This research mainly talks about how someone, by observing the non-congruent patterns on the backs of the playing cards and by working with the dealer on a pre-arranged lay out, can call out the cards as if he possessed the magic power to see through them. During the card-predicting game, one can use the patterns on the backs of the cards as visual clues （Observing whether it was places upside down or not）to help figure out the probability of where the card is going to show up. Suck a mathematical formula is known as the Pigeonhole Principle. Upon an analysis of the formula, we find that when given that the value of n is greater than 24, we can successfully call out a number of cards that is greater than 2n/3 . The possibility of such mathematical studies in other directions is endless. 中文摘要： 本研究主要探討利用橋牌非對稱的牌背，猜牌者經由和傳遞牌的人的一種事先約定的方 式（排法），彷彿（魔術）透視般的將一疊牌的花色逐步猜出。其猜牌過程是利用牌背 圖案的朝前朝後的指示，配合適當的猜牌張的分配，而運用到的數學法則包含鴿籠原 理，分析與討論歸納。最後我們得到一疊由四種花色張數相等所混合的n 張牌，可猜出 的張數恆大於 2n/3 （n>24 時）。後續可研究的方向仍然甚廣。

This is a study about the solution to a chess flipping game. The game is based on a 4*4 game grid. First, place some chesses on the grid randomly to start a game. Move any chess by jumping over one or two neighboring chesses in the same row (left or right), same column (up or down), or on the same diagonal. Chesses which get jumped over should be flipped. The ultimate aim is to make all the chesses upside down. In this study, I try to find the rules of the beginning arrangement that ensure solution. Here are the steps I take: First, break the restriction of the 4*4 grid, and set the coordinate system. Second, find out a few “basic illustrations” that can be solved and moved in order to cope with certain complicated problems. Third, with “basic illustrations”, find the rules applying to games on n*n grid. 這是關於翻棋遊戲的可行解之探討。棋盤是一個4*4 的方格，遊戲開始時在棋盤上任意擺上一些棋子，均是正面朝上，利用相鄰棋子的水平、垂直、斜向跳躍，棋子被跳躍過一次則翻面一次，遊戲目的在於使所有的棋子都翻為反面。我要探討的是關於棋局可解不可解的問題，找出棋盤上可解棋局的規則。研究步驟大致如下：一、打破棋盤4*4 的限制，將棋盤座標化。二、找出若干個可解並可移動的「基本圖」。三、利用基本圖，找出n 列棋盤可解的規律。

任意一個三角形要如何分割成四個彼此相似的組成三角形呢？我們透過嚴謹的數學推理，先對三角形作二、三分割的可能情形進行驗證，並藉由已完成相似二、三分割的三角形，運用「內分」和「外加」的觀念，使相似四分割的討論變得明快，並得以將各式三角形的所有相似四分割的圖示作完整而有系統的呈現。 \r 我們也對「比例四分割」的作圖法與其相關幾何性質，進行猜想與討論，並驗證得出一些結果。尤其對「黃金三角形」經比例四分割後，組成三角形之對應邊長的比值也是「黃金值」，以及使用五條摺痕線的摺紙方式，可以摺出一張黃金三角形紙張的比例四分割，這些研究結果都令我們感到獲益良多。 How to divide a triangle into four similar little triangles？ Possible situations of dividing a triangle into two or three parts could be testified by strict mathematical inferences, and then the concepts of “internal division” and “external addition” could be applied to make our discussion clearly and briefly. With above discussions, figures about four similar divisions of all kinds of triangles could be presented completely and systematically. Some results were come up after making some conjectures and discussions about the geometric constructions and geometric properties of “four proportional divisions”. We learn a lot by these researches especially on the discoveries that the ratio of those corresponding sides in each four similar triangles which form a golden triangle, is also golden ratio; and that we could divide a golden triangle into four similar triangles by using five folding lines.

數字波是探討在直線上的起始點、位移速度、總數相互變化的節點關係。在直線上，將全部格子數做為總數(m)，開始彈跳的點為起始點(i)，每次彈跳的格子數為位移速度(s)，被踩到的格子就是節點。節點是由位移速度和起始點決定，起始點本身可視為節點之一，之後的節點是由起始點加n 個位移速度產生。我們分別以三種型式討論：起始點等於位移速度，總數增加：使起始點和位移速度所代表的數字相同的彈跳。節點呈2、s、s+2…起始點固定，位移速度與總數增加：觀察位移速度和總數的關係。兩節點的和=s+2位移速度固定，起始點與總數增加：探討起始點和總數的關係。發現節點隨起始點有規律的變化在上述討論的型式中，我們再進一步將位移速度分為質數和合數，進而依其因數變化，可觀測到很多特殊的節點變化。The number wave is to discuss the relationship of the starting point, the moving speed, and the variations of total amount. In straight lines, let all the trellises be total amount (m), and let the starting jumping point be the starting point (i). The trellis number of each jump is the moving speed(s). The trodden trellises are knots. And knots are decided by the moving speed and the starting point. The starting point itself can be viewed as a knot. The following knots produce with the starting point and n moving speeds. We respectively discuss them in three types: When the starting point equals the moving speed, the total amount increases. The number of the starting point is the same with the jumping moving point; the knots are 2, s, s+2…. When the starting point is fixed, the moving speed and the total number increase. From observing the relationship between the moving speed and the total number, the sum of two knots is s+2. When the moving speed is fixed, the starting point and the total number increase. After our research into the relationship between the starting point and the total amount, we find the knots have regular variations with the starting point. From the types discussed above, we further divide the moving speed into prime numbers and non-prime numbers. Furthermore, according to the factor variations, we can see a lot of specific knot variations.

After reading “CKSH Communication 20 “, we are interested in Question 5 . We try to explain that in all the No. Crunches, whether we can put the numbers in the No. Crunches any certain position. We specify the question, according to the “Point Symmitry Homing” in mxn No. Crunches, to find the correlated characters of the rules. Finally, we find a “switch” – we can get the better way to rotate the numbers quickly by some programs.從「建中通訊解題」第20 期第5 題出發，本研究嘗試去解釋對所有的數字轉盤而言，是否能將其中的數字歸位到任意指定的位置？接下來將題目特殊化，藉由m× n 數字轉盤的「點對稱歸位」，尋找遊戲規則衍生出的相關性質。最後，利用研究出的性質找到一個「判斷式」可藉由程式設計，快速的找到較佳轉法。

這份研究是關於一個「數形合一」的問題，研究的主要目的是找出同時具有兩種圖形數身分以上的數。研究結果發現雙重多角數必定存在，但個數可能有限個也可能無限多個，有些雙重中心多角數是有限多個，有些雙重中心多角數是無限多個，但令人意外的是同時是K角數和中心K角數的數卻皆是永遠是無限多個，我對其模式進行了探究並加以具體分類，並說明原因。研究過程中發現遞推關係可以大大簡化計算的步驟，可簡潔快速的求出這些數。我也證明了同時是三、四、五角數的數只有 1。此外，我將研究應用在熱門的平衡數(balancing numbers)和NSW數(NSW numbers)等上面，應用多角數的研究解決一些熱門的問題和找出了圖形上不證自明的結果。

此份報告主要是探討一個最優化問題，即是給定特定的移動方式，求出至少要多少操作，才能夠把棋子從某一種分布狀態調整成另外一種。而經過許多例子的嘗試，發現因為原本的問題不帶有數學的式子於其中，故沒辦法用上任何已知的工具進行解題，所以引入座標系以及排列矩陣，讓其含有許多數學式，能夠幫助問題具體化。接著給出好的上界以及得知最大值達成時滿足的條件後，便是開始進行一連串的構造，而這我採取「先猜後調」的策略，簡言之，就是先放置一種看似不錯的分布狀態，再利用討論出來的一些工具，對此狀態進行調整，同時用簡單之圖論方法觀察內部的數學結構，以獲得最優的構造之一。

本文主要研究以數學方法探究物理上的平衡問題，將物理與數學的代數、數論、組合及幾何做了緊密的結合，並導出幾個有趣的結果。文中得出n的因數結構與平衡多項式的關係，以及平衡多項式與平衡問題一般解的關係，並進一步探討其平衡的排序策略及個數。再則將平衡多項式引入複數平面，可以看到許多有趣的幾何性質，並進而回歸至物理的應用。

我們研究的主題是曲率，且以高中所學的函數為主。雖然大學已有曲率公式，但我們將其表示成高中生較易了解的型式，並且以f(x) 的方式呈現。我們在函數曲線上取不共線三點，構成一個三角形，並求出此三角形的外接圓半徑。再將所取三點逼近，所求之半徑即為特定點的密切圓，也就是曲率半徑。而此曲率半徑的倒數，就是所求的曲率，同時我們將公式帶入高中各常見函數，以導出函數上各點曲率。;Our study is about curvature, especially about the fuctions we learn in senior high school. In university, there is a certain formula for curvature, but we hope to change it into a form that can be easily accepted by senior high school students, and express the formula with f(x), the symbol of functions. We pick three incollinear points from the curve of a function, making the three points into a triangle, and figure out the circumradius of this triangle. Then, we approximate the three points to one of them, and the circumradius will also be the radius of the osculating circle of the point. We define the radius as radius of curvature. The reciprocal of the radius of curvature will be the curvature. Then, we use the formula to figure out the curvature of the functions we learn in senior high school.