數學

本研究所探討的主題是一道分割棋盤的題目：\r 給定一個黑白相間的mxn棋盤(設m為縱向邊，n為橫向邊)，將其分割成若干個面積大小都不等的矩形區域，且每個矩形的黑格和白格個數相等。設f(m,n)為符合上述條件的最多分割矩形的個數，則f(m,n)是多少？\r 在研究過程中，我們首先藉由對各棋盤進行分類，利用不等式的運算找出其中的關係，求得當m=n以及mn=k(k+1) 時(其中K為正整數)的f(m,n)值，並構造出其分割方法。而k(k+1)

源自於Thinking Mathematically這本書的一道題目, 關於正方形的切割問題：將一個正方形切成不重疊的正方形, 所得的個數就可被稱作NICE(好的), 問有哪些數是NICE數？ 在平面的正方形切割的問題, 透過分割技巧, 我們得出了重要的結果：除了2、3、5以外的自然數都是NICE數, 並推導出：若k為NICE數, m為自然數, 則k+3m為NICE數。我們將問題推廣至立方體：將一個正方體切成不重疊的正方體, 所得的個數就可被稱作very NICE(非常好的), 問有哪些數是very NICE數？我們也得出重要的結果：大於47的自然數皆為very NICE數, 並推導出：若 是very NICE數, 且m是自然數, 則k+7m為very NICE數。

k(2αβ ,α2 ? β2,α2 + β2 )是大家熟悉畢氏定理的通式解，且一般書籍的証明大都採用代數的手法證明。以國中生而言，上述的代數方對國中生來說不夠直接且較無推展的實用性。因此幾何觀點出發發展另一種思考方式，利用角平線的性質給予畢氏定理比例解另一種全新的詮釋，並賦予比例解中的參數α 、β 在幾何的意義。在推理的過程中，我們得到一個相當有用的對應關係：一個有理數對應到一個直角三角形、兩個有理數對應到海倫三角形，再將此對應關係運用到各種幾何圖形上面，即可證明出他們所對應的通式解。最後我的興趣鎖定在海倫三角形、完美海倫多邊形與超完美海倫多邊形上的做圖方法上，善用我們所發展的對應關係，上述的問題皆可迎刃而解。k(2αβ ,α2 ? β2,α2 + β2 ) is a popular formula in Pythagoras Theory, often proved in algebra approach among books. Nevertheless, in light of junior high students, the aforementioned algebra method is neither direct nor practical. Hence, a different thinking method is derived from geometry perspective, using the straight line concept to reinterpret Pythagoras Theory and define the geometric meanings of α andβ . In the process of logical development, a useful correlation emerges: a rational number correlates with a straight-angled triangle, and two rational numbers correlate with Heron Triangle. This correlation can be applied to all kinds of geometrical diagrams to prove the correlated homogenous solution. Ultimately, my interest lies in the diagram methods of Heron Triangle, Perfect Heron Polygon, and Super Perfect Heron Polygon in order to apply our developed correlations to solve the above mentioned problems.

在 Fibonacci Sequence 中，我將Cassini’s identity 轉換成圖形時發現：邊長為Fibonacci number 的正方形，分割後重新拼成長寬分別為Fibonacci number 前後兩項的矩形，會得到矩形內有縫隙（或重疊）。接著我將Cassini’s identity 的圖形推廣到Catalan’s identity 的圖形，我發現邊長一樣的正方形，拼成的矩形長會變大，寬會變小，矩形內的縫隙（或重疊）面積會以Fibonacci number 平方增長。接下來我再將圖形推廣，邊長為非Fibonacci number 的正方形分割拼成矩形時，我發現若將整數遞迴數列代入Cassini’s identity，圖形將會有規律的方式呈現，且每一種數列的縫隙（或重疊）面積會有所不同；若遞迴數列代入Catalan’s identity，縫隙（或重疊）面積還會再以Fibonacci number 平方增長。所以最後我得到一個通式：只要是遞迴數列[an]的圖形，都會滿足於： 。

我們從日常生活中的酒瓶填塞問題，延伸出圓在相守條件下產生的衍生圖面\r 積極值及相守圓排法等問題，研究過程中我們發現以下的結果：\r 一、 我們可以利用較少個相守圓的排法，以繁殖或增加的方式排出較多個相守圓\r 的衍生圖。\r 二、 當相守圓數量為2個、4個及6個時，相守圓以對角線排列，且衍生圖為正方形\r 時，面積會產生最大值。而當相守圓數量為3 個及5 個時，衍生圖面積最大\r 值會出現在相守圓排成波浪形時，但其夾角並非特殊角。\r 三、當相守圓數量為2個至6個時，相守圓以直線排列，衍生圖面積會產生最小值。\r 四、透過不同層數、個數的研究得知：當相守圓的數量為11個或14個以上時，存\r 在正三角形排法小於直線排法的情形 。而相守圓個數在10個以下時，直線排\r 法面積都叫正三角形排法小。\r 五、我們可以用質單元分割的方式討論更多圓數時，相守圓間的排列方式，使得\r 生多元間的排列變為數字間的加法排列，可以大大降低討論情形的複雜度。\r 六、我們可以用密度的概念，搭配質單元分割的方式，得到不同圓數時，以不同\r 衍生多元搭配波浪形法形成之衍生圖密度最小值。\r 七、我們可以算幾及柯西不等式來驗證特定類型的衍生圖面積極值，也可以列出\r 面積函數來分析不同類型的衍生圖面積極值。

傑克船長和他的海盜們掠奪到許多箱珍寶，每箱含有數量不等的金幣及鑽石。船員們深怕傑克船長又出什麼陰謀，一致同意讓船長任選一半的箱子拿走。\r 當然，傑克不知道金幣和鑽石的價格比，為了保證可以得到一半的利益，傑克希望他拿到的金幣和鑽石都各占一半。傑克的願望會實現嗎？\r 這個問題已用高等數學證明其解，而本研究利用初等數學的方法，除證明傑克需取的最少箱子數外，同時也能更快速的算出取法。\r 類似的結果可應用至分配災區物資等情形，或任何無法轉移、獨立的資源，如各式專長的人才、多功能的機械等，期望可對更有效的分配做出貢獻。

滿足之M點是否為重心之探索

滿足之M 點，我們稱之為Pi(i=1…n)的均值點。當n=3，M 恰為△P1P2P3 的重心 (G); n=4 時，M 亦為三角錐P1P2P3P4 的重心！因此不免引人遐思：滿足之M 點是否皆為其重心？ 我們藉由電腦幾何作圖軟體GSP 協助觀察，掌握了圖形變化間之不變性，再配合向量解析及推理，得以發現均值點、多邊形的重心、以至多面體的重心、及平行多邊形的一般性作法。附帶又發現：任意相鄰三頂點即可決定一平行n 邊形。並進而證實：平行四邊形為四邊形M＝G 的充要條件。但當n≧5 時，平行n 邊形只是n 邊形M＝G 的充分非必要條件！一般而言，具有對稱中心O 的n 個點所構成的圖形必可使M 與G 重合於O 點上。 The point M satisfying is called “the mean point of Pi(i=1…n)”. As n=3, M is the center of gravity (G) of the △P1P2P3. If n=4, then M is also the center of gravity of the triangular pyramid P1P2P3P4. Therefore, I began to wonder if the following assumption stands: The point M that satisfies is always a center of gravity. By using the computer software GSP (The Geometer’s Sketchpad) to observe figures. It is found that when a figure is changing there is still constancy. Furthermore, supported by the analysis based on vectors, general constructions can be established concerning the mean point, the center of gravity of polygon, the center of gravity of polyhedron, and the parallel polygon. Also, I find that any three neighboring vertexes decide a parallel polygon. And thus it is verified that the parallelogram is the sufficient and necessary condition for quadrilateral M=G. As n≧5, the parallel n-sides shape is the sufficient, not necessary condition, for n-sides shape M=G. In general, a central figure of n points having the center of symmetry O can make M and G meet on O.

最後留下數字會是多少？該問題在台灣的全國中小學科學展覽出現多次。而資訊界演算法大師Donlad E. Knuth 在其著作The Art of Programing，CONCRETE MATHEMATICS （具體數學），針對該數列作詳細的說明；但是，不論是歷屆全國中小學科學展覽或是大師著作，對於該問題，都只是談及殺1 留β或是殺α留1。本研究利用獨創α分類、n 及k 分類、d 函數、b 函數及循環、n 及y 分類、碎形數列和演變關係，將約瑟夫問題探討範圍提升至殺α(個數)留β(個數)，直到剩下最後1 個數時就不能再殺了，遊戲終止，倒數第k 個留下的自然數是多少？同時，本研究在殺α(個數)留β(個數)下，指定自然數y 為酋長，酋長不能被殺，殺到酋長時遊戲停止，求剩下的自然數有幾個？會發生什麼情形？The Josephus problem refers to what will be remaining when arranging n natural numbers in a circle and starting killing one and leaving the next one alive. The problem has been on display for many times in Taiwan National Primary and High School Science Exhibitions (as shown in Table 1). And, the information algorithm master, Donald E. Knuth has elaborated on the array in his works The Art of Programming, CONCRETE MATHEMATICS. However, both the past science exhibitions and the master’s works are limited to discussions on cases of killing 1 leaving β or killing α and leaving 1. This research employs uniquely created α classification, n and k classifications, d function, b function and loop theory to extend the Josephus problem scope to killing α leaving β to find out what the remaining natural number is by No. k counted recursively. Meanwhile, this research designates natural number y as the chieftain, which can never be killed. The game is over when the chieftain is to be killed. The problem is to work out how many natural numbers are remaining. And what happened?

Methods of nonlinear time series analysis were compiled for use in the analysis of Electroencephalogram (EEG) tracings of children aged three to seven with varying degrees of autism in order to provide a quantitative means of diagnosing autism and determining its severity in a child. After determining the EEG leads to be used for analysis, the identified methods were coded and saved as functions on Scilab. To test the compiled program, a minimum of five EEG readings per cluster of children diagnosed with mild, moderate, severe and no autism will be obtained. The project was able to identify the mean, standard deviation, skewness, kurtosis and other higher order moments, the autocorrelation function, and the Fourier Series as the time-resolved statistical methods to be used for time series analysis. The nonlinear analysis methods identified include the use of the correlation integral, time-delay embedding and the Lorenz equations. One-way ANOVA testing will then be used on the numerical data obtained from the analysis to determine if a significant numerical differentiation has been obtained between the different clusters of EEG. This will provide a definitive way to medically diagnose autism, pinpointing children afflicted with the disorder and giving them proper treatment.\r Two copies of the "Abstract of Exhibit" (in English) should be sent to the National Taiwan Science Education Center or email to fung@mail.ntsec.gov.tw or yuonne@mail.ntsec.gov.tw before December 31, 2009.