At the website “MathLinks EveryOne,” we found a problem “Snakes on a chessboard,” which was raised by Prof. Richard Stanley. The following is the problem. A snake on the m n chessboard is a nonempty subset S of the squares of the board with the following property: Start at one of the squares and continue walking one step up or to the right, stopping at any time. The squares visited are the squares of the snake. Prove that the total number of ways to cover an m × n chessboard with disjoint snakes is a product of Fibonacci numbers. We call the total number of ways to cover a chessboard with disjoint snakes “the snake-covering number.” This problem hasn’t been solved since it was posted on September 18, 2004, so it aroused our interest to study it. First, we used the way in which we added each block to the chessboard, and therefore we discovered some regulations about the snake-covering number of the1 × n , 2 × n and 3 × n chessboard. Through “recursive relation” and “mathematical induction”, we proved the general term of the snake-covering number of the1 × n , 2 × n and 3 × n chessboard. In the following study, we found a key method in which we added a group of blocks to the chessboard. Finally, we proved the general term of the snake-covering number of the m × n chessboard. Also, we discovered the way to figure out the snake-covering number of the nonrectangular chessboard.在網站“ MathLinks EveryOne ”中，我們找到了一個有趣的問題“棋然上的蛇” ( Snakes on a chessboard ) ，這個問題是由教授 Richard Stanley 所提出。問題如下：在m x n棋盤形格子上，蛇由任意一格出發，但蛇的走法只能往右 → ，往上↑，或停住 ‧ 若此蛇已停住，將由另一條蛇來走，且不同蛇走過的格子不可重疊”證明：將 m × n 棋盤形格子完全覆蓋的總方法數為費氐（ Fibonacci ）數列某些項的乘積。我們將把棋盤形格子完全覆蓋的所有方法數稱之為“蛇填充數” 由於這個問題自從 2004年 9 月 18 日被登在網站上後，還沒有人提出解答，於是引發了我們研究的興趣。首先，我們使用了將一個一個格子加到棋盤上的方法，並發現了 l × n 、 2 x n、 3 × n 棋盤形格子蛇填充數的一些規律。我們使用遞迴關係及數學歸納法來證明 l x n 、 2 x n ， 3 × n 棋盤形格子蛇填充數的一般項。在接下來的研究中我們發現一個特別的方法，一次增加數個方塊 ‧ 最後我們證明了，m x n, ，棋然形格子的蛇填充數的一般項 ‧ 而且，我們也找到如何求出不規則棋盤形格子的蛇填充數。

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！

This study analyzed how short-term solar activities interact with the earth atmosphere, by using two statistic methods: Diffusion Entropy Analysis (DEA), and Standard Deviation Analysis (SDA). Since solar activities influence the Earth atmosphere in its radiating heat and magnetic field, we use DEA and SDA to calculate the exponents, H and δ, of the scaling law in three time series: “the intensity of solar flare” (representing by SOLAR H-alpha flare index), “magnetic anomaly of magnetosphere” and “sea surface temperature anomaly”. The values of H and δ show the time memory and correlative relationship between the event and next event happening in time series. When H = δ = 0.5, events occur in random. When 0.5

正方形和長方形是每一個人都非常熟悉的圖形，但其中卻隱藏了非常多奇妙的“數學之謎”。 所謂「完美長方形」是：在一個長方形中 (長、寬不等)，能否分割出最少大小相異的正方形。 這個研究中，首先用「草圖」的解題方法研究完美長方形，接下來利用「平面圖形」的解題方法可簡化計算的過程，最後利用「對偶關係」證明出：完美長方形的最少階數為 9 階。 進而，我們將這個問題擴展至三維空間，思索在一個長方體中（長、寬、高都不等長） ，能否仿照二維空間，分割出最少大小相異的正方體，而完成這個研究。 Square and the rectangle are figures that everyone knows well very much, but what a wonderful " mystery of mathematics " is hidden among them. What is called the perfect rectangle is whether in a rectangle (its length and width is different ) could cut apart two squares as the least difference in size . In this research, the solution approach of "the sketch map" is used to study the perfect rectangle at first, then the solution approach of "the level figure" to simplify the complicatied calculation of the solving course , and "the dual relation" is finally used to prove 9 orders are the least orders for a perfect rectangle . And then, we expand this question to three-dimensional space, considering in a cuboid (its length, width, and height is different) whether could follow the two-dimensional space model to cut apart two squares as the least difference in size, and finish this research.

超疏水表面(superhydrophobic surface)因其自我清潔效果在近年來引起廣泛的研究及探討，目前世界各大公司皆積極投入自潔性產品(超疏水表面)的研發，但是截至目前為止除了塗料及織物有產品外，其他尚在研發階段。降低表面能及增加表面粗糙度為製作超疏水性表面的兩種方法。本實驗我們針對使用不同方法去仿照自然界植物「蓮花效應」，將二氧化矽粒子適當的分佈在基材表面，以增加粗糙度，製造出超疏水性表面。我們發現鍛燒會使其二氧化矽粒子黏聚在一起，對接觸角的增加無幫助。而在溶膠凝膠的配置過程中直接加入OTS(Octadecyltrichlorosilane)，並利用光散射儀比較二氧化矽奈米顆粒的平均粒徑及粒徑分佈隨時間變化的情形，一段時間後發現會有明顯大顆粒沉澱，塗佈在玻璃基材表面後容易脫落，不適合往後實驗或其他用途。而以氨水催化的溶膠凝膠以不同流速滴入TEOS (Tetraethylorthosilicate，四乙氧基矽)對接觸角的影響也不大。最後我們利用砂紙在基材表面上刷磨，可以使二氧化矽粒子分佈均勻，明顯增加接觸角。塗佈速率的改變搭配疏水性矽烷單分子膜的改質，我們已可以製造出159°的超疏水性表面。Superhydrophobic surfaces are generally made by lowering the surface energy and increasing the surface roughness. In this experiment, we use different methods of spreading silicon dioxide nanoparticles properly on the surface in order to increase the surface roughness and also make superhydrophobic surfaces. In the beginning, we find that the calcinations can cause its silicon dioxide nanoparticles to stick together instead of increasing the contact angles. Then, add OTS (Octadecyltrichlorosilane) directly to manufacture process of sol-gel, and observe the situations of the average length and the spread of silicon dioxide nanoparticles with the time goes by. After a period of time, we will discover that many obvious big particles deposit and spin-coating on the glass surfaces flop easily. Hence, this phenomenon is n’t proper for the following experiments or other uses. However, the contact angles have nothing to do with dropping the sol-gel catalyzed ammonia to TEOS(Tetraethylorthosilicate) by different flowing rates. Finally, it is crucial for us to use the sandpapers to brush on the surfaces because it may cause the silicon dioxide nanoparticles to spread well and obviously increase the contact angles. Combining the silicon dioxide surfaces with the change of spin-coating rate and the cover with hydrophobic SAM, we have made the superhydrophobic surfaces of 159°.

The purpose of this paper is to study the roles of formic acid and formaldehyde playing in the Direct Methanol Fuel Cells(DMFCs).The assumption is now widely accepted that the Oxidation-reduction intermediate of cell like formic acid and formaldehyde will hinder the reaction of DMFCs. At first, we recorded data which measured the efficiency of DMFCs working under the different temperature of fuel, then we recorded data which measured the efficiency of DMFCs working under the different consistency. In the end, we compared the data we recorded before and chose the best reaction environment as standard environment for the future experiment. Then we discover intermediate has positive effect on DMFCs. we separately put formic acid and formaldehyde into fuel, and we discover the efficiency is better than\r before. For example, the volt stability and volt intensity of DMFCs are better. The above conclusion is mainly based on open current volt, equally volt and electric current density which is the standard of evaluation.本實驗主要探討甲酸、甲醛等雜質在直接甲醇燃料電池中扮演的角色，一般認為甲酸等是甲醇在電池中反應的中間產物，大多數人認為這些中間產物會阻礙燃料電池的反應。而我們先針對了甲醇在各種溫度下電池的效能先做出了圖表，並使用不同濃度的甲醇燃料來測量電池的效能並與之前溫度的圖表做比較，選出最合適的直接甲醇燃料電池反應環境作為添加雜質實驗的標準環境。接下來我們在研究過程中發現，其實中間產物可能對電池有正面的效果，我們發現甲酸、甲醛等對直接甲醇燃料電池的電源輸出有正面的影響，例如電壓穩定性與電壓強度的增強。在研究中，我們將甲酸等加入燃料電池的燃料(甲醇)內，模擬甲醇因不當保存而產生的雜質，針對添加物的濃度做些調整，以電池的開路電壓(OCV)、平均輸出電壓、以及單位面積的電流密度作為評比電池效能的標準，並找出甲酸等對燃料電池效能的影響，並進一步找出最適合的電池燃料配置。

由於奈米科技進步，奈米材料應用在產業上具有多功能的性質。本研究使用不同波美度的色料餵食家蠶，以找出最佳的彩色蠶繭色澤，並研究其如何影響家蠶所結出的蠶繭及色料附著在蠶繭上的絲。同時對色料附著的蠶繭進行水洗、光照、微結構的觀察，以試圖找出色料與波美度之最佳組合參數。由本實驗結果得知，利用奈米色料溶液60 ml，在紅色：1.048、藍：1.058、黃：1.039 的參數下，混合飼料30g，可獲得最佳的彩色蠶繭結繭成功率、均勻度較佳、耐褪色與耐洗滌等優點，並且可獲得表面結構光滑且較細的絲徑，約為19.87μm。相對的，一般色料粒徑為微米級，色彩度優於奈米色料，但表面結構較奈米色料粗糙且線徑較粗，約為21.51μm，易於褪色及不耐洗滌。 Because of the great progress of nano-technology, it has the quality of multi-functions to make use of nano-materials on industrial property. The purpose of this study is to find the best colored silkworm cocoons by mixing different consistency of pigments to feed silkworms. At the same time, this study wished to explore how the different consistency of pigments influenced the silkworm cocoons that the silkworms produced. Besides, in order to find the better association between pigments and Baume degrees, this study exposed the cocoons under different lights, washed with different detergents and take observations of micro-structure of the cocoons. The results of this study are as follows: using the nano-pigments 60 ml in different density, that is, red:1.048, blue:1.058, and yellow:1.039, then mixed them with silkworms’ forage 30g , in this way, best successful ratio to get colored cocoons, desired high visual effects in color, well distribution, long duration and strong resistance to detergents agents are obtained. In addition, we can still get glossy appearance and fine cocoons; the wire diameter is about the size of 19.87μm. On contrast, feeding with the ordinary pigments, the degree of colored silk is better than feeding with the nano-pigments, but the appearance of cocoons are rough and the wire diameter is about the size of 21.51μm.

目前市面上缺乏簡單而精準的過氧化氫檢測法，我們參考Fenton Reaction 中，鐵離子對過氧化氫分解的催化模式，利用鐵、鈷、鎳、錳、鋅分子篩，測試他們催化過氧化氫分解反應的效率，從成本、毒性、活化能的多方考量下，選擇以分子篩來固定金屬催化離子，作為我們後續實驗的研究主軸。實驗的初步，我們選擇過氧化氫作為自由基，並著重於過氧化氫的分解反應。利用濃差電池的原理，設計出一套濃度檢測系統，由分子篩作為電極。鑒於粉末狀的分子篩容易流失，我們製備出陶土鐵分子篩，以陶土固定分子篩，並以此作為電極，搭配白金絲，透過能士特方程式，測出過氧化氫的濃度，且藉由電路調控放大倍率，可以直接控制檢測範圍。從實驗結果得知，鐵分子篩在處理過氧化氫的時候，不會有鐵離子溶出的現象，且其催化性在酸性液中可以維持，能不斷的使用，長時間來看，分子篩相當有經濟與環境保護上的價值。We attempted to provide a system for quickly determining the concentration of free radicals. The existing methods or techniques are inefficient or need expensive equipment, therefore, an inexpensive system is being sought for. As a preliminary study, we focused on the decomposition of hydrogen peroxide. Taking the Fenton reaction as reference, we designed a measuring system. This system includes a catalyst containing Fe or Pt ions for catalyzing hydrogen peroxide decomposition reaction. The Fe- and Pt-zeolite were prepared to hold Fe and Pt ions to avoid losing. Because the electrically induced potential would decrease with the decrease in the concentration of hydrogen peroxide, we could measure the concentration of hydrogen peroxide by monitoring the electrical potential. We determined the initial concentration of hydrogen peroxide in water from the initial electrical potential measured through the equation obtained from the calibration line. The practicability of this system has been assured after a series of experiments. We will further develop the technique for measuring other free radicals. We anticipate that this technique will be further developed for measuring other free radicals. Although there are several problems and limitations to be solved and conquered, one thing is for sure: this system is an environment-friendly and cost-effective facility for determining the concentration of free radicals in an aqueous solution.

本研究針對利吉惡地進行探討，研究此區之泥岩含水量、有機質含量、pH值、比重、可溶性陽離子含量、滲水特性和該區之植物種類，並探討坡度、水量對沖蝕率、山脊密度和溝痕形成之影響。 研究結果發現： 一、表層泥岩之含水率較高，深層泥岩最低。中層泥岩之有機質含量較高，表層與深層泥岩較低。各層泥岩pH值約8.1；無植被採樣點之 pH較高，有植被採樣點偏中性。有植被採樣點，其 Ca2+含量較高。 二、此區共發現十九種植物，其中銀合歡、相思樹屬優勢種。 三、坡度增加時，沖蝕率亦增加；水量增加時，沖蝕率、溝痕寬度也隨之增加，兩者呈高度正相關。第一區坡度較緩，山脊密度較大，第二區坡度較陡，山脊密度小，表面較平坦，溝痕較淺。Our research discusses with the contents of water in mudstone, the organic content, pH, the specific weight, the contents of dissoluble cation, dankness and the category of plants in Ligiligi Badland. Dissecting it`s slope, the abrasion of water, the density of mountain ridge and the formation of scuff mark. Outcome of our research： 1.Solum of mudstone is the dampest. Intermediate of mudstone has more organic content than others. Every bed of mudstone`s PH is 8.1. Having plants area is indifferent and having more Ca2+.2.We found nineteen categories of plants. For example Leucaena glauca and Taiwan acacia. 3.The more augmentation of gradient,the more increase of the abrasion of water and the breadth of the density and the abrasion of water too.

長期以來，臨床研究發現酒精會影響人類胚胎的正常發育，但是其分子機 制尚不清楚。在本研究中利用RNA 定位雜交的方式觀察酒精對於胚胎發育過程 中shh、sox9a、sox9b、col2a1、hand2 的影響，發現這些基因的表現均會受到酒 精的抑制。這項結果顯示在胚胎發育過程中，酒精透過對上述基因的影響，造成 神經脊細胞減少，細胞遷移異常，以及干擾軟骨細胞分化的現象，進而造成頭骨 發育的嚴重缺陷。此外，實驗中亦發現生長激素在腦下垂體的表現亦受到酒精抑 制。這項研究的結果成功地從基因的層次深入了解胎兒酒精中毒症候群造成頭骨 畸形及生長遲緩的病理機制。 It was known that prenatal alcohol exposure may cause serious birth defects and developmental disabilities. The molecular mechanism of this fetal alcohol syndrome still remains unclear. As revealed by whole mount RNA in situ hybridization, it was shown that expression of a number of craniofacial cartilage-related genes, including shh, sox9a, sox9b, col2a1 and hand2, were all inhibited in zebrafish embryo by alcohol exposure. It suggests that alcohol exposure may result in reducing neural crest cell production, interfering neural crest migration, preventing chondrogenesis and eventually cause craniofacial defects. In addition, the transcriptional profile of pituitary hormones were investigated by RNA in situ hybridization. It appears that only growth hormone, but not prolactin and thyroid stimulating hormone, was inhibited by alcohol exposure. The reduction of growth hormone transcription was also confirmed by real time PCR. It also appears that the expression of upstream transcription factor pit1 and downstream target gene igf1 remains unchanged. It suggests that the reduction of gh transcription is mediated by a PIT1-independent pathway. The transcriptional profile of alcohol-exposed embryo was investigated by gene microarray analysis. It appears that the expression profiles of a number of development, cellular signaling, cell growth and apoptosis related genes have be affected by 1.5% alcohol treatment. It was noted that a number of retinal-specific genes were all repressed significantly. It consists with histochemical observation that alcohol exposure results in loss lamination and disturbed differentiation. This study help us understanding the molecular mechanism of fetal alcohol syndrome.