魔術猜牌
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.
凸n 邊形等分面積線數量之分布探索
(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.
黑棘蟻聚落的生物時鐘
This study is to investigate whether colony of the spiny-weaver ant, Polyrhachis dives, have biological clock so as to observe the locomotion activities of the ants in the nest and find out if the Light period will interfere the rhythm. The conclusion is the colony of the ants get the rhythm is 23.8 hours during in L:D=12:12.There are ants not significant difference between large colonies and small colonies. While in Dark (D:D)the ants appears free running with 23.1 hours as the rhythm, so, the colony of the ants has obvious light-rhythm movement, showing that the biological clock will act on group and being controlled by light period. 本研究是在探討黑棘蟻 (polyrhachis dives) 聚落是否有生物時鐘(biological clock)，以觀察黑棘蟻在巢裡的活動情形，找出週期並探討光是否會影響週期。結果觀察出黑棘蟻 的聚落在有光的時候(L:D=12:12)以23.8 小時為週期，沒有光的時候(D:D)黑棘蟻仍呈現自由律動(free running)，以23.1 小時為週期，所以黑棘蟻的聚落有明顯的日週律動，顯示生物時鐘能作用在聚落上，且受光週期之調控。
磁剎車系統探討
本研究探討運用磁場來達到非接觸煞車的功能，本實驗採用兩種方式來探討磁煞車力，分別為馬達有外加電流及沒有外加電流的情況。首先本實驗提供一穩定的電源使鋁盤轉動，觀察加上磁場及把電源切掉後鋁盤轉速的變化。實驗發現，當馬達沒有外加電流時，磁煞車力與轉速及磁場平方皆成正比；馬達有外加電流時，電流差會與轉速平方差成正比。探討磁煞車力與厚度及介質的關係，實驗結果發現，渦電流常數與厚度成正相關，且當兩片鋁片中夾有介質時，渦電流常數較小。 This experiment is based on the magnetic brake’s practical uses and braking forces. We want to calculate the braking force, and also examine the factors that cause the braking force to differ.We attached a metal disk to a motor to make the disk rotate, then we control the distance between the magnet and the metal disk, therefore measuring the relativity of the distance and the rotational speed. We discovered that when the metal disk received a large quantity of the magnetic field (close distance), the breaking force and the rotational speed increased. On the other hand, when the metal disk received a small amount of the magnetic field (far distance), the breaking force and the rotational speed decreased. The magnetic braking force will convert into kinetic energy, thus, by using this connection and also by increasing the electric current to measure the resistance, we calculated the magnitude of the magnetic braking force. Hence we perceived an inverse ratio between distance and the braking force, that is to say, the closer the distance, the stronger the magnetic braking force; the further the distance, the weaker the magnetic braking force.
線蟲補捉菌Arthrobotrys musiformis 黏液相關基因之選殖與功能界定
線蟲捕捉菌Arthrobotrys musiformis 是一種可經線蟲誘導產生捕捉網來捕捉線蟲的真菌，本實驗即針對A. musiformis 的捕捉網黏液相關基因：Manosyltransferase（AH73）, β-1,3-glucan transferase（AH102）, fimbrin（AH121）及mannose-specific lectin precursor（AH338）進行選殖與功能界定，希望建立這方面的研究基礎，將來能應用在松材線蟲的生物防治上。首先我們大量培養A. musiformis，萃取菌絲體的DNA；接著進行聚合?連鎖反應 (Polymerase Chain Reaction，PCR) ，利用專一性引子對 (primer) 大量增幅AH73、AH102、AH121 及AH338之基因片段；增幅後的產物經過純化、選殖，定序並進行分析比對，確認增幅之序列無誤後，以 Digoxigenin (DIG) 標示當為探針，篩檢A. musiformis 的Fosmid Library﹔目前已成功選殖出AH73 之可能基因，完成AH73 之探針製備，並以其篩檢A. musiformis 的Fosmid Library﹔呈雜合正反應之選殖株 (clones) 將以散彈槍方法（shotgun）定序，作序列組合，探索相關的基因；接下來用 Rapid Amplification of cDNA Ends（RACE) 做出互補DNA (complementary DNA , cDNA) 全長度後；最後建構基因缺失株，驗證此基因所調控的生理以及生化機能。 Nematode trapping fungus Arthrobotrys musiformis can capture nematodes by producing adhesive nets when nematodes go through. Many kinds of nematodes, including pine wood nematode (Bursaphelencus xylophilus), can be captured. Pine wood nematode causes serious pine wood disease. Therefore, A. musiformis has the potential of biocontrol in pine wood nematode. Our research focused on adhesion and adhesive relevant genes of A. musiformis ：Manosyltransferase (AH73), β-1,3-glucan transferase (AH102), fimbrin (AH121), and mannose-specific lectin precursor (AH338). We try to clone these genes and carry out functional analysis. In order to achieve this goal, we used specific primers derived from previously obtained complementary DNA (cDNA), by Polymerase Chain Reaction (PCR) to amplify these genes and gained adequate quantity of genomic DNA products. After sequencing and verifying of the identity of the genomic DNA, we use Digoxigenin （DIG） to label them and use them as probes to screen the constructed A. musiformis Fosmid Library. Currently, the Southern colony hybridization is undergoing. The positive Fosmid clones against the specific probes will be sequenced completely by shotgun library to monitor the existence of adhesion related gene cluster. After working out the full length cDNA of these genes, we will use them to construct replacement vectors to knockout the adhesion related genes, creating mutants and further verify their functions through genotype or phenotype bioassay.
竹嵌紋病毒及其衛星核酸5'端非轉譯區與複製競爭關係之探討
RNA 病毒在複製過程中容易產生錯誤，導致其族群具中有很大的遺傳歧異度，累積的錯誤再加上選汰的壓力造成往後之變異。由於RNA 基因體之病毒變異較大，使得RNA 病毒在單一寄主上具有quasispecies 的特性，提供病毒產生新基因體的機會以適應環境或演化成新病毒。例如流行性感冒病毒與之前造成恐慌的嚴重急性呼吸道症候群病毒(severe acute respiratorysyndrome,SARS)以及禽流感病毒 (avain influenza virus) 皆為RNA 病毒，意味著RNA 病毒知不穩定性，並容易造成一些目前我們無法及時反應的危害。大部分的植物病毒又為RNA 病毒，本研究將以竹嵌紋病毒 ( Bamboo mosaic virus , BaMV )及其衛星核酸 (satellite RNA, satBaMV)為材料，進一步探討核?酸序列之變異對其族群在複製競爭上的影響。
幽靈雷劈數的推廣及其性質研究
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！
生生不息-正五邊形的繁衍法則
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。