NICE數-正方形與正立方體的切割
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。
耍「薛骰」-Sicherman Dice 的探討
George Sicherman discovered that it is possible to take a couple of 6-sided dice re-labeling them with different positive integers (1,2,2,3,3,4) and (1,3,4,5,6,8) having the same probability distribution as rolling a standard pair of 6-sided dice. Such unique pair of dice is calling Sicherman dice. The secret behind the Sicherman dice can be studied by combining the powerful mathematical tool “Generating functions” with the symbolic manipulation software “Derive 6”, The same procedure may be applied to studying the possibility of the generalized Sicherman dice along the consideration of :\r (1) Adding more dice. (2) Changing the number of faces. To this end, we introduce the concept of the Sicherman Bound. For a given integer n, the number of n-sided Sicherman dice is finite. We computed manually such numbers for n?50 based on the method of “Elimination of negative terms”. Sicherman Dice 就是一對點數配置與正常骰子(6 面正立方體,點數為1到6)不同的骰子,它所拋擲出的每一種不同點數和(2,3,4...,12) 的機率恰好與一對正常的骰子相同。這種骰子是美國的Col. George Sicherman 所發現的。 Sicherman 更進一步指出:在不使用Sicherman Dice 的情形下,不可能找到一組大於或等於三顆的非正常骰子,它們拋擲出的每一種不同點數和的機率恰好與一組同數量的正常骰子相同。本研究的目標在於1. 尋求計算「Sicherman Dice 的組合和正常的骰子有相同的出現機率」的方法2. 證明Sicherman 結論的真偽及是否適用於其他正多面體(4 面/ 8 面/12 面/ 20面) 的標準骰子3. 修正Sicherman 的結論,並定義Sicherman 極限(Sicherman Limit)。在假設n面正多面體(n 為自然數, n ? 50 )存在的情形下,探討每一個正多面體的Sicherman 極限4. Sicherman Dice (Crazy Dice) 的延伸探討(1) 不同面數骰子的組合,是否可以找到面數組合相同,但點數配置不同的Crazy Dice( 如4 面與6 面的標準骰子組合,找到4 面與6 面的Crazy Dice)(2) 多個面數相同或不同骰子的組合,是否可以找到面數、個數及點數配置皆不同的Crazy Dice ( 如3 個4 面標準骰子組合, 找到2 個8 面的Crazy Dice)在研究的過程中,我發現以下的現象:(1) Sicherman Dice 的產生,是生成函數因式重新組合的結果(2) Sicherman Dice 是否存在,則視上述重新組合的結果是否有負項產生由於上述的觀察,我使用自行發展的「負項消去」法來檢驗Sicherman 結論的正確性及求得n 面正多面體其對應的Sicherman 極限。同時我也和Col. George Sicherman 取得聯繫, 討論當年他發現Sicherman Dice 的經過及其結論的限制條件,作為本研究未來發展的參考。
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.
約瑟夫問題
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?
Nonlinear Time Series Analysis of Electroencephalogram Tracings of Children with Autism
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.