George Sicherman discovered that it is possible to take a couple of 6-sided dice re-labeling them with different positive integers (1,2,2,3,3,4) and (1,3,4,5,6,8) having the same probability distribution as rolling a standard pair of 6-sided dice. Such unique pair of dice is calling Sicherman dice. The secret behind the Sicherman dice can be studied by combining the powerful mathematical tool “Generating functions” with the symbolic manipulation software “Derive 6”, The same procedure may be applied to studying the possibility of the generalized Sicherman dice along the consideration of :\r (1) Adding more dice. (2) Changing the number of faces. To this end, we introduce the concept of the Sicherman Bound. For a given integer n, the number of n-sided Sicherman dice is finite. We computed manually such numbers for n?50 based on the method of “Elimination of negative terms”. Sicherman Dice 就是一對點數配置與正常骰子(6 面正立方體，點數為1到6)不同的骰子，它所拋擲出的每一種不同點數和(2,3,4...,12) 的機率恰好與一對正常的骰子相同。這種骰子是美國的Col. George Sicherman 所發現的。 Sicherman 更進一步指出：在不使用Sicherman Dice 的情形下，不可能找到一組大於或等於三顆的非正常骰子，它們拋擲出的每一種不同點數和的機率恰好與一組同數量的正常骰子相同。本研究的目標在於1. 尋求計算「Sicherman Dice 的組合和正常的骰子有相同的出現機率」的方法2. 證明Sicherman 結論的真偽及是否適用於其他正多面體(4 面/ 8 面/12 面/ 20面) 的標準骰子3. 修正Sicherman 的結論，並定義Sicherman 極限(Sicherman Limit)。在假設n面正多面體(n 為自然數， n ? 50 )存在的情形下，探討每一個正多面體的Sicherman 極限4. Sicherman Dice (Crazy Dice) 的延伸探討(1) 不同面數骰子的組合，是否可以找到面數組合相同，但點數配置不同的Crazy Dice( 如4 面與6 面的標準骰子組合，找到4 面與6 面的Crazy Dice)(2) 多個面數相同或不同骰子的組合，是否可以找到面數、個數及點數配置皆不同的Crazy Dice ( 如3 個4 面標準骰子組合， 找到2 個8 面的Crazy Dice)在研究的過程中，我發現以下的現象：(1) Sicherman Dice 的產生，是生成函數因式重新組合的結果(2) Sicherman Dice 是否存在，則視上述重新組合的結果是否有負項產生由於上述的觀察，我使用自行發展的「負項消去」法來檢驗Sicherman 結論的正確性及求得n 面正多面體其對應的Sicherman 極限。同時我也和Col. George Sicherman 取得聯繫， 討論當年他發現Sicherman Dice 的經過及其結論的限制條件，作為本研究未來發展的參考。

吹長笛時，按同一按鍵，以大小不同的力量去吹，會引發不同頻率的泛音，而通常越用力吹，引發泛音的頻率越高，所以我們想了解為什麼越用力，泛音的頻率會高，其間的關係究竟是什麼？風經過管口會產生各種頻率的噪音，其中某些特定頻率的聲音會因為會在管內形成駐波而放大，所以我們只能聽到某些特定頻率的聲音。當風速增加時，會在管口形成渦漩逸放的紊流現象。其渦漩頻率與流速成正比（註一）。我們以塑膠管實驗。發現以特定的風速引發該基音後，繼續增加風速，當風速達某一定強度時，才會躍遷為下一個泛音的頻率。這個現象告訴我們：在一封閉管下，風速與泛音的關係並非「線性遞增」，而是越「躍遷遞增」的關係。另一個實驗測量不同管長、其諧音之頻率的關係，我們可以得知，越短的管子，因為相鄰兩泛音間頻率差較大，越不易激發更高階泛音。經由這些實驗結果，我們能夠推論：當管子越長、基音頻率越低時，諧音間頻率的差距相對越小，繪出的風速－頻率關係圖應更加顯示了風速與頻率呈正比關係。未來我們可以以閃頻器觀測紊流渦漩的產生，再變化至不同吹入角度，及各式管口造形，這些實驗能協助我們更進一步了解樂器的發聲原理，甚至開發一個以聲音頻率測量風速的儀器。註一：林婉如、張?文2006 國際科工程組佳作作品。When we press the same key and blow a flute using different strengths, we can get different overtones. Usually, the harder we blow the flute, the high the frequency we get. We want to understand why we get a higher frequency when we blow harder into the flute and to understand the relationship between them. When wind passes through the mouthpiece, many kinds of noises will be produced. Some of the frequencies will expand because they will form standing waves in the tube. Therefore, we can only hear certain frequencies. As wind speed increases, a turbulence of the vortex shedding will be formed. The frequency of the vortex shedding and wind speed will be in a direct ratio. We experiment with plastic tubes. When we increase the wind speed and get certain magnitudes, the frequency will jump to the next overtone. The phenomenon shows that the relation, in a closed tube, between wind speed and harmonics is not a linear increase but a transition increase. In another experiment, we measured the relationship between wind speed and different lengths of tubes. We can infer that the shorter the tube, the higher high-frequency harmonics can be produced. Through these experiments, we come to the conclusion that the longer the tube, the lower frequency of the fundamental tone we get and the discrepancy in frequency between harmonics is smaller. Then we make a diagram between wind speed and frequency that indicates that there is a direct ratio between wind speed and frequency. In the future, we can use “” to observe the production of turbulence. Then we can switch to different angles when we blow into flutes. Otherwise, we can experiment with different shapes of mouthpieces. These experiments can assist us to understand more how the instrument sounds. We can develop a device measuring wind speed with frequency.

The temperature of the water was commonly higher and both of electrodes were oxidized during electrolysis by using alternating current, but the above happenings were only found at the positive electrode by using direct current. It can be explained by the principle of the microwave stove. The exchange of the current causes the water molecules to release heat. The strength and weakness of electrolytes, the length and width of the electrodes, and the frequency of the alternating current can affect the rate of electrolysis of water. Commonly speaking, the stronger the electrolyte is, the faster the rate of electrolysis will be. NaOH and HNO3 are strong electrolytes, but the amount of the gas is zero during electrolysis by using alternating current. Maybe both of electrolytes react with stainless-steel electrodes to form some kinds of protective layers to affect the conduction of current. When copper bars, carbon bars, and iron nails are used as electrodes, either the alternating current or direct current is used, the amount of the gas is very small. Maybe these electrodes react with oxygen produced during electrolysis to form oxidized layers to block the conduction of current. Long and wide electrodes produce more amount of the gas. The amount of the gas increases when the frequency of the alternating current increases. The longer the distance between electrodes is, the smaller the amount of the gas will be. The smaller the angle between electrodes is, the smaller the amount of the gas will be, too. When the number of multiple electrodes in parallel increases, the total amount of the gas almost increases. The amount of the gas is smaller at the farther electrode. The amount of the gas at the electrode at the same distance becomes smaller when the number of multiple electrodes in parallel increases. At the same voltage, the effective current of the alternating current is about 0.707 of that of direct current. So we can predict that the total amount of the gas elect rolyzed by alternating current must be about 0.707 of the total amount of the gas electrolyzed by direct current. When we used stainless-steel electrodes and the electrolyte- H2SO4, we found the ratio was about 0.4286. Maybe the system of the electrolysis of water doesn’t obey the ideal linear system of Ohm’s Law and some part of electrical energy is wasted by increasing the temperature of the water and the oxidization of electrodes. 交流電解普遍水溫較高且兩極都有被氧化現象，直流電解則只有正極有上述現象，可能是交流電有類似微波爐的原理，電流交替時造成水分子震盪發熱。電解質強弱、電極種類、電極長短粗細、交流電頻率會影響電解水速率：強電解質較快，但NaOH 、KNO3 雖是強電解質，在交流電解時，氣體產生量幾乎是零，這可能是他們與不銹鋼電極反應形成某種保護層而影響導電；以銅棒、碳棒、鐵釘為電極時，不管是交流電或直流電，氣體體積都很小，這可能是這些電極和產生的氧氣反應，形成氧化層阻礙了導電；長的和粗的電極氣體產生量較多；交流電頻率越大，則電解所產生的氣體量也隨之增加；電極之間的距離越大兩極的氣體體積越小；兩電極之間的角度越小，兩極的氣體體積越少；多重電極並聯的個數越多，總氣體體積約略越大，距離越遠的氣體體積越小，同距離的氣體體積隨並聯的個數越多氣體體積越小。在相同電壓下，交流電輸出的均方根電壓(電流)為直流電電壓(電流)的0.707 倍(1/√2)，所以推測交、直電解水的氣體總體積比值也應為0.707，但我們以不銹鋼為電極、H2SO4 為電解液下比值為0.4286，這可能是本電解水系統並非為遵守歐姆定律的理想線性系統，且電解水時部分電能被消耗在水溫的升高及電極的氧化上。

The purpose of our experiment is to analysis a specific ring shaped airplane called 'Round Wing' to know its characteristic. We've done several experiments to find its characteristics.\r First, Unlike other airplanes, Round Wing needs little time to recover its stability by comparing duration of flight.\r Second, as the eccentricity of the ring increase from 0.5 to 0.95, the stability and duration of flight are increased too. Also the size of body increase 1, 2, 3 times, the duration of flight is increased to 184%, 204%, 222%.\r Third, when Round Wings are attached each other by 2, 3, 4 they flew with high stability than before and stayed in the air much longer.\r Conclusion, Round Wing has unique characteristic (like high stability, and long duration of flight). And if additional power plant added, it can stay in the air very long. Also it can be used for leisure, patrol, broadcasting, and geological purposes.

面對一個終身學習，多元化學習的社會，學習的方法不再侷限於閱讀或求教於師了，取而代之的是電腦科技的變革。上屆學長利用線上代理人融入遠距教學的作品令我印象深刻，但總覺得少了些趣味及互動性，於是本實驗嚐試將虛擬實境加入線上代理人及遠距教學，利用空間魔法師建構出來的環境，讓使可者能夠融入其中，悠遊在虛擬的世界，並配合鍵盤的滑鼠的控制移動，點選建構出來的虛擬物件，透過超連結連結至網路上的知識網頁，配合線上代理人的互動及帶領，結合了主動、趣味、互動與便利，開創未來的學習方法，一個講求效率、速度及克服空間無時無刻都能學習的環境。With the arriving of 21st century，things are widely different，including the ways of learning. Because of the revolution of multi-media，computer technology will change the traditional way of learning. I was impressed by the master piece 「Long-distance with Agent」 which was done by the schoolmates last semester. But I feel their work still have much room for improvement，so I combine to use the virtual environment with the advantages of their work. Learners can move and click the objects in the virtual reality space，and they can be linked to the knowledge website through the Internet. Learners can enjoy the interesting ，fast，interactive and efficient learning environment. This work is done by the Space Magician Ver3.0， which is used to design a 3D virtual reality environment. And the users can enjoy the net-surfing and through the hyperlinks to get the information.

(一)我們再虎山溪找到了--網石蠶稚蟲、蜻蜓幼蟲(春蜓科、晏蜓科、勾蜓科、弓蜓科)、豆娘幼蟲(細蟌科)、錐螺等生物，代表虎山溪的污染程度並不嚴重，水質尚屬良好。(二)水棲生物具有定居性，可以反應局部地域性的狀況，也可以顯現長期污染及各種污染對生物的綜合效果。(三)水中附生的藻類長生長在固定的石頭上，經過水質環境變化的長期衝擊，能適應水質的藻屬自能在此繁殖，使族群擴大，不能適應水質的藻屬便會減少，甚至消失。(四)矽藻藻屬指數(GI)＝(曲殼藻屬Achnanthes＋卵形藻屬Cocconeis＋橋彎藻屬Cymbella/小環藻屬Cyclotella＋直鏈藻屬Melosira＋菱形藻屬Nitzschia)(五)矽藻藻屬指數(GI)值的大小與水質的好壞成正相關，其值越大，水質越好，實際對應關係如下：A:極輕微污染GI＞30B:微汙染30＞GI＞11C:輕度污染11＞GI＞1.5D:中度污染1.5＞GI＞0.5E:嚴重污染GI＜0.5(六)以虎山溪目前的水質而言：水棲生物與矽藻藻屬指數(GI)在水質污染的判定上沒有顯著的差別，可以作為互相參考的依據，若能再參酌理化參數，在水質的判斷上會更加更完整。(七)矽藻藻屬指數(GI)，在水污染的分析上有其方便性；在採樣時，對河川生物、棲地所造成的破壞性也較輕微，對學生而言，使用的器具較容易掌握，所以，是監測河川水質一個理想的研究方法。(八)根據矽藻藻屬指數(GI)顯示：虎山溪為輕度污染的溪流，應加強宣導前往該處休閒遊憩的遊客們有水質保護的觀念，對於上游農家也應一並宣導，避免水質進一步惡化。

細胞脂肪分化是造成肥胖、骨質疏鬆、和糖尿病的重要前置因素。我們若要維持良好身材又想省去減肥藥的問題，那麼我們必須了解造成脂肪化的原因，才可能擁有好的預防之道。利用人類骨髓間質細胞可以分化成骨質與脂肪等細胞的特性，我們研究氧化壓力對間質細胞\r Ras基因轉換表現後骨質與脂肪分化的影響。結果發現以添加超氧根 (O2-，l5nM)形成氧化壓力，可促進正常Ras基因表現的間質細胞朝骨質分化;相反地，超氧根會促進\r Ras基因突變而不表現的細胞，朝向脂肪分化的現象。進一步研究其作可原理，發現氧化壓力可促進 RaS蛋白質啟動細胞外訊息活化酵素(ERK)，接著驅動骨質轉錄因子(CBFA1)表現，再到骨鈣蛋白質與骨結節形成。而抗氧化酵素(超氧根轉化酵素;SOD,5OOU/ml)的作用，可以抑制正常Ras基因細胞氧化壓力下骨質分化的進行;但不能防止氧化壓力促進Ras基因突變細胞，朝向脂肪分化的作用。總結而言:Ras基因的表現與否，是決定脂肪分化的關鍵切換點;也是影響氧化壓力對間質細胞朝骨質分化的樞紐。這種基因與氧化壓力互動影響骨質與脂肪分化的剖析，將有助於提醒人們:使用抗氧化劑來調節抗衰老、肥胖、和美容時，必須是在不同情況和不同基因體質的人，有所不同。 \r Human\r mesenchymal stem cells are able to differentiate into bone, muscle, cartilage or\r fat tissues. Our preliminary study with human mesenchymal cell line (HS-5) showed\r that HS-5 cells could differentiate to bone, cartilage and muscle but not fat cells\r as determined by histochemical staining of phenotypes. We have further studied the\r influence of oxidative stress on the switch between bone and fat cell differentiation.\r Results showed that oxidative stress started with exogenous superoxide, produced\r by the interaction of xanthine oxidase and hypoxanthine, promoted the differentiation\r of osteogenic lineage showing expression of osteocalcin and bone nodule formations.\r The mechanism was investigated and superoxide was found to induce ERK (extracellular\r regulated signal kinase) activation; and then the expression of osteogenic specific\r transcriptional factor (CBFA1). A plasmid containing ras-mutant (Ser 17 Asn) which\r can inactivate the expression of ERK was transfected into the HS-5 cells for studying\r the influence of oxidative stress on ras-mutated mesenchymal cells. Surprisingly,\r it was found that oxidative stress did not promote osteogenesis but it enhanced\r adipogenesis from the ras-mutated HS-5 cells. Further studies indicated that superoxide\r neither induced ERK activation nor CBFA1 expression, but it did enhance expression\r of adipogenic specific transcriptional factor (C/EBPα) and lipoprotein lipase in\r the ras-mutated mesenchvmal cells. Taken together, the study model to induce the\r bone cell differentiation from human mesenchymal stem cells may be employed to make\r bone cells for tissue engineering.

本研究主要是以台北市中心的氣溫觀測站，以及市區外圍東邊的汐止，南邊的屈尺，以及西邊的山佳等三個氣溫觀測站，從1998 年至2004 年的七年期間的氣溫資料中，來探討台北地區的熱島效應現象。這個研究的結果發現，台北市區與周遭鄰近地區確實會因為熱島效應的影響而產生1~2℃的溫差。這個溫差在白天時不甚明顯，在中午過後，便由台北市中心逐漸向郊區遞減，形成類似同心圓狀的分布。溫差最大值發生在夜晚，使得台北地區晚間宛如一座夜晚增溫的城市，同時夏季的熱島效應又較冬季顯著，氣溫又逐年遞升，造成夏季台北市區的高溫屢創新高，將是未來都市發展的危機。This study is focused on the urban heat island effect of Taipei Basin, northern Taiwan. The hourly temperatures of Taipei meteorological station and three rural sites eastern Hsi-Chih, southern Cyu-Chih and western San-Chia, were compared from 1998 through 2004 to illustrate the temperature differences between city center and surrounding country areas. The results show that a difference of 1~2℃ exists between city center and surrounding country areas, indicating the extent of urban heat island effect in the Taipei Basin. The daily temperature contours show a clear high-temperature bull’s eye at the city center during the midnight, implying the high latent heat trapped by the dense and high rise buildings. This phenomenon is not sharp in the high noon due to the reduction of temperature difference between city center and surrounding country areas. The heat island effect is most prominent in the summer than that of the winter. Along with the increasing temperature-difference trend through years, the summer times often experience record-breaking heat waves and pose great risks for the city development in the future.

In the previous year study, we assumed that the fish basin as a column-shape model in order to exam and explain how the water spouting. This year, we established a reality-like situation of the fish basin to construct our theory base. At this point, round-waves and up-down waves are found as the beginning point of the water spouting. The sound from the basin is the frequency of the vibrations. Meanwhile, this process enables us to understand the causes of four, six, and eight waves. In addition, we tried to change the basin's positions, the capacities of water, the densities of liquids, and the sizes of the containers to demonstrate the theory. Furthermore, we used similar containers, such as high-heel cup and big magnetic bowl to experiment. The results are also corresponded with the theory. These results can also be used to explain the Chinese aeolian bell, clock, shaking bell, etc. That is a major innovation for us. Finally, we obtain better understanding of the abstractive theory though the simulation of the computerized photographs. 魚洗，也稱龍洗，傳說是中國唐宋時代皇帝洗臉的臉盆，內底四條魚紋，麟尾畢具。洗內盛水後，用手摩擦其兩個雙耳，立即發出響亮的嗡鳴聲，並出現美麗了漣漪，水珠四濺，摩擦越快，聲音越響，波浪翻騰，水珠越烈。對此在上一年度的研究中，是將洗盆視為圓柱狀，然後由圓柱狀所建立的模型去解釋水跳的原因，初步瞭解了水跳的原因。今年我們以洗盆真實的情況建立理論模型，在殼體產生共振的圓形駐波及上下駐波，此為水跳產生點，此時殼體發出的聲音就是共振的頻率。也瞭解了有四個、六個、八個、十個波腹的原因。並以改變殼體不同位置、水量、液體密度及洗盆大小等變因應證模型，結果與理論相符合。繼而以高腳杯、大磁碗等類似魚洗盆構造的容器，加以實驗應證，也能符合此理論。甚可推到中國古時候的編鐘、搖鈴、風鈴等器材。對於我們總認為碰撞就產生聲音的觀念實是一大突破。最後經由電腦動畫的模擬，讓此抽象的理論更容易瞭解。

含羞草會因為受到各種不同外界刺激(光照、外力等等)，而造成膨壓改變。外 觀上表現出葉片閉合或有葉柄下垂的現象。其中含羞草處在黑暗的環境下一段時 間後，會做出睡眠運動。當它重新暴露於光照之中，將會需要一段時間以恢復原 先葉片張開的樣子。這個實驗是研究不同類型的環境因子(主要是光照和溼度)在打 破睡眠運動之後，對其葉片復原時間所造成的影響。我們針對上述環境因子在不 同狀況時，進行我們的實驗上百次，進一步得到了多項的數據。也在實驗中，為 了精確了解整個恢復的程序，而將恢復程序程序做成書面說明。以下是我們所紀 錄與分析完數據後的結果，以及我們在實驗期間，進行一些不同的實驗嘗試，所 發現不同於表面所見的驚人事實。 The mimosa can accept plenty of stimulation (light, force, etc), which results in the change of turgor pressure, and on outward appearance, it shows the phenomenon that the leaves become closed or that the stalks get pendent. Among the stimulation, the mimosa will undergo nyctinasty when it is left in the dark for a period of time, in order to enable it to be exposed to the light once more, which requires certain time. This experiment is based on how different kinds of elements of the environment (primarily light and hydro level) effects the rehabilitation time after the nyctinasty is broken. A large number of data are gained after experimenting on it for hundreds of times. Steps of habilitation are also made into illustration in writing, so as to understand the whole steps accurately in the experiment. Below are our records and analysis based on the data, including a few special experimental tries during our working time, in which some surprising facts that were discovered are different from what are seen on the surface.

When we were learning about organic compounds at school ,there was a unit discussing the isomers of alkane .Our teacher made us practice drawing all the structural formula of the isomers from hexane to nonane .We were much interested in the subject .However ,we often missed or duplicated some isomers .Thus , we began to think if it is possible to find a way by developing programs to let the computer calculate the exact number of the isomers of alkane . After discussion ,we set up a complete coding system .We numbered the isomers in the way that computers could decode and then wrote them in C language. Through computer execution ,the numbers of the isomers from C1 to C20 all match those on the reference website. According to the same concept , we also find a way to calculate the number of alkane with one substituted group . In the future,our goal will be focused on the research of multi- substituted alkane and cyclokane. In addition , the ionic crystal accumulation model are so variable. Take the double face-centered accumulation of NaCl for example, when the ion pairs are extended to the infinity , the potential energy of attractive field will approach a constant which is named as the Madelung Constant. We also managed to write a computer program with C language to approach this convergence with three models, including cube , octahedron , and sphere . The result turned out to be that the data of the sphere was less stable . In the other two models , when “n” is up to 43 layers , the data is identical with that on the reference website to the eight decimal point . 在學校裡學習有機化合物有關烷類異構物這個單元，老師讓我們練習畫出己烷~壬烷的所有異構物結構式，這引起我們極大的興趣！但常一不小心就漏掉或多出幾個，我們開始思考：能不能找到一個方法並設計成程式，讓電腦執行以找出烷類異構物？經過討論，我們建立了一套完整的編碼系統，將各異構物以電腦可解讀的方式編號，並以C 語言寫成程式。透過電腦執行，各碳數化合物自C1至C20都與參考網站吻合。依相同觀念，我們也設計出烷類含一個取代基的異構物數目。將來努力的目標為：多取代基及環烷類之研究！另外，離子晶體堆積模型變化多端，以NaCl 雙面心堆積為例，其引力場位能，當離子對延伸至無限大時，這個值將趨近於一個常數，又稱為馬德隆常數。我們嘗試以C 語言設計電腦程式，用三種模型（正立方體、正八面體、圓球）來逼近並求得這個收斂值。執行結果是：圓球數據較不穩定；而另二種模型到n=43 層以上，其數值大小與參考網路上的數值，在小數點以下8 位完全相同。

筆者將K?Cr?O?置於明膠溶液中煮沸，加入AgNO?水溶液，使之造成空間震盪反應。產生Liesegang ring。並且在研究空間震盪的過程中，筆者在研究過程中同時發展出一個程式，可以完成空間震盪實驗完成之後，由其震盪條紋的時空關係，得知ksp與擴散係數。或者可以在已知ksp和擴散係數的條件之下，預測出空間震盪實驗完成時的情形。We put K?Cr?O? into glutin, boiling it. After it concretizes, add AgNO? aqua into the glutin of K?Cr?O?. It is made to be space oscillation reaction with the Liesegang rings. And when we research for the process of space oscillation, we develop a program which can know the Ksp or diffusing coefficient after finising an experiment in Lablratory. Or we can predict the statement after the experiment on the condition of the Ksp and diffusing coefficient.