In the last two decades, there has been increased interest in the utility of quantum scale semiconductors. These fluorescent single crystals can be as small as 1 nm in diameter, and their size and shape has been shown to be controlled by the duration, temperature, and ligand molecules used in their synthesis. Quantum dots, provide clear benefits over the organic dyes currently used for tracking biological processes. Yet, as the production of quantum dots is often very costly, the search continues for finding an industry-ready synthesis for a quantum scale semiconductor which would have high yield, optimal durability, high luminescence, and a broad absorbance range. Silicon, in particular, has been of great interest as it is the second most abundant element on the Earth’s crust and is generally a bioinert and electrochemically stable element. We report the synthesis of water-soluble, luminescent silicon nanoparticles with potential applications to bioimaging. Through a solution state top-down approach, the synthesis of hydrogen capped silicon nanoparticles was achieved in various organic solvents. The surface of the nanoparticles was capped with the functional organic molecules rendering the Si-QDs both air and water stable. Cell studies performed with our silicon nanoparticles and human monocytes show the direct applications these particles could have for tracking biological processes and the progression of cancer in the human body. In attempt to shift the luminescence of these particles, alterations of experimental methodology was also explored in the areas of reaction solvent and heating time. Through these changes, shape control of silicon nanoparticles was achieved in the form of silicon nanorods. The synthesis of this new shape of silicon at the quantum scale was confirmed by ultraviolet spectroscopy, photoluminescence, and transmission electron microscopy. The results of this study indicate that the use of silicon nanocrystals for biomedical applications is feasible.

NO.60-02 2021 APR | 科學研習期刊目錄 本期專題 文化與數學數養 以數學建模在STEAM教育中創價數學文化 | 楊凱琳 從數學史談核心素養之涵育：以「畢氏定理」為例 | 蘇意雯 怎識數學真面目？談數學文化與素養 | 劉柏宏 數學建模—操作實務與建構理論的橋樑 | 蕭志如 數學素養導向評量簡介 | 吳正新 談數學素養－生活中的數學記憶 | 李宗祐 教學現場 有數學感的分數教學：半條蛋糕平分給四個人 | 李源順、林旭霓 素養導向的數學課室樣貌―以國小三年級「小數」概念啟蒙教學為例 | 溫世展 素養導向的國小數學課室樣貌―以高年級「分數除法」為例 | 陳玉珊 「數戰棋」奠基模組融入數學教學之設計 | 孫德蘭 科學新知 看見格子點－高中數學的探究與實作 | 張宮明 朝三暮四－談Haberdasher問題的摺紙解 | 李政憲 特約專欄 繪本跨領域創造力探究學習活動 | 劉淑雯、楊紹圓 「今天飛不飛？」－金門高中生活議題之探究實作課程分析 | 丁于真、李育賢、黃琴扉 森棚教官數學題-圓形軌道碰碰車 (配合3/14國際數學日Pi Day) | 游森棚 大氣河長江水 | 盧孟明 科普活動報導 臺灣科學節創意實作科學營推廣與省思 | 洪偉清 女孩們，FUN科學! | 黃真瑱、呂玉環、林琦峯、蘇萬生 科教館GO好玩 「點實成今」-中小學科展一甲子特展紀實 | 蘇珮婷 總召集人的話 最近在看「好音樂的科學」，這是一本同時從科學理性及音樂感性的角度讓我們理解聽眾跟音樂產生共鳴的機制—情感連結，當然還有很多好東西要深究此書才能享用到。科學如同音樂般，透過多層次的轉譯，透過不簡單的語言，讓受眾在情感上受到挑戰、啟發，才有機會與情感對準頻率產生共振，才有主動探究的力道。試想一條琴弦，琴弦的材料、鎖緊的張力以及固定的方式都會對於彈奏的效果有所影響；這時需要數學建模把弦的振動方程式推導出來，再來就是有系統的求解，從不同的振動模態跟音樂之間的關係開始有了連結。 數學建模需要豐富的生活經驗作為後盾，數學建模歷程對於問題解決是個很好的學習典範，又數學建模與STEAM相關學科的整合是實踐數學文化多種價值的很好策略。「文化中的數學」與「數學中的文化」交錯互動耦合成為「數學文化」，您知道多少種畢氏定理的證明方法?人類發展過程中的數學扮演的角色可以透過「柏拉圖多面體」的美與「七橋問題」的趣味來說明。結合數學史的脈絡，展現不同時空文明間的問題解決方法，可以協助學習者將抽象觀念具體化，在學習數學過程中，也能認識這些問題在我們人類文化中代表的意涵。 想知道如何建立「帶著走」的數學能力嗎?如何建立素養導向評量工具?這些目標都是讓學生從真實情境中，學習思考、提問以及建立問題解決的能力，進而產生學習遷移，達到終身學習的目標。生活中的數學記憶則透過生活中的許多實例，如鉛筆工廠的倒三角形容器、酒杯堆疊的二階等差數列、祈福摺紙、牛奶盒的堆疊，讓學生理解生活中的數學應用；有空時也該搜尋自然界中無所不在的數學元素，讓數學之美提升對數學的相關概念及精神並內化到自己的所學。 熟悉代數運算的大人們遇見算術問題的解釋，就突然間發現需要回到童年時期的思考。國小的數學教育是很具挑戰性的，如何在先備知識不多的情境之下逐步建構學生的知識體系?半條蛋糕平分給四個人，要如何表達有感的分數教學?如何透過生活情境、學習需求及操作有感的學習啟動小學三年級的小數啟蒙教學?還有高年級的分數除法，結合數感理論的讓學生說、讓學生畫圖、讓學生舉例等教學策略，引導學生有感的數學學習。 課堂上要吸引學生的目光需要好的任務設計，如何透過活動及遊戲在競爭中增加挑戰性，還要從玩樂中增強學生數學的準備度?看見格子點---高中數學的探究與實作是思考與探究的很好例子，很值得讀者拿起鉛筆，跟著做一遍，動腦歸納動手畫。切割及拼布問題--朝三暮四，是講縫紉商透過適當切割正三角形拼布，再重新拼組為正方形的問題；透過討論四種不盡相同的正方形切割重拚為正三角的組法，並比較不同切割方式的差異性，非常有趣，看了好想跟著動手裁切拼裝。 繪本跨領域創造力探究學習活動透過第一線教師的學習活動分享，讓讀者有典範學習之效。「今天飛不飛？」一文讓讀者理解使用航空站公開資訊結合生活議題，做出不同條件的估算，以達探究與實作之效。森鵬教官照例給了趣味有挑戰性的圓形軌道碰碰車，三輛車從十二等分圓周上的指定出發點及指定方向出發，在特定條件及設定的規則下，這三部車要經過多久才能同時回到原來的出發點。 地上的河流、空中水氣形成的大氣河是全球水循環的基本現象，值此久旱不雨的今天，讀此文章特別有感；對於大氣河的發生及地點的氣象預報準確度提升，有助於因應變動加劇的未來。舉辦活動是需樣樣兼顧、統籌運轉且勞心勞力的工作，臺灣科學節創意實作科學營的舉辦，綿密的合作夥伴以及各方資源的投入都值得參考學習。科教館透過執行科技部女性科技人才培育計畫，入校推廣科學實作的課程以及推動高年級女學生探究與實作課程的教學實況。中小學科展六十年了，有許多好作品值得學習，「點實成今」-中小學科展一甲子特展紀實呈現整個策展的理念以及所欲表達的探究、好奇及融入生活；展現創新思考、解決問題與合作學習的科展精神。 這一期的科學研習月刊是數學含量很高的一期，讓我們跟著動手玩玩、動腦想想，也趁機省思數學文化及數學學習的各個面向。 總召編輯委員 - 李旺龍 關於本刊 出版單位：國立臺灣科學教育館 發行人：劉火欽 總召集人：李旺龍 編輯委員： 物理科吳仲卿/陳耀榮/李柏翰/盧玉玲 | 化學科古建國/許良榮/王伯昌/林如章/周金城 | 生物科王美芬/蕭世輝/陳建志/郭淑妙 | 地球科學許民陽/王郁軒/李文禮/謝隆欽 科技科張玉山/汪殿杰/林育沖/趙珩宇 | 數學科李源順/鄧家駿/溫世展/張宮明 | 跨領域學科李名揚/連信仲 | 特約專欄 游森棚/黃琴扉/陳正改/劉淑雯 策劃：曾聰邦 主編：吳中益 本月專題特約主編：李源順 編輯：佟冠誼 網頁設計編輯：施曉恬 投稿規範請來信詢問：article@mail.ntsec.gov.tw

Purpose of Project: To save energy and to help the underprivileged with a cooler that uses no electricity to make their lives better. Procedure/method followed: STEP 1: Collected 28, used, 2 litre plastic bottles. STEP 2: Chose the window with the best wind flow. Measured the size of the window and the room chosen. STEP 3: A sturdy thick polystyrene board was cut to the size of the window. Holes were drilled to the rim size of the bottles spacing them according to the body size of the bottles. STEP 4: Bottles were cut in half. STEP 5: The bottle necks were slid through the holes with the necks open to the inside of the room and the bodies open to the outside. STEP 6: Fixed a thermometer in the room and measured the temperature and recorded it. STEP 7: Fixed the Powerless Shack Cooler with the necks of the bottles open to the inside and the bodies open to the outside of the room. STEP 8: The temperature variation was checked and recorded every 30 minutes for 3 hours. STEP 9: Another room of the same size and window was also chosen. Fixed a thermometer and temperature variation was checked and recorded every 30 minutes for 3 hours. This served as the control of the experiment. Data/results: The room temperature decreased over time inside the room where the Powerless Shack Cooler was installed onto the window. But the control room maintained the initial room temperature although slight fluctuations in the room temperature were observed over time. Conclusion: The hypothesis was supported. As the air molecules moved through the bottles, it bounced off each other, and off the walls of the container, holding the air. A small volume of air passed at a high velocity. When the molecules moved faster the collision became more often. These collisions and the push increased air pressure. When the container’s space was getting smaller, the molecules picked up speed and the temperature went up. When the air was released out into the room, the volume suddenly expanded. The intermolecular spaces became larger; so less agitation and vibration of molecules took place. The molecules moved slowly. The room temperature reduced. Air inside the room became cooler. During the adiabatic expansion, air molecules used heat energy from the room and converted it into kinetic energy for faster movement.

高分子發光二極體乃是利用電子和電洞在發光工作層結合而形成激子，激子在形成後隨即以光的形式衰退，而發出光來。一般而言，高分子發光二極體是一種三明治的形式，電子注入層/發光工作層(高分子)/電動注入層，而通常我們會選擇加入電子傳遞層和電動傳遞層，來增加其發光效果。本研究係針對聚苯胺做為電洞傳遞層對發光二極體之影響，分別對有電洞傳遞曾和無電洞傳遞層之發光二極體作導電度測量、循環伏安法、電位-電流測量、亮度-電位測量。發光二極體中所須之高分子層，須利用旋轉塗佈機，將其均勻的披覆在載體上，由於本校並無此儀器，所以將離心機改造成簡易旋轉塗佈機，並觀察其披附之效果。另外，在封裝元件時，係因學校缺乏真空蒸鍍的儀器，所以將利用電鍍法把鋅鍍在ITO玻璃上，或將鋁片和ITO玻璃緊靠在一起，針對此二替代方法，本研究將探討其所製出原件成效。Polymer light emitting diode (PLED) utilizes the energy gap between the layer of electron and electric hole that emits the light due to the decay of solitron. PLED is always in the sandwich from, that means the conducting polymer is contained between metal as an cathode and indium tin oxide as an anode. It is used the addition of another electric transfer layer and electric hole transfer layer to improve the lighting efficiency of PLED. The purposes of this study are to discuss the effect of polyaniline as the electric hole transfer layer. The study methods are conductivity measure, the cyclic voltametry, the luminance-voltage curve. The simplified spin coating machine was designed to coat the polymer onto the ITO. The zinc was electroplated onto the graphite as an cathode.

Purpose of research. Microbial fuel cells (MFCs) are bioelectrochemical systems that generate electrical energy by exploiting the extracellular electron transport (EET) capabilities of electrochemically active bacteria (EAB) (Logan 2009). This investigation aims to identify genes involved in driving bacterial EET with a new procedure that enables rapid screening of a side array of genes. These insights may lead to improved MFC performance through enhancing reactor design or genetic engineering EABs (Alfonta 2009). Procedures. MFC metagenomic analysis. Twelve MFCs incubated with four different bacterial samples were operated for approximately one year. The bacterial DNA from before and after incubation was extracted and the 16S rRNA regions were PCR amplified and sequenced. The bacterial community changes were analyzed using the QIIME program to identify bacteria that were being selected. Fosmid Clone Isolation. An E. coli fosmid library (Mewis et al. 2013) that contained genes from EAB inferred in the previous step was incubated in three MFCs. After a 48 hour enrichment period, biofilm samples from the MFCs were extracted and individual clones were isolated and screened in the MFCs individually. An E. coli DH5α strain with no insert DNA was incubated separately as the control. DNA sequencing. Fosmid insert DNA from high-performing clones were extracted, purified using gel electrophoresis, constructed into sequencing libraries and sequenced. Bioinformatics Analysis. The sequences were constructed into larger contigs using the Velvet algorithm package. The open reading frames (ORFs) were inferred and translated into amino acid sequences and annotated with proteins identified from the KEGG, and SEEDs databases using Metapathways 2.5. Results. The changes in bacterial communities from the metagenomic analysis revealed increases in relative abundance in numerous genera from Firmicutes and Bacteroidetes. The MFCs incubated with the fosmid clones generated about 4 times more peak power than the MFCs incubated with the E. coli DH5α. Polarization curves generated for the MFCs demonstrated that the fosmid clones were able to sustain a higher current. Incubation of pure cultures of individual clones yielded four clones with significant performance improvements over the control strain. Protein data from Metapathways outputs reveled both novel and previously reported EET genes encoding for Type IV pilus structures, c-type cytochromes, soluble cytochromes, flavoproteins, and porins. Taxonomy inferences of the gene inserts by the Green Genes database reveal the genes most likely came from the same EABs that were inferred from the metagenomic analysis. Conclusions. The increased performance of the fosmid clone-powered MFCs suggest that the clones carried genes that enhanced their performance in the MFCs. This is further confirmed by polarization curves generated for the MFCs. The results of the taxonomy inferences suggest that the bacteria being selected for in the environmental samples carried genes that enhanced their performance in the MFCs, and that these genes were successfully identified in the subsequent steps. The results of this study demonstrate that using a gain of function approach to rapidly screen a wide array of genes in a gene library may be an efficient method to identify genes that enhance power generation of EABs in MFCs.

本研究針對台灣產水生植物，蓴菜之構造與生長環境、蓴菜對腸胃道常見致病細菌之抑菌效果以及主要成分暨化合物分析。由本研究結果得知，崙埤湖內之稀有浮葉型水生植物蓴菜，其生長環境為無汙染之乾淨偏酸性水源，最適合生長之生深為50-160 ㎝；水溫則為22-25℃；而蓴菜之地下根莖對表皮金黃葡萄球菌（Staphylococcus aureus）具有輕度之抑菌效果，經由分離純化得知為BS-1：沒食子酸（Gallic acid）；另外，由蓴菜之葉片分離出十種成分分別為BS-2 （Kaempferol-7-O-Glucosids）、BS-3 （Quercetin-7-O-glucosids）、BS-4（5,8,4’-Trihydroxyflavone-7-O-glucosids）、BS-5 （3,5,8,3’4’-Pentahydroxy flavone）、BS-6（Vitamin E: d-Tocopherol）、BS-7 （Glyceride）、BS-8 （Phenolic A）、BS-9（Quercetin）、BS-10（Kaempferol）、BS-11（Phenolic B）。其中發現BS-8 對神經膠腫瘤細胞株有18.42%之抑癌效果，另外，BS-2、BS-3、BS-5、BS-10、BS-11 等成分，呈現良好之美白作用。This investigation is to analyze Brasenia schreberi Gmel., a native rare floating water plant in Taiwan, focusing on the plant’ s structure, its growth environment and, most importantly, the effect of chemical compounds it produces on restraining the common pathogenic bacteria in human stomach. The result indicates that the most suitable growth environment for Brasenia schreberi Gmel. is in slightly acid, pollution-free water such as that in the lake Lung Pi in northern Taiwan. The ideal water depth for its growth is 50-160 cm, and the water temperature is 22-25°C. The impractical BS-1 (Gallic acid) extracted from the izome of Brasenia schreberi Gmel. by separation and purification has a light effect on restraining Staphylococcus aureus, a bacteria in the stomach. From the epidermis of the blade of Brasenia schreberi Gmel., ten other ingredients are also isolated, including BS-2 (Kaempferol-7-O-glucosids), BS-3 (Quercetin-7-O-glucosids), BS-4 (5,8,4’-Trihydroxyflavone-7-O-glucosids), BS-5 (3,5,8,3’,4’-Pentahydroxyflavone), BS-6 (Vitamin E: d-Tocopherol ), BS-7 ( Glyceride ), BS-8 (Phenolic A ), BS-9 (Quercetin), BS-10 (Kaempferol),and BS-11 (Phenolic B). BS-8 is found to resist cancer C6 ( Glioma ) by 18.42%, while BS-2,BS-3, BS-5, BS-10, and BS-11 show an outstanding effect on skin-whitening.

The purpose of this project is to create a project that can improve the quality and the production of the merchandise and also to prevent a 100% of mistakes in the electronic components which are complicated to detect or not detectable for us when we are making a circuit. Other of the purpose of this project is to prevent the bad production in the products which are elaborated in a company and then to check if the employee is working well, in that way the company will be able to avoid the bad production and it will be more profit for the company thanks to the products.

1. Purpose of the research: Recovering bioethanol from fruit wastes by using brewing yeast and enzyme Viscozyme L\r 2. Procedures: Four different experiments were conducted in our project.\r Experiment-1: Samples in RM by the addition of only S.Cerevisia.\r Experiment-2: Samples in RM broth meduim by the addtion of 100 μl/g ViscozymeL and S. Cerevisiae\r Experiment-3 : ViscozymeL with different concentration\r Experiment-4: to determenine whether the enzyme with high concentration inhibited the fermentation.\r Analysis of ethanol: HPLC method was used for the analysis;\r HPLC method: (high performance liquid chromatography)\r 3. Data\r HPLC measured initial and final concentrations of glucose and ethanol by the addtion of S. Cerevisiae.\r Initial glucose concentration and final ethanol concentration was the highest in Grape pulp 12 and 9 g/L respectively. Despite the glucose concentration was the lowest, the ethanol concentration was quite high (7 g/L) in the mixture with orange. The lowest ethanol was produced in the mixture with pomegranate.\r Concentrations of ethanol and glucose from the samples in RM broth meduim by the addtion of 100 μl/g ViscozymeL and S. Cerevisiae\r After the 24 hour incubation by the addition of enzyme glucose concentration incresed by 1, 4 and 2 respectively in sample 1, 2 and 3. The ethanol concentrations in comparison to first experiment were higher especially in the sample1 (grape pulp).\r Effect of ViscozymeL with different concentration on glucose production.\r The glucose concentration generally increased by the addition of 200 micro L of enzyme as the time proceeded. In grape pulp sample glucose concentartion didnt increased by the addition of 500 micro L enzyme.\r The remaning glucose and produced ethanol concentrations by the addition of viscozyme L with two different concentartion and S. Cerevisiae\r As the concentartion of enzyme added incresed the amout of ethanol also inreased in sample 2 and 3 but in the sample 3 the fermentation was inhibited.\r 4. Conclusions\r In this study sugars in fruit wastes that are regularly not recylecled were fermented successfully into ethanol by using S. Cerevisiae.\r Generally we get more ethanol from the samples when the enzyme was used. As the graph5 shows the ethanol concentration generally increased as the enzyme concentration increased. But especially in the sample1 (mixture with pomegranate) at 200 and 500 micro liter concentrations, ethanol production were 2 and 0 g/L respectively. In the light of this daha we conluded that the enzyme with high concentartion might inhibit the fermentation. When the activity of enzyme was inhibited by keeping the pretreated enzyme in boiling water, the fermentation restarted and recovered more ethanol; 8 and 12 g/L at the enzyme concentration of 200 and 500 microliter respectively.

此研究著重於機器瓢蟲在不同的操控變因下所走出之路徑是否存在著某些性質。對於轉向次數k→∞且轉向角θ為任意角時，我們計算各收斂點P於坐標平面上恰形成圓C：(x- 1/(1-r2)2+y2=(r/(1-r2))2。將瓢蟲的轉向點P1、P2連線，圓心C與收斂點P連線，則P1P2與CP之交點S的軌跡形成長軸長為圓C半徑(r/(1-r2))的橢圓，且此橢圓的焦點為P1(1, 0)與C(1/(1-r2), 0) 。各轉向點Pn(n∈□)位於一個方程式為R=mrθ-π/α，m=OP=√1/(1-2rcosα+r2)定角為cot-1(㏑r/α)之等角螺線上；同時繪出轉向次數k在不同值時，瓢蟲行進終點之軌跡，以驗證當k愈來愈大時，各終點形成的軌跡會趨近於一個圓。當k=2時，圖形為蚶線並證明其經平移後之極坐標方程式為 R=r+2r2cosθ。最後我們展示行進公比r→1- ，r=1，r→1+時所呈現的終點軌跡，並對此軌跡所呈現出的意象與自然界連結，而其實質關聯性則有待未來研究。

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。

流體旋轉時，外圍及底部流體，因槽壁及槽底摩擦力的影響，流速較慢，相對的壓力也較大，導致外圍的水流會轉入中心。發現本實驗的渦流為強迫與自由漩渦組成。實驗中，探討f(轉動器的頻率)、H(總水深)、y(?入深度)、R(轉盤半徑)四者與角形數間的關係。若y、R 愈大、H 越小，隨著f 的增大，可觀察到的形狀邊數越多；反之，若y、R 愈小、H 越大，則f 愈高，所形成的圖形半徑愈大，易超過轉盤，不易觀察。依白努利方程式，外層水流的流速較慢，而內層水流的流速較快，故外層壓力大而內層壓力小，水會由外往內流，而此渦動流於轉動液面產生的剪力，可能為產生N 邊形漩渦的主要原因之一。流體旋轉系統中，因轉動而產生流體離心力與內外層壓力差交互作用下，於某特定相關的因素條件下，形成特定角形數漩渦，是本實驗的重要發現。When fluids are in rotation, fictitious force given by the container brings about the relative decrease of speed of the bottom and outer layer of water, which causes its pressure to increase, and water to spin inward, resulting in a vortex motion with N-corner polygons formed at the surface of the rotating plate. During this experiment, we discover that the vortices consisted of free and forced vortex and the polygons vary as control parameters f(rotation frequency), H(height of fluid), y(depth of the plate), and R(radius of the plate) change. The larger y and R are,the smaller H is, the more corners show up as f increases. On the contrary, the smaller y and R are,the larger H is, few polygons are identified since the rotating radius of polygons are larger than the plate. According to Bernoulli’s principle, smaller velocity of the outer-layer water causes water pressure to increase and water to spin inward. During this process, shear force is developed at the surface of the rotating fluid, which we believe is the main cause of N-corner polygons. In a rotating system, the interaction of centrifugal force and differential pressure causing a certain Ncorner polygon to be formed under different controlled parameters is our main discovery.

Two soldiers walk on a checkerboard. They can only walk one step once a time and two directions, front and left, are decided randomly. The gunshot is the column and row where a soldier is located, and one will die if he enters the gunshot area of the other. To treat the probability of winning, we first study the cases of 1×n, 2×n, 3×n, and 4×n rectangles iterately. Then we establish a general form of the probability of winning in a general n×k rectangle by using recurrence technique and generating function, respectively. Finally, we extend to the general n×m×k cuboid case to obtain the first soldier’s probability of winning.在一個長方形的棋盤中，兩士兵行走，每一次只走一步，而且上和左兩個方向是隨機的，射程範圍是所在的此行和此列，而進入他人射程範圍則死亡。探討其獲勝機率，從1×n 、2×n、3×n、4×n 矩形的情形逐步研究，並分別運用遞迴式的技巧及生成函數，導出 n×k 矩形中先走士兵獲勝機率的一般式。更進一步地，我們也獲得了n×m×k 立體空間先走士兵的獲勝機率。