在此文中我們研究： 一個 n×4 的長方形網格中，所有從(1, 1)或(1, 4)出發，在不重複且經過每個格子點的情況下，走到(n, 1)結束的所有路徑總數分別為 Tn 、Un。 In our research, we study a n× 4 rectangular network lattice, of which all the routes are starting from (1, 1) (resp.(1, 4)), and ending at (n, 1) (situations are considered only on the conditions of passing every spot and not being repeated ). And we set the sum of all the different routes as Tn (resp. Un).

一個有3 個旋鈕，每個位置的號碼數分別是a、b、c 的密碼鎖，如果有兩個位置的數字正確就能打開，最少需要猜多少次才能保證打開這個鎖。在本論文中，我們將密碼鎖三個位置的號碼數分成：a＝b＝c＝n、a＝b＜c，a＝b＞c 和a＞b＞c 四個部份來討論。前兩部份的研究已經找到最少次數開鎖的方法 ，後兩部份則是給了一個演算法可求出開鎖次數的上界。If a combination lock with three rotate wheels can be opened when two wheels are adjusted to the correct numbers, then how many guesses does one need to make before he or she can actually open this lock? Let us say a , b and c respectively represents the numbers that should show on each wheel. In this paper, we divide the numbers into shown on the three wheels, and they are a = b = c = n , a = b c and a = b < c . The research on the first two combinations has already given us the method we can use to open he lock with the least number of trials. On the other hand, the latter two offer us an algorithm that can be uses to obtain the upper bound of tries needed to open the lock.

中文摘要 植物體在生長發育的過程中，有許多必要的營養素，一旦缺乏這些營養素，植物的生長 狀況會產生問題，不但會抑制植物生長，植物對外界的免疫力也會跟著下降。在台灣，水稻 不僅是台灣人賴以維生的糧食，更是台灣人引以為傲的作物。但台灣高溫多濕的海島型氣候， 適合多種稻作病害的發生，紋枯病及白葉枯病就是危害台灣稻作主要兩種病害。這些病原菌 的感染直接限制了該土地中植物的生長和生物的存活，間接的影響人類的生活。 本研究中，以水稻為實驗對象，我們主要探討六種水稻生長所必須的營養素：鈣、鎂、 鐵、鉀、氮、磷和水稻之間的關係。我們將水稻種植在缺乏某種特定元素的培養基中，觀察 其生長的異狀，以便於得知某特定元素對於水稻生長發育的影響為何。之後，再將缺乏各種 營養素的水稻及營養素供應正常的水稻，分別感染白葉枯病病原細菌及紋枯病病原菌，觀察 受感染的水稻之損傷程度，並檢測受感染植株中過氧化氫酵素活性的變化，試著歸納出其防 禦機制和營養素之間的關係，期望對其防禦機制能有進一步的了解。 英文摘要(Abstract)\r When plants grow up, there’s a lot of necessary nutrition. Once plants lack of this nutrition, some problems will occur. It can cause plant growth retardation and decrease the immune ability to the surroundings. Rice is a main cereal in Taiwan, and we are all proud of it. However, rice is frequently attacked by many pathogens , such as Xanthomonas oryzae pv. oryzae and Thanatephorus cucumeris. The infection directly limits plant growth and survive of organisms, but also indirectly effect life of human. In this research, we study the effect of nutrition deficiency such as nitrogen , phosphorus, potassium, iron , magnesium and calcium deficiency. First , we cultivate the crops on the media which lacks of some specific nutrition. After intensive observes on the plants, we can slightly discover the relationship between these nutrition and the plants. After the plants are infected by the pathogens (X. oryzae pv. oryzae and T. cucumeris) in two weeks, we record the spread of the lesion size resulted from the infection. The changes of Peroxidase activity after infection are also studied. We hope we can get a deeper understand about the immune system of rice.

「賽局理論」其實就是一種科學、理性行為在一個會相互影響的狀況，在一些比賽中是可以應用的。因為要贏得比賽，必須要有實際上的能力(速度、金錢等等...)、技術(這方面就可以運用賽局理論)還有重要的....運氣！例如:統一集團7-11與全家便利商店,為了營業額和吸引消費者上門,在於商品部份的價錢與優惠和一些促銷方案計畫,如何讓店家可以創造出更高的業額與價值,同時間也滿足消費者的需求。而「賽局理論」就很清楚得剖析了生活上這些類似的問題，能藉此理論找到最好的方法，又兼顧著個人的最大利益的同時並創造雙贏。在生活中我們常常面臨各式各樣的賽局，只是我們都沒有發現，小至與人互相擋到要往左還是右邊讓的問題，大到國際談判、軍事競賽和登入月球…等問題，若能進一步收入研究了解賽局理論，必能藉此為我們找尋到最佳的解決模式。

當輕敲啤酒杯時，會發覺酒杯發出之聲調隨氣泡漸漸消失而有所改變。為了解氣泡是如何影響頻率，我測試了幾種含氣泡之飲料，以探討當氣泡漸漸消失時，杯子發聲頻率之變化。實驗發現杯子內飲料之氣泡漸漸消失時，裝啤酒與可樂杯子的發聲頻率逐漸增高，然而裝沙士杯子的頻率卻逐漸降低。為解開此相互矛盾的現象，我設計了將液面上之泡沫及液面下之氣泡分開檢驗的實驗。實驗結果發現，液體中氣泡的存在會使杯子發聲頻率變高。而當液體表面受到擾動時，會降低杯子的發聲頻率。若液面上存在泡沫時，杯子的發聲頻率也會變低。這表示裝盛含氣泡飲料杯子音律之變化，須同時考量液內含泡量與液體表面之效應。此結果可以成功的解釋為何啤酒、可樂與沙士於氣泡漸漸消失時，杯子頻率會變高或變低的現象。Tapping the side of a glass of beer as the bubbles escape, one can find that the pitch will change. In order to know how the bubbles would influence the frequency, I survey the pitches of a wine glass with various drinks that would generate bubbles. As bubbles getting away from the glass, my experimental data shows that the frequencies of the tone did get higher when the glass contains beer and coke. However, I surprisingly find a different result when the same glass contains sarsaparilla. Hence I design a series of experiments to understand the possible mechanism. The data suggests that when the drinks contain bubbles in it, the frequencies of the tones will be higher. When the liquid surfaces were disturbed, the frequencies of the tones will become lower. If there were foams above the liquid surfaces, the frequencies of the tones will also be lower. This finding proposes that people needs to consider both the bubble bulk status and surface effects. The consequences of the competitions between these two effects can successfully explain how the tones are changed in the cases of beer or sarsaparilla or coke.

Vacuum packaging is a packaging technique intended to extend the shelf life of food via the removal of air from an enclosed package prior to sealing. This process limits the growth of aerobic bacteria or fungi due to oxygen deprivation. In this work, we present a novel do-it-yourself vacuum packaging device using the exchange of water and air between two bottles to continuously generate a vacuum-suction effect. The sizes of bottle and vacuum bag were investigated for its impact on the vacuum generation in a plastic bag containing smoked fish sausages. Large commercial 3.1-litre PET bottle generated more vacuum than the smaller ones. An equilibrated vacuum pressure of a smaller plastic bag was lower than that of a larger size. With 3.1-litre PET bottles, the vacuum pressure for 3”x5”, 5”x8” and 6”x9” bags was equilibrated at 8, 10, 18 mmHg, respectively. Sausages packaged by our device last for 14 days when they were kept in -20oC refrigerator, which was comparable to those packed by the commercial vacuum packaging system for household use. This project demonstrates an application of simple science in a real life situation as well as a promotion of environmental protection idea as the electricity is not used in the vacuum generation process and the disposed plastic bottles can be reused.

Autonomous car is a very new concept, being a car without any driver. Several concurrent software process data using Artificial Intelligence to recognize and propose a path which the car should follow. The goal of the project is that a driverless car can reduce the distance between the cars, lowering the degree of road loadings, reducing the number of traffic jams, avoid human errors, and allowing people with disabilities(even blind people) to travel using an autonomous car. Theoretically a car without driver in the future should be much safer, because human reaction speed is higher than 200 ms, and the computing power of the newest computers allows traffic calculations even to 10 ms. The necessary power is provided by three multi-core laptops that process with Artificial Intelligence in order to recognize traffic signs, traffic lanes , traffic car fingerprints, processing the data from a 3D radar, using particle filters to localize car in a GPS map, the management of database with traffic signs, magnetic sensors, acceleration sensors, a distributed software, a supervisory system and the software which drives the stepper motor to turn the steering wheel (acceleration and braking). Currently the software is able to recognize the traffic signs, register them in a database using Google Maps. The fields record the sign and direction of travel from that area. Each car participating in the traffic and using this software will register new signs detected and the will modify the degree of confidence of recognition for other users. Another software component is able to recognize the demarcation lines between lanes, with three cameras to calculate exactly or using probabilities where it is on the road, where the roadsides are and to propose a new direction even in the absence of traffic signs for the next seconds. Another part of the software is trying to use Artificial Intelligence to detect other car fingerprints from webcam images. The calculation was performed on 3 computers, requiring distributed processing. I developed a management information system based on semaphores that allows data processing and supervision from 3 different computers. This project presents a hardware version of a LIDAR – a 3D radar and a software for creating a 3D environment in which the car navigates and using it the car will take decision to avoid obstacles. The LIRDAR contains a total of 16 avalanche photo-detector mounted on a stepper motor that spins at a frequency of 10 Hz. The information provided by my radar is about 576.000 pixels at resolution of 10 bits. The 3D radar helps the entire software system to increase the confidence of decision.

探討原正方形經拼湊組合成的長方形面積會減少一平方公分可能的原因，另外，由角度與相似三角形邊長與邊長比值相等的原理來觀察。實際上由於原正方形切割成兩小角形，及兩梯形後，小三角形與梯形組合後並未成為一大三角形，以致於全部組合後，有面積重疊的地方。我們推論出的公式：原正方形邊長甲，切割長度乙，則面積差量 =拼湊後長方形的面積－原正方形的面積=〔（2 × 甲－乙）×（甲－乙）〕－（甲 × 甲）。而我們發現.任一正方形邊長甲都可以找到一最佳切割比例，及長度乙≦甲/2，且以最接近甲/2 的長度時，使得拼成後的長方形面積與原正方形面積的差量為最小，並可將拼成後的長方形歸類為凸出型（以⊕表示）與凹陷型（以Θ表示）兩種類型：最後我們歸納出一規則，即相同面積差量的每一列中的每一數恰好是前兩數的和（最前面兩數除外），也就是說，對每一列數任取連續三數（A，B，C）就是一組最佳組合，C 為正方形邊長甲，A 為切割比例乙，而面積差量＝B×（2B－A）－C×C＝B×（B+C）－C×C 最小。另外，亦可連續四數為一組合（A、B、C、D）， D 即為組合後長方形長，B 為寬，因此面積最小差量為：B×（B+C）－C×C＝B×D－C×C，可用來預測出不同的正方形邊長甲，它們所得到的面積最小差量相同的最佳切割值。.另外，我們可以面積差量＝1（如底色為黃色那一列）正方形邊長為依據，我們將其面積差量＝1 的正方形邊長甲×倍數，即可預測得到最小面積差量＝倍數×倍數。反之，我們亦能根據面積差量最小值所分解成平方數a× a（倍數×倍數），預測得到最佳正方形邊長＝如底色為黃色那一列正方形邊長甲×a（倍數）。同時，經由我們整理出的規則，預測面積差量為平方倍之正方形邊長，並依據面積差量＝1 的正方形邊長，來預測得到長度為倍數之最佳正方形邊長的凹凸類型

本文是利用整數分拆及排容原理研究以1x2 矩形鋪mxn 矩形的排法邏輯及總排法數，並進一步探討下列兩種情況：雙邊不含缺陷線及至少有一缺陷線的排法數。\r Based on the Apart of Number and the Inclusion- Exclution principle, this research intends to analyze the logic and final result of permutation by laying an 1x2 rectangle over an mxn rectangle. From the analysis, this study will go further to explore the possible permutation in the following two situations : when there is no fault line on both sides and when at least one fault line exists.\r

由於塑膠不能在自然情況下順利分解，所以我們在塑膠中添加其他成分使塑膠可以較易分解。我們選定常見的塑膠—熱塑性的耐綸-66。在聚合物的製作過程中添加葡萄糖、澱粉、洋菜粉末以及甲基纖維素，並觀察加入添加物的塑膠在結構上是否有變化？其塑膠在線型時之張力是否有增強？耐酸鹼性是否有變化？由實驗結果我們可以得知含有甲基纖維素之耐綸-66 所能承受之張力強度最高，且其彈性係數也比無添加物之耐綸-66 高出近2 倍；而含可溶性澱粉之耐綸-66 所能承受之張力最小，且彈性係數也最低。此外，進行生物分解的實驗可發現，含洋菜粉末的耐綸-66 分解的速率最快。使用400 倍的光學顯微鏡可發現含有洋菜粉末的耐綸-66 表面與其他耐綸-66 複合材料差異較大，值得進一步研究。;Because plastic cannot be decomposed naturally by itself, therefore, additives needed to be added to facile the decomposing process. Let us choose one common material: thermoformed Nylon 66. During the formation process, add the following additives: glucose powder, methyl cellulose, soluble starch and agar powder. Observe whether adding additives would allow changes to occur structurally, or would the elasticity be improved when exist in a linear state, or even it would form a better pH resistance property. Most importantly, observe whether the decomposing rate has increased or not. According to the experiments, when Nylon 66 contains methyl cellulose, it can sustain the highest tension. Its coefficient of elasticity is 2 times as large as the original one. In terms of the data, we can also observe that when Nylon 66 contains soluble starch, it has the lowest ability to sustain tension. Besides, it has the lowest coefficient of elasticity. And when Nylon 66 contains glucose, it has the highest rate in the process of decomposing. As we look at the surface of polymers under 400 diameters, we can observe that the Nylon 66 with agar powder has some filiferous substance. But we have not confirmed what the matter is. As to decomposing rate, we found that when Nylon 66 contains additives, it could accelerate its decomposing rate. And the one with agar powder has the highest rate of decomposing.

這個研究起源於一個平分圓的問題：在平面上2n +1個點(n∈N)，其中任三點不共線，任四點不共圓，任取三點可以畫出唯一的圓，若一半的點在圓內，一半的點在圓外，則此圓為平分圓，Federico Ardila 教授在America Monthly 111 期[2]中發表了一篇論文，證明平分圓的個數為n2個。我們研究的目的是：如果將圓改成拋物線，則平分拋物線的個數是否為一定值? 若為定值，則為多少個? 我們的研究題目是：平面上2n +1個在正常位置上的點(n∈N)，平分拋物線的個數為何? 我們將研究的主要結果分述如下： 一、證明在平面上2n +1個點(n∈N)，平分拋物線個數為定值。 二、證明在平面上2n +1個點(n∈N)，平分拋物線個數為n2個。 接著推廣至：若平分拋物線改成(a ∨ b)拋物線，則個數為何? 我們將研究的主要結果分述如下： 一、證明在平面上2n +1個點(n∈N)，(a ∨ b)拋物線個數為定值。 二、證明在平面上2n +1個點(n∈N)，(a ∨ b)拋物線個數為2(ab + a + b +1)個。 This study originated from a question of “The Number of Halving Circles＂: Setting 2n +1 points in the plane is in general position if no three of the points are collinear and no four are concyclic. We call a circle halving with respect to those 2n +1 points if it has three points of those 2n +1 points on its circumference, n −1 points in its interior, and n −1 in its exterior. Then we call this circle “Halving Circle.＂ Professor Federico Ardila issued a paper in the America Monthly 111 [2]. The goal of that paper is to prove the following fact: any set of 2n +1 points in general position in the plane has exactly n2 halving circles. The purpose we make the study of is: If we turn circles into parabolas, how many Halving Parabolas are there? The title we make the study of is: Setting 2n +1 points in the plane (n∈N) , how many Halving Parabolas are there? We show our main effect below: 1. Proving that 2n +1 points in the plane (n∈N) , the number of Halving Parabolas is constant. 2. Proving that 2n +1 points in the plane (n∈N) , the number of Halving Parabolas is n2 . Spread: If we turn Halving Parabolas into (a ∨ b) Parabolas, how many (a ∨ b) Parabolas are there? We show our main effect below: 1. Proving that 2n +1 points in the plane (n∈N) , the number of (a ∨ b) Parabolas is constant. 2. Proving that 2n +1 points in the plane (n∈N) , the number of (a ∨ b) Parabolas is 2(ab + a + b +1) .

The purpose of this research was to challenge misconceptions about what attracts teenaged boys to girls. It has been observed that some teenaged girls hold stereotypes about boys and believe boys are only attracted to superficial qualities like beautiful hair. The aim is to investigate differences between what girls believe boys find attractive about them and what boys actually find attractive about girls. Hypothesis: There is a significant difference between what teenaged boys find attractive in girls and what teenaged girls think boys find attractive. Procedure Background reading was done to discover which characteristics are commonly used to measure attractiveness among teenagers. A survey questionnaire was used for gathering data. To gather information for the construction of the questionnaire, 4 focus groups were interviewed. The groups were asked to determine indicators of attraction and repulsion. Two separate questionnaires were designed (one for boys and one for girls), based on focus group information, with similar questions which could be used to compare results. Questions were related to personality and looks. The research was representative as 399 teenagers participated in the final survey. Results were collated in Excel tables. Percentages were calculated by using a scientific calculator and Excel. A bivariate analysis was created to examine the main question: ranking qualities that attract boys to girls. A comparison was made between boys’ and girls’ opinions from different age categories. Mean averages were calculated for this question. Other questions were analysed using mode averages translated into percentage of each gender. Graphs were designed using the values calculated. Observations/Data/Results 72% of boys prefer a girl with average looks and an awesome personality but only 37% of girls believed that to be true. The average boys from 12-14, 15-17 and 18-20 year old categories agree that confidence, humour, a nice body, politeness and intelligence are the five most attractive qualities in a girl they would consider dating. The order of these five change across the age groups but are constantly the top five, however confidence and humour always trump a “nice body”. The average 12-14 and 15-17 year old girl overrates a nice body and underrates politeness and intelligence but confidence and humour feature in their top five. The average 18-20 year old girl agrees with the boys priorities. When defining what makes a girl beautiful, 68% of boys highlight personality traits and 59% of boys mentioned physical features. When noting what repulses boys about pretty girls, more than 50% of both sexes agree that smelling bad was most off putting. 52% of boys mentioned taking drugs as unattractive yet only 23% of girls recognized this as repulsive. Conclusions The data supports the hypothesis. Confidence and humour are more important to the average boy than a nice body across all age groups. The younger the girl, the less she understands what attracts boys. 63% of teenaged girls believe that looks are more important to boys than personality. This is clearly not true. 72% of boys value a great personality more highly than good looks.