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.
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.
漩渦也有形
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.
有毛!沒毛!哪個好!?探討石田螺及其螺殼上附生藻類與環境因子之關係
This research is about two ponds in the Ｂ park’s and the Ｄ park’s snail(Square Mystery Snail：Sinotaia quadrata) in Taipei city of Nei-hu District for research object, carry out the study of the following research proceed: 1.Discriminate the algae species that are growth on the snail shell and which is a kind of interaction with the snail; 2.The influence of the snail and algae with difference of temperature, salinity, pH value and dark ; 3. The factors affect algae growth on snail shell; 4.Use the variation of snail and algae to be a biological incator. The result manifestation: the algae that are growth on snail shell have two kinds, one is Cyanophyta and the other is Cladophora sp. The interaction between algae and snail belong to communalism, but under the condition of lacking of food, the snail then will eat the Cladophora sp. which grow on the shell of other snails. The temperature adapts aspect, upper limit of the feat existence of the snail should be low in 28℃. When over than 28℃, Cladophora sp. as the most strong, Cyanophyta is secondly, and the snail then is most poor. For the maximum tolerance of the salinity, the snail is about 4.375?, Cyanophyta is about 5.0?, Cladophora sp is then about 7.0?; Under the different salinity for the tolerance , the Cladophora sp. still the most strong, Cyanophyta is secondly, and the snail then is most poor. Under the dark environment, the speed of Cyanophyta begin to be bleaching is very fast than the Cladophora sp.. In the tolerance of pH value range: The snail is about pH=5～10, Cyanophyta is about pH=7～8, Cladophora sp. is about pH=6～8; When the pH value range is in the pH=5～8, the speed of the Cyanophyta occur changing is very fast than Cladophora sp.. The algae are growing on snail shell very different between two ponds, the main reason is water pH value dissimilarly: When pH value over than 8.5, there is no Cladophora sp. to grow on the snail shell, after the pH value to decrease, Cyanophyta then will compare early than Cladophora sp. to grow on the snail shell. Calculate by the classification of the freshwater biological incator : Two organic pollution degree of the ponds may be lain in theβ-mesosaprobic to theα-mesosaprobic, and the polluting degree of the Ｄ pond is more seriously. As for two ponds, have already faced what level of eutrophication? Belong to actually which stage of pollution grade? Not only added the classification data of floating and fixative algea in two ponds, and also according to the parts of chemistry analysis method measure of the data makes the substantial evidence, then could carry out the more accurate and thorough study in the days to come steadily studying process.本研究是以臺北市內湖區兩個綠地公園(Ｂ公園與Ｄ公園)池塘內的石田螺(Sinotaia quadrata)為研究對象，進行以下研究目的之探討：1.鑑別石田螺螺殼上藻類的種類及其與石田螺的互動關係；2.溫度、鹽度、酸鹼值及黑暗等環境因子的差異，對石田螺及螺殼上附生藻類的影響；3.影響藻類附生於石田螺螺殼上的因素；4.將石田螺及螺殼上附生藻類的變化作為監測環境因子或水質變異的指標現象。結果顯示：附生於石田螺螺殼上的藻類有藍綠藻(Cyanophyta)與剛毛藻(Cladophora sp.)兩類；與石田螺的互動關係應屬於片利共生(communalism)，但在缺乏食物的情況下，石田螺則會採食同伴殼上的剛毛藻。溫度適應方面，石田螺適宜生存的溫度上限應低於28℃，超過28℃水溫環境的耐受程度，是以剛毛藻為最強，其次是藍綠藻，而石田螺則為最差。對於環境鹽度最大耐受度方面：石田螺約為4.375??，藍綠藻約為5.0??，剛毛藻則約為7.0?；在不同鹽度環境下，鹽度的耐受程度，仍以剛毛藻為最強，其次是藍綠藻，而石田螺則是最差。在黑暗環境下，藍綠藻褪色產生白化現象的速度明顯地比剛毛藻要快了許多。在環境酸鹼值耐受的範圍方面：石田螺約在ｐＨ＝5～10 之間，藍綠藻約在ｐＨ＝7～8 之間，剛毛藻則約在ｐＨ＝6～8 之間；而酸鹼值範圍在ｐＨ＝5～8 時，藍綠藻產生變化的速度明顯地比剛毛藻還要快。而兩樣區池塘水體酸鹼值的不同，應是造成石田螺螺殼藻類附生現象差異的主要原因：當酸鹼值超過8.5 時，螺殼上就無剛毛藻附生，當酸鹼值降下後，藍綠藻則會比剛毛藻早出現在螺殼上。藉由淡水生物指標的分類推測：兩樣區池塘水體有機污染程度，可能介於β-中腐水性(β-mesosaprobic,βm)至α-中腐水性(α-mesosaprobic,αm)的範圍之間，而Ｄ池塘受污染的程度應會比Ｂ池塘還要更嚴重些。至於兩樣區池塘水體，已面臨了何種優養化的程度？究竟是屬於哪一個階段的污染等級呢？除須補充水體中浮游性及附著性藻類分類的詳細觀察資料外，仍必須參照部分水質化學分析法所測得的數據作佐證，才能在日後持續地研究過程中進行更精確及深入的探討。
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.