最後留下數字會是多少？該問題在台灣的全國中小學科學展覽出現多次。而資訊界演算法大師Donlad E. Knuth 在其著作The Art of Programing，CONCRETE MATHEMATICS （具體數學），針對該數列作詳細的說明；但是，不論是歷屆全國中小學科學展覽或是大師著作，對於該問題，都只是談及殺1 留β或是殺α留1。本研究利用獨創α分類、n 及k 分類、d 函數、b 函數及循環、n 及y 分類、碎形數列和演變關係，將約瑟夫問題探討範圍提升至殺α(個數)留β(個數)，直到剩下最後1 個數時就不能再殺了，遊戲終止，倒數第k 個留下的自然數是多少？同時，本研究在殺α(個數)留β(個數)下，指定自然數y 為酋長，酋長不能被殺，殺到酋長時遊戲停止，求剩下的自然數有幾個？會發生什麼情形？The Josephus problem refers to what will be remaining when arranging n natural numbers in a circle and starting killing one and leaving the next one alive. The problem has been on display for many times in Taiwan National Primary and High School Science Exhibitions (as shown in Table 1). And, the information algorithm master, Donald E. Knuth has elaborated on the array in his works The Art of Programming, CONCRETE MATHEMATICS. However, both the past science exhibitions and the master’s works are limited to discussions on cases of killing 1 leaving β or killing α and leaving 1. This research employs uniquely created α classification, n and k classifications, d function, b function and loop theory to extend the Josephus problem scope to killing α leaving β to find out what the remaining natural number is by No. k counted recursively. Meanwhile, this research designates natural number y as the chieftain, which can never be killed. The game is over when the chieftain is to be killed. The problem is to work out how many natural numbers are remaining. And what happened?

光催化氧化反應以半導體金屬氧化物為催化劑，進行有機性空氣污染物之快速分解反應。其原理係將半導體材質（如：二氧化鈦，TiO2），在適合之光能量照射下，將半導體激發成為具有氧化/還原能力之催化劑，可加速氧化還原反應之進行，迅速分解有機污染物。 研究動機在於利用TiO2在紫外光的照射下，將H2O 分解產生自由基，使其和水中的重金屬離子進行氧化還原反應，期待可以還原水中的重金屬藉以降低水中重金屬離子濃度，同時藉由使用界面活性劑對奈米微粒具保護作用，可回收重金屬奈米微粒。 由實驗結果得知在紫外光照射下，TiO2 使用量0.5 克，AgNO3(aq) 0.01M，照光24 小時其電導度值上升最多且在溶液表面觀察到銀白色銀金屬薄膜生成而所測得銀金屬析出量明顯增加。探討超音波振盪對TiO2還原力的影響得知，超音波震盪的時間越久，所上升的電導度值愈多。 探討常見的界面活性劑（陽離子型及陰離子型）對TiO2還原力的影響：從數據中可觀察到，加入陰離子界面劑時，電導度值明顯上升；而加入陽離子界面活性劑後，電導度值迅速下降，照光後電導度值也不理想。 探討日光及不同頻率的紫外光照射光源對TiO2還原力的影響：發現紫外光的波長愈短，銀金屬析出的量愈多。 探討除氧處理之溶液對TiO2 還原力的影響得知，除氧處理後所配製的AgNO3（aq）0.01M， 經照光24 小時後電導度值明顯上升，且在溶液表面觀察到大片銀白色銀金屬薄膜生成而所測得銀的析出量也大幅增加。 ;In the experiment, we used the properties of T i O 2 that can catalyzed by UV rays and breaking the molecules of water and produce free radicals, which free radicals can redox metallic oxide as accelerator to analyze organic pollutant briskly. We use different shinning time and to find the best effect of the redox reaction . So we want to use this attribute to begin the redox reaction with the metal ion (for example: Ag＋) in water, expecting to reduce them. And then we can use this method to recycle the metal and to reduce the pollution in rivers. Throughout shining UV rays in 24 hours, we can find out the best effect of TiO2 reducing the metal ion solution. We can also find that we use ultrasonic first; the more redox it will have. In this research ,we can observed that if we put more anion surface-active agent, the more redox it will have. We find the effect of UV rays is better than visible light. The most important is that we deoxidize the metal ion solution, we can get the best effect of the redox reaction . In our research，we can't get the satisfing result of the copper sulphate (CuSO4) by TiO2 accelerating the redox reaction under UV rays.

桃園虎頭山是位於林口台地南端的低海拔(約240~260M)丘陵地,表土層屬於紅土層,紅土主要成分為細沙(直徑>50μm，約佔50%)、粉沙(直徑2~50μm，約佔30%),其餘以黏土為主,加上少量含鐵礦物,採樣地點字圖二中的三聖宮旁的登山步道,其坡面面向東南方,對位處北緯25度的桃園屬於向陽波,經日曬適當時間後所產生的龜裂現象,是我們探討的對象;We have focused the study on the sun cracks found in the Mts.Hu-To are located in the southern part of the Mesa Lin Kou, which are 240m to 260m above the sea level. Mainly red clay, the surface soil is a composition of fine sand(diameter>50μm,up to50%)and silt(diameter>2~5 μm,up to 30%),including clay and a limited amount of iron minerals. The soil sample was gathered at the trail beneath the Hil Gue-Lun(241m,see pic.1)from the southeastern latitudes.

The purpose of the research was to find the relationship between the fruit fly’s attraction to light (or heliotaxis) and their genes. Fruit flies, because of their short life span of approximately two weeks, and their simple requirements for survival, are ideal candidates for the study. With a specially designed device "Mi Gong" for the experiments, fruit flies could be separated and cultivated in accordance to their preference to specific wavelengths of light. They received cross-breeding and were studied to learn the ratio of the resulting offspring in their preference to the light. It was observed that the majority of the fruit flies were more attracted to violet light. It was also found that the next generation of the fruit fly inherited higher sensitivity to the light than their parents. Knowing that there could be potential margin of errors in the experiments, the results of the study demonstrated that light, as an external source, had limited impact to heliotaxis of fruit fly. The study suggested that insects with heliotaxis, including fruit fly, inherited heliotaxis in their genes. The study also pointed out the potential benefits of employing heliotaxis in many areas especially in the agricultural development. Furthermore, heliotaxis can be utilized to study the difference in the behavior of nightlife insects before and after the invention of manmade light source.這項研究，主要是在探討果蠅的趨光性和遺傳的關係。很多種昆蟲都具備趨光性，我們之所以以果蠅作實驗，主要是因為其生活週期短，大約兩週，且易於培養。實驗內容是以「迷宮器材」來鑑別果蠅對某一特殊色光的偏好，將該群果蠅分離出來培養，令其繁衍後代，觀察其後代對此一色光偏好的比率，我們所用的果蠅大致上對綠色光較敏感，而其子代的敏感度較親代高。因此，我們推論趨光性會遺傳，且受光影響，但並非全由光影響，故這可能是實驗中的誤差成因。雖然此實驗是以果蠅為對象，但亦可假設其它具趨光性的昆蟲，也可能是因遺傳而導致其子代有趨光性，而我們可利用其趨光的一些特性，減少農作物因害蟲類的果蠅而損失，並且希望可以得知,在人造光源還未出現在世上時，夜行性昆蟲是否有差異。

台灣溫泉資源豐富，不同的溫泉泉質適合不同的溫泉藻類生長。溫泉藻屬於極端生物，其應用資源極待研究開發。本研究根據細胞型態、藻膽蛋白種類分析以及rbcL 基因定序等結果推測北投溫泉藻屬於一種溫泉紅藻，其種類近似於Galdieria sp. (Rhodophyta)。此外，本研究利用葉綠素螢光分析儀於野外實測北投溫泉紅藻於不同光強度下，光合作用能力的差異，發現適應於不同光強度的溫泉紅藻，其電子傳遞速率並無顯著的差異。推測其原因可能與北投溫泉紅藻的phycocyanin 含量會隨著光量有所調整，藉此達到最佳的光合作用能力有關。另外，本研究發現隨著光強度的上升，溫泉藻的NPQ 值有上升的趨勢，然而低光區的溫泉藻有較高的NPQ 值，此與一般理論不符。利用HPLC 分析北投溫泉紅藻的類胡蘿蔔素種類組成，其種類分別為Lutein、α-carotein、β-carotein 與Zeaxanthin，未發現Violanxanthin 與Antheraxanthin，北投溫泉紅藻並不俱有葉黃素迴圈。因此根據研究結果，我們認為NPQ 值與色素的關係仍有討論的空間Taiwan is resourceful in hot springs. Various hot spring algae are adapted to different types of hot springs. Hot spring algae applications are still under developing. In this work, based upon the morphology, analysis of phycobiliproteins, and rbcL sequences, the hot spring algae of Peitou is found to be Galdieria maxima (Rhodophyta). In addition, the dependence of the photosynthesis of Galdieria maxima on the light intensity was measured by Diving-PAM. The results show that the light energy availability efficiency of Galdieria maxima adapting to different light intensity exhibited different, although the electron circulate rate differed insignificantly. This suggests that the concentration of phycocyanin in Galdieria maxima may vary to attain optimal photosynthesis. Furthermore, the NPQ of Galdieria maxima increase with the light intensity. However, contradict to the theory; even under the same light intensity, the NPQ of Galdieria maxima was higher at low light zone. The carotene composition of Galdieria maxima was analyzed using HPLC and found lutein, α-carotene, β-carotene, and zeaxanthin. Violaxanthin and antheraxanthin were not present. Therefore, Galdieria maxima do not exhibit xanthophyll. Based on the results of this study, the correlation between NPQ and pigment still needs to be investigated.

給定一個p∈(0,1)，令k0=0,p0在(0,1)間隨機分布,定義 k1為能使的最小正整數k，而;相同的，對於給定的kn-1,kn為能使的最小正整數k，。若存在kn使得，則稱p∈In;若對於所有的n與kn，，則稱p∈I∞。如此區間(0,1)可分解成集合I1,I2,…I∞。

This experiment mainly aims at three kinds of solution - Dextrose, Saccharose, and Fructose. By changing its temperature, density, length of tube, as well as different wave length factor of polarized light, we observe the influence of the direction of polarization by those factors. The experimental result showed as follow. The Dextrose and the Saccharose can cause the polarized light with the rotary direction of clockwise, so both are ‘dextrorotatory’. The Fructose can cause the polarized light with the direction of counterclockwise, so it is the ‘laevorotatory’. For the Dextrose, when the\r temperature is lower than 20℃, the direction of polarization has changed observably, but doesn’t have any rule. When the temperature is higher than 20℃, the direction of polarization increase slowly. For those three kinds of solution, when\r density increased, the polarization increased observably. When the polarized light passed through the solution with longer path, the direction of polarization has more change. When the wave length of the polarized light changed, the direction of polarization has been changed observably. When the wave length of the polarized light is shorter, the direction of polarization change increased.本實驗主要針對葡萄糖、蔗糖、及果糖等三種旋光性溶液，改變其溫度、濃度、容器管長、以及不同波長的偏振光等因子，觀察這些因素對偏振方向所造成的影響。實驗結果顯示：葡萄糖與蔗糖會使得偏振光的偏振方向以順時針旋轉，屬右旋性之光學異構物；果糖會使得偏振光的偏振方向以逆時針旋轉，屬左旋性之光學異構物。若溶液為葡萄糖，當溫度低於20℃時，偏振光的偏振方向會有明顯的改變，但無規則可尋；當溫度大於20℃時，偏振方向旋轉角位移則以非常緩慢的方式增加。當此三種溶液之濃度增加時，偏振光的偏振方向有明顯遞增的現象。此外，當容器長度越長（即偏振光在介質中的行程越長）時，偏振方向的改變亦越明顯。當偏振光的波長改變時，偏振光的偏振方向有明顯的變化，且當偏振光的波長越短，偏振方向的改變越大，似乎與波長呈反比，但此結果與理論值（即旋光度與波長平方成反比）仍有一些差距。

汽油給人的第一印象想必就是危險易燃，並且很容易揮發，在運輸及保存上有許多缺點，最簡單來說，汽油的高揮發性造成能源的散失、汽油的易燃性則造成了許多意外。而萌生研究製成汽油固態燃料的念頭。

在 Fibonacci Sequence 中，我將Cassini’s identity 轉換成圖形時發現：邊長為Fibonacci number 的正方形，分割後重新拼成長寬分別為Fibonacci number 前後兩項的矩形，會得到矩形內有縫隙（或重疊）。接著我將Cassini’s identity 的圖形推廣到Catalan’s identity 的圖形，我發現邊長一樣的正方形，拼成的矩形長會變大，寬會變小，矩形內的縫隙（或重疊）面積會以Fibonacci number 平方增長。接下來我再將圖形推廣，邊長為非Fibonacci number 的正方形分割拼成矩形時，我發現若將整數遞迴數列代入Cassini’s identity，圖形將會有規律的方式呈現，且每一種數列的縫隙（或重疊）面積會有所不同；若遞迴數列代入Catalan’s identity，縫隙（或重疊）面積還會再以Fibonacci number 平方增長。所以最後我得到一個通式：只要是遞迴數列[an]的圖形，都會滿足於： 。

We investigate the machinery producing successive Simpson’s paradoxical reverse. Taking advantage of algebraic and geometric techniques, we obtain the following results. Take playing baseball for example. In our study, we find that Simpson’s paradox only occurs when the hitter’s hits over 3 times in one game. Set n equal to the times I will hit in one game. If my batting average in each game is at least(n ?1)/2 times higher than the others’; then I am sure that my total batting average would not be invert by the others. In order to find how many the lattice points in the triangle, we use Pick’s formula. But sometimes, the Pick’s formula is not appropriate to triangles whose vertex are not all lattice points. So we develop New Pick’s formula to estimate the number of lattice points in such kind of triangles. Besides, we also find an iterative algorithm to produce successive “Simpson reverse” phenomenon by using C++ language, and we can therefore produce as many “Simpson’s set of four sequences” terms as we like(not beyond the computers’ upper limit).Moreover, if both sequences of ratios converge, then they must have the same limit.我們探討了一般人乍看之下顯得頗弔詭的辛普森詭論。我們配合GSP 作圖，用解析幾何、設立直角座標系和C++ 程式的運算，找出在特殊情況下或一般情況下所產生的辛普森數列組和特殊的性質，並且以棒球場上的打擊率為例子來做印證。通常一場棒球賽中，每個人平均上場3 次～4 次，經過我們的討論，發現要發生逆轉的機會只有在上場達到4 次或以上時才會發生。?了求出在直角座標系中可以滿足的格子點個數，我們用了Pick公式，但?了更準確的估計，我們引進了虛點的概念，重新推導出了新Pick 公式。另外，我們還發現，假設兩個人上場比賽，若打了2 場，且每場最多上場打擊K 次，其中的一個人的打擊率只要是另一個人的(k-1)/2倍以上就保證不會被逆轉。我們又找到了連續產生辛普森逆轉的演算法，利用C++ 寫出程式，經由演算法和遞迴式，製造出項數可任意多(只要電腦能夠承受)的辛普森數列組，且我們發現若兩個比值數列接收斂，則極限趨近於同一個數值。