried to prove it. The other results are as follows: With the initial circle of six circles moved, in certain circumstances, the six circles merge into three. Further in studying this coincidence leads to an algebraic method to solve the Malfatti's Problem. Applying six circles theorem to the odd-number-sided polygons exists the same characteristic. It indicates that the inscribed circles will form a cycle. However, it hasn’t been successfully proven. The even-number-sided polygons show no similar results.

Motivated by Napoleon theorem, we study the properties of the triangles obtained by moving the midpoint of each side of a given trianle along the perpendicular bisector of corresponding sides, and extend the results to the case of quadrilaterals. On the other hand ,we consider the method of erecting a regular M-gon to each side of a given N-gon and joint the N centers of these M-gons to form a new N-gon. (abbreviated as CRG method),and get the following results. 1. We characterize some kinds of N-gons that can be transformed to regular N-gons via CRG method. 2. Of M,N are nature numbers with M|N, then it is possible to find a N-gon that can be transformed to a regular N-gon by CRG method. \r 3. If a polygon P is symmetric with respect to a fixed point or a fixed line, then P can be transformed by CRG to a polygon with similar symmetries. 4. If a polygon P is transformed by CRG to ′P,there exists a commonpoint G such that ΣGA=0 andΣGB=0, where A and B runs through vertices of and P′P, respectively. 本研究將拿破崙定理加以延伸。先探討由各邊中點沿中垂線延伸得出之三角形的性質並推廣至四邊形之情形條列式報告成果。另一個推廣是將給定的多邊形的每邊外接一個正多邊形，再以這些外接的正多邊形的中心為頂點造出一個新的多邊形。我們發現此幾何變換具有以下性質：（1） 「哪些多邊形能被變換成正多邊形呢?」，我們觀察出能被變換成正多邊形的多邊形其限制條件隨邊數增加而增多，並進一步區分了哪些多邊形可以被變換成正多邊形。 （2） 在將非正N邊形做變換時，不一定須外接正N邊形才能得到正N邊形，我們區分出可外接哪些正多邊形而得到正多邊形。 （3） 對給定的多邊形作此變換時，若原多邊形有點對稱或線對稱等性質，則新多邊形也將具有相同的性質。 （4） 此變換得到的新多邊形會與原多邊形共重心，亦即新舊兩多邊形內到各自的頂點向量和為0的點會是同一點。

正方形和長方形是每一個人都非常熟悉的圖形，但其中卻隱藏了非常多奇妙的“數學之謎”。 所謂「完美長方形」是：在一個長方形中 (長、寬不等)，能否分割出最少大小相異的正方形。 這個研究中，首先用「草圖」的解題方法研究完美長方形，接下來利用「平面圖形」的解題方法可簡化計算的過程，最後利用「對偶關係」證明出：完美長方形的最少階數為 9 階。 進而，我們將這個問題擴展至三維空間，思索在一個長方體中（長、寬、高都不等長） ，能否仿照二維空間，分割出最少大小相異的正方體，而完成這個研究。 Square and the rectangle are figures that everyone knows well very much, but what a wonderful " mystery of mathematics " is hidden among them. What is called the perfect rectangle is whether in a rectangle (its length and width is different ) could cut apart two squares as the least difference in size . In this research, the solution approach of "the sketch map" is used to study the perfect rectangle at first, then the solution approach of "the level figure" to simplify the complicatied calculation of the solving course , and "the dual relation" is finally used to prove 9 orders are the least orders for a perfect rectangle . And then, we expand this question to three-dimensional space, considering in a cuboid (its length, width, and height is different) whether could follow the two-dimensional space model to cut apart two squares as the least difference in size, and finish this research.

對4 × 4 方格表中計數問題的二個解題方法(1..解方程式的方法， 2.分割圖形的方法)作分析和研究後，首先我推廣分割圖形的方法來証明 : “好的n × n 方格表” 存在若且惟若n 為偶數。同時証明這種“好的n × n 方格表”內所有n2 個數的總和f(n) 為n(n+2)/4。當討論一般的n×m 方格表時，發現分割圖形的方法盲點，無法繼續推廣來証明。再經過深入分析與推廣解方程式的方法，藉由n×m 變數方格表，我們終於找到構造所有“好的n × m方格表”的方法。同時計算“好的n × m 方格表” (n≦m)內所有mn 個數的總和f(n,m), n≦7和証明好的nxm 方格表會有2(n+1)行一個循環的現象。We first studied two solution methods (1.solving equations,2.dissecting diagrams.) for calculations on 4x4 checkboard. Using the method of dissecting diagrams, we proved that``good nxn checkboard'' exists if and only if n is even. Furthermore, the sum f(n) of those n2 numbers in a ``good'' nxn checkboard is equal to n(n+2)/4.In studying the more general nx m checkboards, we found that the method of dissecting diagrams does not work, However, by extending the method of solving equations, and by considering nx m variable checkboards, we obtained a way of obtaining all ``good nxm checkboards.'' By way of computing the sum f(n,m) (n≦7) of those mn numbers in a ``good nxm checkboards,'' periodicity in every 2(n+1) rows is observed.

(一) 本研究首先導出ΔABC等分面積線移動所包絡出的曲線方程式，其圖形是由等分面積線段PQ(其中P、Q皆在ΔABC的周界上)的中點所構成，具有3 條曲線段(分別為3 條雙曲線之一部分)的封閉曲線，形成內文所謂的「包絡區」。利用包絡區的區隔，我們找出：1.當P 點在包絡區內，則有3 條等分面積線。2.當P 點在包絡區周界上，則有2 條等分面積線。3.當P 點曲線段的端點或在包絡區外，則有1 條等分面積線。(二) 以三角形的研究當基礎，擴展到凸n 邊形(不包含點對稱圖形)，我們發現：等分面積線數量之分布，仍然與包絡區息息相關，且1.凸2m +1邊形最多有2m +1條等分面積線。2.凸2m邊形，必發生內文所謂的「換軌」。因此，最多只有2m ?1條等分面積線。3.包絡曲線所分割出的區域，於相同區域其等分面積線數量相同，且相鄰兩區域數量差兩條。(三) 若凸n邊形有k個「換軌點」，則此n邊形過定點等分面積線至多有n ? k 條。(四) 若凸n 邊形為點對稱圖形(如正偶數邊形、平行四邊形)，則所有等分面積線皆過中心點。1) Our study got a curve equation of bisectors of a triangle. When a bisector is moving, we get three curves. They’re constructed by the midpoints of PQ. The three parts of the three curves make a closed curve which we called “the Envelope Area”. We found out:\r 1. When Point P is in the Envelope Area, we can get 3 bisectors. 2. When Point P is on the curves of the Envelope Area, we can get 2 bisectors. 3. When Point P is outside of the Envelope Area, we can get only 1 bisector. 2) Based on our study of triangles, we found that in Convex polygons(not including Point Symmetry Convex polygons), the distribution of bisectors is related to the Envelope Area. 1. We can get at most 2m +1 bisectors in a 2m +1 Convex polygon. 2. We can get at most 2m ?1 bisectors in a 2m Convex polygon, and the bisectors on the curves will “Change the Track”. 3. Envelope curve will divide a Convex polygon into several areas. The same area has the same numbers of bisectors, and the near areas have less or more 2 bisectors. 3) If a Convex polygon has k points to change the track, it will have at most n – k bisectors.\r 4) In a Point Symmetry Convex polygon (ex. Regular 2m convex polygons and parallelograms), all the bisectors will come through the center point.

在2003 年台灣國際科展之作品說明書「Concatenating Squares」中【9】與2004第三屆旺宏科學獎「SA3-119 ：與特殊型質數之倒數關聯的兩平方總和的整數分解」成果報告書中【10】，就已有令人驚訝的結果。在2005 第四屆旺宏科學獎「SA4-298 ：分和累乘再現數產生的方法及其性質探討之推廣與應用」成果報告書中【11】，更是以逆向思維進行研究推廣，創造出許多新穎且引人注意的美麗數式，由原創性的觀點來進行逆向思維的研究。正由於這些再現數充滿奇巧，且與不定方程（代數）、簡單的數整除性分析（數論）以及同餘式理論都有所關聯，我們要發展前所未有的同步關聯與研究分歧， 且是最新的發現，更以豐富的想像力、創造力與推理能力提出令人耳目一新的重要結果，得到許多從未見過世面的美麗數式，回眸觀賞時內心充滿了數學之美！;Number which when chopped into two（three）parts, added / subtracted and squared（cubed） result in the same number. Consider an n -digit number k , square it and add / subtract the right n or n .1 digits. If the resultant sum is k , then k is called a Kaprekar number. The set of n -Kaprekar integers is in one-to-one correspondence with the set of unitary divisors of 10 n m1. If instead we work in binary, it turns out that every even perfect number is n - Kaprekar for some n . We wish to find a general pattern for numbers these numbers cubed or 3-D Kaprekar. We also investigate some 3-D Kaprekar of special forms. In addition, some results relating to the properties of the Kaprekar numbers also presented. This study indicates the “interesting” and “pragmatic” natures of the research project. We have developed the original results based upon his initiatives and has thus created a new horizon through the research project. This is the proudest achievement for this study. Generalization of Some Curiously Fascinating Integer Sequences：Various Recurrent Numbers！

本研究是藉由數學手法探討；如何由一疊36 張四種花色的撲克牌中，尋找出保證可猜中最多張花色的方法。研究過程是以在適當的猜牌時機，以鴿籠原理、邏輯推理、二進位、分析與歸納……等數學原理與方法，搭配巧妙的策略運用而達到目的。猜牌方法：先約定好猜牌規則，助手將36 張牌背圖樣相同但非對稱的撲克牌，以旋轉牌背的方向傳達訊息。在本研究中得出利用數學原理與方法可「經由巧妙的猜牌方法保證可以猜中26 張花色」，並提供後續研究者利用本研究之結果繼續深入探討與研究。 The study is mathematically based with reasonable explanations behind it. We are to correctly guess as many cards as possible from a deck of 36 cards, with random numbers and four different suits. We will apply mathematical methods, such as pigeonhole principle, logic inference, binary system, and analytical reduction, upon right timing. Using careful arrangement of the principles and reasoning, we can reach our ultimate goal. To state guessing: Conference between the guesser and the assistant about the guessing rules, the assistant will have 36 cards with the same exact pattern on the back but not symmetrical. The pattern of the cards will be different when rotated 180o. The only communication between the two is by rotating cards. In the process we will obtain mathematical theory and methods assuring 26 cards correctly guessed, and the study is for further and deeper discussion.

正方形兩條對角線的交點（即中心點）距四頂點等長，也與四邊等距。如果將正方形的頂點比擬成它的「手」，兩對角線的交點當成它的「心」，則兩個正方形頂點間、中心點間、或頂點與中心點間的線段相連（或重合），就如同「手」或「心」彼此相連。本文即探索當多個正方形間「心手相連」時，衍生圖形間的面積關係。而四個正方形中某幾個頂點相接（邊未重疊），恰圍出兩個三角形的圖形則是本內容討論圖形的主體架構，我們以此架構向外作出「層出不窮」的正方形，再配合中心點連接成四邊形，將推導出這些四邊形與基準正方形（Reference Square）間的面積關係。In a square, the lengths from the intersection point （center point） of two diagonal lines to the four apexes are the same, and so are they from that point to the four sides. If the apexes are “hands” and the intersection point of two diagonal lines is the “heart” of a square, the connection or overlap of two squares’ apexes and apexes, center point and center point, or apexes and center points is just like the connection of hands with hearts. In this article, hence, we are to explore the relation in area of derivative graphs formed by several squares connected “heart in hand.” When some apexes of four squares are overlain without sides overlapped, two triangles are created. And that’s the theme we are going to discuss. Furthermore, we extend the operation to infinitely overlain squares and frame out quadrangles referring to the center points of some squares. Then, the relation in areas of these overlapped squares and the Reference Square would be deduced.

In this work I make the analysis of the possibility of the existence of the number system with non-natural base & its investigation. The question examined in my work is totally opened:\r ‧ making the list of new characteristics, rules of the translation of the numbers, and also rules of the simple calculating operations, checked the operations of subtraction & division;\r ‧ checked the Euclidean algorithm, its characteristics by means of estimating the coefficients;\r ‧ found the practical appliance of new method in compiling & solving of the tasks.\r Investigation I’ve suggested stipulates for independence of new system & its appliance in type of tasks, that is beyond the course of school program & also beyond the whole system of school education.

This study was to explore the nature of two basic constitutes of the regular pentagon,With these two constitutes, the regular pentagon could be multiplied into any times. We used four multiplication methods (m2 = 2m1 + n1 、n2 = m1 + n1 、m2= k2m1 、n2= k2n1、a2 = a1 + 1、a2 = a1 + ) to show how the regular pentagon could enlarge and to verify that the enlarged regular pentagons derived from computer did exist. By integrating these four multiplication methods, we were able to arrange regular pentagon of any length of side, and evidenced the equation was ( If the side length of a regular pentagon is a form of m,n is the number of A,B respectively ) We further proved that the first multiplication method could be developed into a new modified method, which could divide a regular pentagon with a given side length into a combination of A and B. But only when the x and y of side length of a regular pentagon could be divided by a natural number, k, and made x/k into an item of the Fibonacci Sequence and y/k a successive item. When we tried to verify if any regular pentagon could be constituted by other smaller regular pentagons, we also found that it was un-dividable only if the length of pentagon side were ( the number of A, B were the 2n and 2n-1 item of Lucas Sequence). Otherwise, any regular pentagon might be able to be constituted by other smaller regular pentagons. 本研究是以正五邊形的兩個基本組成元素(B)作為討論對象，利用此二元素可以將正五邊形做任意倍數的放大。我們共使用4種繁殖法則(m2 = 2m1 + n1 、n2 = m1 + n1 、m2= k2m1 、n2= k2n1、a2 = a1 + 1、a2 = a1 + ) 來說明正五邊形的放大情形，並利用此4 種繁殖法驗證電腦運算出的放大圖形確實存在。利用這4 種繁殖法則的改良與整合，已達到能排出任意邊長之正五邊形的目標，並能計算並證明出其通式為。 (若正五邊形的邊長為形式,m、n代表、的個數) 更特別的是，我們能用第一繁殖法反推出一種方法，將給定邊長的正五邊形利用簡單的切割方式分成由A、B 組合成的形式，但只有正五邊形邊長之x、y 值可同除以任一自然數k 而使 x/k 為費波那契數列之一項且 y/k 為其後一項者才可以使用。 將此想法推廣至一個正五邊形能否由比他小的其他五邊形組合而成時，我們也發現當正五邊形之邊長為時（其A、B 個數為盧卡斯數列之第2n,2n-1 項），不可分解，否則應該皆可將一個正五邊形分解成比它小的其他五邊形組合（我們也可以利用這些質形檢驗出其他正五邊形是否也為質形）。但其分解形式，不只一種，而我們推測只用兩種較小的正五邊形就能達成，我們期待能找出一或多種分解方法，能將正五邊形分解成標準的分解形式。