我們的題目是”座標平面上任意三點與x軸的最短距離和”。我們一開始直接由尺規作圖著手，把距離和最小時的那點設為p，但始終無法找到一個方法精確畫到p點。後來變轉入公式的研究，我們設第一象限中A B 和C三點，固定A B C的y座標為d，並討論不同d時的情況，求出B點的範圍。另外我們也從橢圓方面著手，但此部份目前還待努力。;A B and C are three points in the plane, and L is a line in the same plane. Our target is to find a point P in L make the sum of PA+PB+PC minimum. At first we undertake by drawing pictures but we can’t find the solution we want in any case. We make L to be X-axis make A B C’s Y-coordinate to be d, and then we discuss various situations of d, and figure out the range of B. We also analyze the situation in oval, but still have no conclusion.

In any , we prove that there exist cycles which have any length between 3 and 3n and paths which have any length between their smallest distance and longest Hamiltonian paths in any two different nodes; for any two nodes, there exist varied Hamiltonian cycles, making the two nodes locate on any possible counterpart position(only limited by the distance between the two nodes). In , there are 2n internally-disjoint spanning cycles, and 2n-1 internally-disjoint spanning paths. Besides, we also prove has no more than 2n disjoint spanning paths, and calculate its wide diameter. 本報告證明在環狀網路 中，存在有長度3到 3n 的所有迴圈；任何相異兩點都有各種不同的長度的路徑:從最短的距離到最長的漢米頓路徑；取定任意兩點，存在有各種不同的漢米爾頓迴圈，使得兩點位於所有可能的相對位置上(僅被兩點之間的距離限制)。在 中，也具有2n 個彼此不相交、經過所有點的迴圈，以及2n-1 個彼此不相交、經過所有點的路徑。除此之外，也證明了，在兩相異點間，具有個數不超過2n 且互斥的路徑，且這些路徑經過所有點。我們也估算了它的寬直徑。

Most football managers are not aware of the need for analysis of soccer data, which is one of the dynamic sports. In this study, we developed a statistical model with performance indicators and score indicators of individual soccer player based on various event data including dynamic features such as goal, assist, pass, etc. In this study, the correlation between the dependent variables and the explanatory variables, and each explanatory variable was confirmed through the correlation analysis to solve the problem of multiple communicability from the regression model and to analyze the statistically significant preliminary model. In addition, we analyzed the correlation between individual rating of the players and the data recorded in the soccer games, and found that there has been a problem with the rating of the soccer players evaluated by the reporters and the soccer statistics site. To solve this problem, we developed a model that best fit the performance indicators of individual soccer player using the linear regression model and the beta regression model. The performance index model of the athletes was developed by comparing the R-squared value and the mean absolute percentage error of two models, the linear regression model and the beta regression model, and found out the beta regression model is better model to use. By using the estimated regression coefficients of the regression model we made new PI model. Score Index, which is the attractive point of soccer, was developed by comparing Poisson regression model and negative binomial regression model based on AIC value, and the one using negative binomial regression model was found to be better. Through the model developed by this study, it is possible to collect the event data recorded by individual athletes for each soccer game, and obtain the PI & SI index which are the athlete performance index models. This allows us to evaluate each team's players objectively, analyze the team's deficiencies, and provide tools to find players, who can fill in the missing positions of the teams. This study can also be utilized to grasp the performance of athlete in real time by simulating the resultative model.

給定一個簡單圖G（simple graph ），令V(G)、E(G)分別為G的頂點與邊所形成的集合。對於兩個不同的頂點u, v∈V(G)，若存在一條邊連結頂點u, v，則將此邊記為uv，以uv∈E(G)表示。給定函數f: V(G)→ℕ，若對任意的邊uv∈E(G)，函數f皆滿足f(u)≠f(v)，則稱函數f為圖G的一個著色函數（proper coloring）。對於圖G，任意給定每個頂點v∈V(G)一個可用的顏色清單L(v)⊂ℕ，其中清單內各有四個可用顏色（|L(v)|=4 ，每個頂點的顏色清單可以不相同），若總是存在一個著色函數f，滿足∀v∈V(G)，f(v)∈L(v)，則稱圖G為『可四元列表著色（4-choosable ）』。對於平面圖（ plane graph），針對長度較小的圈（cycle）進行限制，我們設計充分條件，使得滿足條件的平面圖為可四元列表著色。

本研究探討正凸多邊形正則鑲嵌及阿基米德鑲嵌，在限制每一格相鄰格子中至多(或至少)有 格被塗色的情形下的最大(或最小)塗色格子數問題。研究利用塗色格子位於邊線角落、非角落的邊線、鑲嵌內部的共用邊數差異、及與塗色格子總數間的限制條件，採用賦值法解析塗色格子數的最小上界或最大下界。接著建構具最大(或最小)塗色格子數的塗色方式，以歸納法推導塗色格子數，證明其與賦値法解析結果相同，證得存在該塗色格子數。研究結果可應用至貼磚或印染鑲嵌圖案設計、LED點燈遊戲設計、供給-需求組合配置最佳化、LED廣告面板或色差控制等。

編號1~mn 的mn 個物件已隨機置入m× n 階的矩陣中，另外有一行m 個空格的暫存區供物件暫存用。我們探討將這mn 個物件移至目標區並按照1,2,…, mn 的次序排列，所需的移動步數；每一步的移動中，只能移動每一行最頂層的物件到其他行(含暫存區)的最頂層或目標區。在這篇報告中，我們給出了一個適用於n ? m ?1時的移動方法，此方法在一般的情形下，所需的移動次數未必是最少；但是在最不利於移動的情形下，我們證明此方法所需的移動步數為最少。There are mn objects, numbered from 1 to mn, put on an m× n matrix randomly, and there is another column with m blank spaces for temporary storage purpose during moving. In each step of moving, we can only move the top object from one column to the top of another column or to the target pile. The total steps needed to move these mn objects to the target pile in increasing order from the bottom to the top is studied in this article. A general method for solving this problem when n ? m ?1 is given, and we prove that it provides an optimal solution in the worst cases. However, it may not always provide the minimal steps in all cases.

In this project, we defined a mid-tangent point with respect to a fixed point X and a tangent at a point Y on a planar curve C as a point on the tangent that is equidistant from X and Y. We studied the locus of mid-tangent points of conic sections. We found that the locus of mid-tangent points of most conic sections are non-linear curves. However, we observed and proved by using Euclidean geometry that the locus of mid-tangent points of circles are straight lines. The mapping defined by mid-tangent points was studied further. The similarity between a mid-tangent mapping and a stereographic projection was displayed as a one – to – one correspondence function. We also extended the concept of mid-tangent points to three dimensional space and found that the similarity with the stereographic projection was retained in higher dimensions. Finally, we studied the locus of mid-tangent points of a sphere to create a mapping of the sphere to a plane.

本研究接續去年的研究主題”驚奇的數”，邊長為平方數的三邊形數亦為四邊形數的問題。解決這個問題後，利用Pell方程式與矩陣計算來求哪些邊長的p邊形數亦同時為四邊形數。處理方法分為兩類：第一類可以使用矩陣計算來討論，已討論出附帶方程式部分的初始解情形，並嘗試改進矩陣計算的漏解問題以及對數據做詳細分析、歸納。目前已有兩種方法：1.放寬附帶方程式初始解的限制，也就是縮小遞迴式的係數；2.伸縮雙曲線為一套固定的方法，可以解出原矩陣計算所遺漏的解。第二類無法使用矩陣計算，利用因式分解的技巧處理，發現結果與切比雪夫多項式有著密切關係。