棋子排列的平均值
本研究由下述問題開始：將n1 個黑色棋子和n2 個白色棋子排成一列，規定第一個棋子必為黑棋；對於每一種排列方法中，同色棋相鄰處記為1，異色棋相鄰處記為-1，所有1 和-1 的總和記為 t (n1,n2 )。對所有可能的排列方法所算出來的t( n1,n2 ) 值求其平均值，記為a (n1,n2 ) 。我們先由觀察各種n1 和n2 值，得到這平均值的可能公式，隨後並嚴格證明其正確性，證明方法也經過多次精鍊到十分簡潔的方式。以此為基礎，我們並做了各方向的推廣，研究涉及下列各點：(一) 利用組合數探討原來的問題。(二) 在第一個棋子不限定為黑棋的假設下，求平均值a( n1,n2 ) 。(三) 將棋子由兩種增加到多種。(四) 改變棋子排列以及相鄰的方式。經由研究，我們發現，每一次愈將問題推廣時，愈能找出清晰的概念涵蓋並印證先前的想法。Our study starts with the following problem. Suppose n1 black chesses and n2 white chesses are arranged in a line under the condition that the first chess is black. For any arrangement of these chesses, an adjacent pair of chesses having the same (respectively, different) colors is associated with a value of 1 (respectively, -1). Let t(n1,n2 ) denote the sum of these values. The purpose of this problem is to calculate the average value a (n1,n2 ) of these t (n1,n2 )which runs over all possible arrangements of the chesses described above. We begin from observing various values of n1 and n2 and find a possible formula for the solution. We then give a rigorous proof for the formula. After some refinements, simple proofs are also established. Based on this, we also make some generalizations. In summary, the research includes the following: 1. Study the problem by using binomial coefficients. 2. Calculate a(n1,n2 ) when t( n1,n2 ) runs over all possible arrangements in which the first chess can be black or white. 3. Increase the types of chesses from two to many. 4. Variant the arrangement method of the chesses from a line to other configurations. During the study, we find that whenever we extend the problem to a more general case, we make the ideas for the original problem clearer.
超聲波在液體的探討
本實驗一開始主要探討超聲波在水中的基本性質，如：指向性、衰減性…等。實驗發現，超聲波的衰減會同時與其指向性以及衰減性有關。 接著希望利用超聲波在水中的物理性質，近一步測量超聲波在水中的聲速，實驗中則利用駐波以及聲光效應測量。在駐波法測量聲速的實驗中，用洗淨機當作聲源，內部放置量筒，量筒內盛水後放入木屑，並使聲波在其中產生駐波即聲浮現象，求出波長後反推聲速，測量出的聲速誤差值僅有1.13%，而在使用聲光效應測量聲速的實驗中，使1.65 MHz 的超聲波在自製的壓克力容器內部所裝的水中產生駐波後，以波長650 nm 的紅光雷射通過，在遠處屏幕即產生似於光柵繞射現象，藉著屏幕上的繞射條紋反推該液體聲速，測量出聲速誤差均在5%以下。 在觀察聲光效應實驗中，發現過段時間後有氣泡產生，由文獻上，得知此現象為超聲空蝕現象(Acoustic cavitation)，就設計實驗測量聲場中聲壓分部，並利用蠟紙觀察氣泡的成長。實驗發現聲場中的聲壓強度以及液體的表面張力和蒸汽壓會影響到產生空蝕的臨界值及產生氣泡的數量。 ;At the beginning this experiment explores the ultrasonic base in the water, including its velocity, the physics property of liquid, direction, and attenuation etc. . . At first, we use methods of standing wave to measure the velocity of sound under the water, using an ultrasonic cleaner as the sound source and putting some wooden powder in the water. As the standing wave accrues/produces, the powder will “stand still.” To measure the length between two grains of powder, in this way we can calculate sound velocity. Another method we use is diffraction of optics. Put 1.65 MHz source and water in a transparent container; then using laser through it. At the board much diffraction light stripes are created. By this way, we can estimate the velocity. Following these ways can calculate velocity precisely. In these experiments, some bubbles create in the container are discovered. We learn it is so-called “Acoustic Cavitation” based on the reference paper. Besides, we design experiment to know the bubbles’ growth and the number of the bubbles is connected to the physics property of liquid. We use different kinds of liquid with different vapor pressure and surface tension. Finally, we know when it has the smaller surface tension and bigger vapor pressure, the liquid makes bubble velocity grow faster and larger amount of bubbles are produced.
「轉環」的餘地
在生活的觀察中，我們注意到人們在轉動呼拉圈時似乎是行一種「以軸轉動一個半徑遠大於軸半徑的環」的運動，在查過相關資料後，並沒有發現比較完整的探討。本研究的目的，是要找出在圓環被轉軸所驅動的運動模式中，影響環轉動頻率的各個因素，諸如：環半徑、環質量、轉軸半徑與環轉速之間的關係。根據我們所做的實驗，對相同的一個環而言，以半徑較小的軸用固定轉速轉動時，即使環的轉速變快，但始終與轉軸的轉速相等。由此我們推斷：無論環與軸之間的半徑關係為何，在環能穩定轉動的情況下，兩者的轉動週期將會相等。另外，在實驗的過程中，「軸驅動環」所引起的軸晃動一直困擾著我們，但這也引發了一項應用：如果原本穩定轉動的環和軸振動，則振動將被放大，藉此設計可以作為地震感測器。亦可作為儀器的保護裝置或是指向裝置。While playing a hula-hoop, we noticed that it seems to be a motion that the axis rotates a circle whose radius is larger than axis’. By checking relative theses, we found that there is no better research having fully discussed about this topic. The purpose of this research is to find out the motion pattern that a circle is rotated by the foce of an axis and the factors affecting rotation, such as radius and mass of circles, the radius of axes, and the frequency of axes and circles. According to our experience, no matter which height the circle stay at, or how fast the frequency of axis is, the frequency of circle will be the same. As a result of this, we guess that if it can be a stable circle, the frequencies of the axis and the circle shall be the same. Another confusing fact is the vibration of the axis, but it enables a new application: if a vibration affects a circle-axis system, the vibration will be enlarged. By this application, we are able to design an earth-quack senor, or protecting or pointing instruments as well.
紫蝶幻影
The main purpose of this experiment is to discuss the characteristics of iridescent colors of Taiwanese Euploea’s wings, inclusive of the relations between the colors of wings and squamas. According to the results from scanning electron microscope, we discovered that the iridescent colors had a close relation to nanostructure and arrangements of squamas. We inferred that both the nanostructure and the arrangements would influence the formation of iridescent colors and the basic colors on wings. In addition, the basic colors on wings are related to different types of scales. To compare with the diverse formations of different sorts of Taiwanese Euploea’s wings, we took SEM pictures of Elymnias hypermnestra as well, discovering that its iridescent colors had similar relation with scales. And there was the regulation that Elymnias hypermnestra had only one type of scales at iridescent area, and two different scales at not-iridescent area as well as Euploea’s. 本實驗目的為探討台灣地區紫斑蝶蝴蝶翅膀幻色的特性，以及翅膀幻色與鱗片的相關性。由結果得知，幻色實驗中利用掃描式電子顯微鏡發現紫斑蝶幻色的形成和其鱗片的細微結構與排列方式有密切相關。我們推論紫斑蝶的鱗片細微結構與排列皆會影響其幻色的形成，而顏色的不同則與不同類型的鱗片相關。除此之外，我們亦對同具幻色的紫蛇目碟進行拍照分析，發現其幻色亦與鱗片有相關性。紫蛇目蝶的幻色區具有單一種鱗片構成的規則性，非幻色區則有兩種鱗片，與紫斑蝶相同。
漩渦之美
我們常可以在自然界中發現漩渦的存在，但其存在的形體與性質也不盡相同，為了研究漩渦的結構與形體，筆者分析出多種會對漩渦產生影響的因素：開始放流的水而高度、放流洞口大小、有無破壞漩渦結構的阻礙、單孔落流漩渦與雙孔落流漩渦、還有流體的黏滯度對漩渦的影響， 但漩渦是一個不斷改變的流體，非常難以觀察，且自然界的漩渦也不是說出現就出現，所以必須設計一個簡易實驗器材來觀察，並用數位攝影機紀錄下來，再慢慢分析，而我們也可以在這個實驗中了解漩渦的結構，和體會到漩渦所表現出自然界的力與美的一面。‧We can always find in nature of different swirl’s forms and properties. To study the swirls, we analyzed such factors, as the beginning water level, the size of the hole, the presence of obstruction that will destroy the structure of swirls, differences between single-hole-swirls and twin-hole-swirls, and the viscosity of fluid. Because swirls change all the time, it is very difficult to observe. We designed a device .The procedure was recorded with a digital video camera and analyzed it. The study helps us understand the structure of swirls and admire the beauty of swirls.
太陽能發電環境評估與追蹤器探討
本研究首先探討台灣各地的日照時數與世界重要都市的比較，發現台灣南部日照時數皆超過2000 小時，適合發展太陽能。接著，?了增加陽光的能量密度而加設弗瑞奈透鏡，雖然能順利的使照度放大三百餘倍，但歲日照角度的影響甚鉅，?了克服角度的問題，我們決定開發自製的追蹤器來改善角度的問題。太陽能板需要改變仰角跟傾角（雙軸調整），由光感應器判斷及自動控制程式，判斷隨時辰與季節變化的太陽角度。當搭配奈米塗料、弗瑞奈透鏡與追蹤器，總輸出功率可增加約50%。太陽電池表面玻璃會阻擋藍紫光的吸收，但本研究在太陽能板上塗佈奈米塗料，發現能增加短波長的吸收；經實驗後奈米等級表面具有自潔效應，可防止灰塵雨滴的堆積影響光線吸收，具有開發價值。This project first compares Taiwanese locations with other places in the world on average daylight times. It was discovered that southern Taiwan has the longest average daylight time all over 2000 hour sand therefore most ripe for solar power development. To increase the energy density of solar Fresnel lenses were incorporated. Although this has the advantage of magnifying illumination by three hundred percent, the alignment angle for the solar panel will have a significant impact on performance. We then designed and built a automated tracking device with illumination sensors to control the elevation and inclination of the solar panel which adjusts the angle according to environmental conditions such as time of day and season. When solar cell collocate Nano coating, Fresnel lens, tracking device, its power can promote almost150%. The glasses on the solar cell will interfere solar cell absorbing blue and purple light, but we lay on a Nano coating and we find Nano coating can improve solar cell to absorb short wave; and surface o Nano have lotus effect, it can prevent dust and rain effecting solar cell absorb lights, and it is worth developing .
液態導體的磁效應
本文所探討的議題為電解質溶滿通以電流後所產生的效應與機制。本實驗所採用的方法為電解與電鍍，運用這兩種方法，來比較電解液在不同狀況下所產生的結果；經過多次的實驗，累積了許多實驗結果，使我們可以得到更精確的數據 · 在此次實驗中，我們發現電解液在相同的電壓下，通以電流後的穩定性與金屬的活性有關，活性越大越不穩定；反之，活性越小越穩定。另一個發現為，只有單一極性離子移動的情形，可通過的電流，比陰陽離 r 同時移動時為大；但因通過的電流大使電解液反應劇烈，產物時時覆蓋電極使電流下降。所以就穩定性來說，是以陰陽離子同時移動為佳 · 在展望方面，希望可以發展到液態磁屏避的設備，可減少設備過重之問題 ·This is a study of how electric current effects the electrolyte solution. The experiment was conducted through two methods: electrolysis and electroplating, the results of which were compared. The experiment of the same designs hi been conducted repeatedly and, as a result, accurate data were collected and accumulated.One of the two major findings from the experiments was that, when under the same voltage, stability of the electric current varied with the change of activity of the metals; the greater the activity of the metals, the less stability of the current, and vice versa. The other major finding was that, with the movement of dipole-ion, a greater amount of current would go through the solution than that which would go through with the movement of cathode and anode; however, the greater amount of current would cause intense reaction of electrolyte solution, hence merging the electrode and reducing the current. So as long as stability is concerned, the movement of cathode and anode is preferable.It is hoped that more sophisticated experiments designed on the basis of the similar principles will eventually lead to the construction of equipment of liquid-magnetic shielding of smaller weight and size.
蝌蚪游泳能力之探討
本研究主要探討蝌蚪之游泳運動特性，及游泳速度(V)與尾鰭長度(SL)、尾鰭高度(SH)、身體質量(M)、尾鰭擺動頻率(TBF)、擺動幅度(AMP)之關係，並分析蝌蚪游泳之體軸變化及流場變化。祈能了解蝌蚪之游泳運動特性，進而探討其適應環境之機制。研究結果顯示：黑眶蟾蜍蝌蚪體重(M)愈重，則鰭長、鰭高亦隨之生長，並呈現高度相關性(R2=0.9381、R2=0.9809)。另外，尾鰭生長時之長度增加較多。蝌蚪體重(M)與鰭長(SL)、鰭高(SH)之迴歸方程式(M=0.027SL＋0.342SH－0.078，R2＝0.9832)。黑眶蟾蜍蝌蚪之游泳速度，會隨著尾鰭擺動頻率之增加而提高。尾鰭長度愈短之蝌蚪，增加游泳速度時尾鰭擺動頻率增加較多。蝌蚪游泳速度(V)與鰭長(SL)、擺動頻率(TBF)之迴歸方程式(V=0.480TBF＋4.804SL－4.381，R2＝0.9110)。不同尾鰭長度蝌蚪之擺幅對體長之比率並無明顯變化，其擺動幅度(AMP)的範圍介於0.45(BL)至0.56(BL)之間。蝌蚪游泳時各部分體軸之擺動幅度自吻端開始(P=0)至P 為0.24 時逐漸遞減，且在P 為0.24 時呈現最小擺幅，但P 超過0.24 之後直至尾鰭部分卻又大幅遞增，其最大值出現在尾鰭末端(P=1)。蝌蚪游泳是以尾鰭快速向中心軸擺動，產生較大的前進動力，過了軸線則慢速擺動，以減少阻力。This investigation is to explore the swimming habits of tadpoles- the relationship between their swimming velocity, length and height of their tails, mass, the frequency at which their tails movement, and the amplitude of the tail’s movement, as well as analysis their body axes, and the flow distribution of the water, in order to understand how the swimming patterns of the tadpoles are affected by the changes in their environment. The results of this investigation have shown that as the mass of the tadpoles increases, both the length and the height of their tails also increase according to the R values of the tail increases according to the R values of 0.9381 and 0.9809. However, it is observed the length of the tail increases at a faster rate than its height during the tadpoles’s growth. The formula which models the regression relationship between the tadpole’s mass, tail length, and tail height are found to be (M=0.027SL+0.342SH-0.078,R=0.9832). It’s also noted that as the length of the tadpole’s tail decreases, the velocity and the frequency of the tail would increases (the length of the tail is inversely proportional to the tadpole’s velocity and tail frequency). The formula which models the regression relationship between the tadpole’s velocity, tail length and tail frequency is (V=0.480TBF+4.804SL-4.381,R=0.9110) The different frequency model by tails of different lengths do not appear to have an apparent relationship with the tail length, given that the amplitude is between 0.45(BL) and 0.56(BL). As the tadpole swims, the angle between its oscillating body axes decrease as the P values increases from 0 to 0.24, their force the angle is at a minimum whom the P is at 0.24.Yet when P exceeds 0.24 the angle would increase dramatically. The maximum value is observed when P=1.The tadpole’s swimming motion mainly relays on the rapid oscillations of the tail about the centre of mass (body axis)-producing a stronger driving force, and slowing down towards the end of each oscillation to minimise the friction forces acting on the tadpole, which in furn, decrease its velocity.
探討「避開矩形框」的配置方法與推廣
一、若Mn×n(s)表示在n×n 的正方形棋盤中，排列s 顆棋子在方格內，且每一方格最多只能排1子，其中s 顆棋子的配置需滿足兩個條件：1. 並無任意4 子可以形成矩形框的4 個頂點。（此矩形框的邊需與棋盤的邊平行）2. 在沒有棋子的方格中，無法再加入棋子。二、若Vn×n×n(a1,……,an) 表示在n×n×n 的正方體棋盤中，每層的棋子個數分別為a1,……,an，且s= a1+……+an，其中s 顆棋子的配置需滿足兩個條件：1. 並無任意8 子可以形成長方體的8 個頂點。（此長方體的邊需與立體棋盤的邊平行）2. 在沒有棋子的方格中，無法再加入棋子。本研究即在Mn×n(s)與Vn×n×n(a1,……,an) , s= a1+……+an 中探討s 的最小值、最大值及變化情形，並分析其配置方法。之後推廣至長方形Mn×m(s)及長方體Vn×m×k(a1,……,ak) , s= a1+……+ak。最後根據其研究結果設計一個「避開矩形框棋」，並加以分析出致勝的策略。一．If Mn×n(s) indicates in the n×n square chessboard, we put s chesses to line in the square and each square only can put one chess. Then the station of s chesses must satisfy the following two conditions：1. No any 4 chesses can form the tops of the rectangular frame ( The sides of rectangular frame must be parallel to the sides of chessboard )2. If there’s no chess in the square, we can’t add any chess. 二．If Mn×n×n(a1,……,an) indicates in the n×n×n square chessboard, the chess number in each layer are a1,……,an and s= a1+……+an. The station of s chesses must satisfy the following two conditions： 1. No any 8 chesses can form eight tops of the rectangular cube ( The sides of rectangular cube must be parallel to the sides of cubic chessboard ) 2. If there’s no chess in the square, we can’t add any chess. This research try to explore the minimum, maximum and variation of s which in Mn×n(s) and Mn×n×n(a1,……,an), s= a1+……+an, and analyze its station. Then we will extend the research to rectangle Mn×m(s) and rectangular cube Vn×m×k(a1,……,ak), s= a1+……+ak. Finally, according to the result of research we wish can design one “avert rectangular frame chess“ and analyze the strategies to triumph.