N 元二次不定方程式的整數解探討
傳統的畢氏定理三元二次不定方程x² + y² = z²有一組漂亮的整數解為(m² - n²、2mn、m² + n² )；中國數學家嚴鎮軍、盛立人所著的從勾股定理談起一書中記載四元二次不定方程x² + y² + z² = w²的整數解為(mn、m² + mn、mn + n²、m²+ mn + n² )，這組解被我們發現有多處遺漏，本文以擴展的畢氏定理做基礎修正了他的整數解公式，並推廣取得N 元二次不定方程的整數解公式。
There is a beautiful integer solution formula for the Pythagorean theorem equation, x² + y² = z² , such as (m² - n² , 2mn ,m² + n² ). The “m＂ and “n＂ of the solution formula are integer number. A book written by two Chinese mathematicians, Yen Chen-chun and Sheng Li-jen who expanded the Pythagorean theorem equation to the four variables squares’ indeterminate equation, x² + y² + z² = w² . They claimed that they found its integer solution formula, such as (mn , m² + mn , mn + n² , m² + mn + n² ) for any integer “m＂ and “n＂. But we found it losses many solutions. This paper corrected their faults due to the expanded Pythagorean theorem built by ourselves. Further more, we derived a general formula of N variables squares’ indeterminate equation. Now, we can get integer solutions of the equation, (for all natural number “n＂) easily by choosing integers m1 , m2 , m3 ,……, mn−1 up to you.
蜘蛛數
We understood the definition and meaning of spider number by reading〝Wonders of Numbers〞. It interested us so much. So, we took further step to study the situation of extreme value when the gap sometimes lie on the line and sometimes on the circle or even on both. That is to say, we explored the relation between spider number and the gap when the spider number is maximum or minimum. New research for the application of spider number involves several directions. First, we design a new game called〝Stepping Land Mine〞with the rule of spider number. Give you a net with several hidden gaps, trying to find the right positions of gaps. Second is the further result for a different type of net about regular n-polygon. Third is a tactic for a net with destroying of the strategy points. In this situation, the gaps amount on the circle and on the line are fixed. At the same time, consider the situation of circles and lines designing the tactic of placing the gaps to attain the maximum of the destructive effect. 在本文中我們定義一個蜘蛛網上的蜘蛛數，若在蜘蛛網中加入缺口後，會影響蜘蛛數的大小。我們探討蜘蛛網上的缺口，該如何分配才能夠得到蜘蛛數的極值(最大值及最小值)。先觀察一直線和圓上缺口如何放置蜘蛛數有極值，再探討許多條直線及圓上的情況，進而推展至許多同心圓及通過圓心的許多條放射線的缺口，該如何放置，蜘蛛數才會有極值發生。
死亡巧克力—切切割割好計謀
三角形的邊上取任意多個點，我們可以把這塊大三角形沿著切割線切割成較小塊的三角形，但切割線必須是點（或頂點）和點的連線，而且必須切割三角形，同時可以切任意大小的三角形，如圖（1）與圖（2）。但不可以一開始就取走整個三角形。定義拿到最後一塊三角形的人獲勝，而在多邊型中的玩法與在三角形中相同。 我們分A、B、C三種規則來討論，其中A規則即是上面提到的玩法，B規則大部分的玩法和A規則都相同，唯一不同的地方在於：A規則中，只要有一方取到剩下的圖形為三角形，另一方就可以直接取走剩下的三角形，而B規則規定即使剩下的圖形已經是三角形，也必須取到剩下的圖形邊上都沒有分點為止。C規則是限制玩家一次所能取的三角形數來進行遊戲。 我們完成了A、B、C規則中三角形與多邊形的必勝策略，並找出必勝策略之間的關聯。 ;Given any numbers of points on the sides of a triangle, the players can cut this triangle into pieces. Each cutting line has to be one, linked between two points given from two different sides. And the player can’t have to cut smaller triangles out of the original triangle. The out-cut triangles can be chosen randomly without any restriction in size, just like what’s shown in picture（1）and（2）. Meanwhile the first player can’t cut the original triangle exactly all out in the very beginning process. We define the player as the winner, who gets the last triangle. And the above way we play can be applies to any multi-side shapes. We discussed the question respectively in three rules, A, B, and C. Rule A is what we mention above. Rule B is generally the same as rule A, except for the only difference：The rule A , if there is any triangle left , the next player can get it directly, but while in rule B, the every next player has to cut out smaller triangles until no point is left on sides. Rule C proceeds on conditions that there is a limitation to a certain number of triangles cut out at a time. We has finished the winning tactic respectively in rule A, B, and C in the games with a triangle and multi-side shapes. Furthermore, we find the connection between the winning tactives.
二次函數上正三角形建構之研究及探討
在拋物線上置掛正三角形看似簡單，其實不然。本篇文章研究在二次函數的各種不同情況下，可做正三角形的分佈以及其個數。
1. 在一條拋物線上時，最多只能作正三角形。
4. 在三條對稱軸相等的拋物線和共頂點開口大小不同之拋物線上，本篇文章證明一定能找出正三角形落在它們之上。但由於最多有四個分界點，要解四次方乘組過於繁複，於是本篇文章對分界點作了一些估計，找出了分界點的極限值。
5. 本篇文章證明了對於給定的正n 邊形，存在一1 元n-1 次方程式可以通過它所有頂點。
Building a regular triangle on a parabolic curve looks easy . In fact , it doesn’t . This Article researches regular triangles distributions and its numbers in different conditions.
1. On one parabolic curve can only build regular triangles , squares and other regular polygons can’t be built.
4. For three parabolic curves which has same symmetrical axis or three concurrent parabolic curves, we prove that it can build at least one regular triangle on them .But because it can have at most 4 boundary points, to solve quartic equation is to complicated. So we do some estimation of boundary points, and find out some limits.
5. This Article prove that for given regular polygons , there exists a one dimension n-1 orders equation can pass all its apexes.
「圖形板」的圖形軌跡之探討及其延伸
Starting from the problem in AMC competition of Australia, we try to find out the locus and its length when a point in a regular polygon rolls in a circle. The result is that the locus has a wonderful and regular cycle.Next, we discuss the regularity of the cycle when a regular polygon（n sides） rolls in another regular polygon. Furthermore,we discuss the the equation of the locus by changing the radius and the angle of rolling. we find out the argument function of the locus of a point inside when a a regular polygon（n sides）rolls in another regular polygon （m sides）： ， Aj is the summits of the regular polygon（m sides）, Bjcorresponds Aj when a point inside the regular polygon (n sides) rolls, ) And then, we do some moving simulation with some computer math software, such as Cabri Geometry、Mupad, etc. We discuss the regularity of the locus and its equation of a point inside when some special cycloids, like asteroids, cardioids, etc, roll in a certain condition. Moreover, with the result of research 2, we create the “plate＂ and apply for a patent on it. We hope to study math by playing games.
從澳洲AMC 競賽題出發，嘗試探討一正n 邊形中的一點在單位圓內滾動軌跡及其軌跡長度，發現該軌跡均會產生奇妙的循環規律。
接下來，推廣探討正n 邊形在其他正多邊形中滾動時循環的規律，並利用旋轉半徑及角度之間的變化深入探討其滾動軌跡方程式，發現正n 邊形繞正m 邊形滾動時其內部一點軌跡參數式為，其中， Aj 為 正m 邊形之各頂點、Bj 為正n 邊形中內部一點旋轉時對應 Aj 之點，。
進一步想嘗試使用數學電腦軟體如：Cabri Geometry、Mupad 等對以上研究去做一些動態模擬，並再探討一些特殊擺線如：星狀線、心臟線…等，在條件下相切滾動時，圖中某一點的軌跡規律性及其方程式。另外，應用研究二中的結果，創造出寓數學於遊戲的「圖形板」，並申請了新型專利。
Bezier曲線與蚶線間之關聯性的探討與推廣
在這篇報告中，我們以貝斯曲線的做圖原理建立出一種新的曲線－環狀貝斯曲線，進而得到不少有趣的結果。我們發現有名的古典曲線－蚶線，也是屬於二次環狀貝斯曲線。軌跡方程式為：，此時，係數恰符合二項式定理。之後我們推廣至n次環狀貝斯曲線的軌跡方程式：，也符合二項式定理。
在複數平面上，給定z0、z1、z2三點，我們定義出一個二次變換 ，若，，可映射成蚶線的圖形；若z∈實數，則可映射成拋物線。利用此結果類推我們找到一個複數平面上由 z0、z1、...、zn 所決定的n次變換將以原點為圓心的單位圓，映射成n次環狀Bezier曲線。
In this essay, we use the method of forming a Bezier Curve to establish a new curve, circular Bezier Curve, and find a lot of interesting results. We discover the famous classical curve ＂limacon＂, which belongs to the Quadratic Circular Bezier Curve. The locus of Quadratic Circular Bezier Curve is, where. Its coefficients match the binomial theorem. Then we apply it to the locus of nth-circular Bezier Curve:, and it also matches the binomial theorem.On the complex plane, we define a quadratic transformation corresponding to three points—z0,z1 and z2 as .If , where , a limacon is mapped. If z is a real number, a parabola is mapped. With this result, we will find a nth transformation defined by z0、z1、...、zn on the complex plane. It will form a nth-circular Bezier Curve with unit circle centering on the origin.
高中各學期成績與指定考科相關性
在升學主義越來越興盛的社會中，考試成績成為人人關心的重點，這\r 次研究就是藉由數理資優班同學的各學期在校成績和指定考科成\r 績，透過迴歸分析，找出各學期成績與指考成績之間的關係，並利用\r 圖表來解釋各種科目在各學期的課程，在高中三年所學的重要性，在\r 藉由此結果，希望能對目前老師的教育重點及學生學習方式能有所幫\r 助，亦可了解學生在高中求學過程中，哪些階段對指考成績較有正面\r 影響，進而強化該學習階段，以有助在指定考科時能充分發揮所學。\r \r In a society that emphasize on degrees, examination scores become the\r spotlight, and the ultimate goal for a high school student who had worked\r so hard for three years is to achieve high scores in the J.C.E.E. In the\r three years of high school, each subject has different topics each semester,\r but which semester has the most decisive effect on the J.C.E.E. score?\r This research is to study the effect of each semester on the J.C.E.E. by\r analyzing the grades of a science and math talented class in Senior High\r School using Regression analysis to find out the connections between\r term grades and the J.C.E.E. Then finding out which term grades had the\r most decisive effect in each subject. By using the result, we hope it can\r help teachers in their teaching and students in their learning. Also, it can\r provide the information about which stage in high school has positive\r effects on J.C.E.E. grades, therefore enabling students to emphasize on\r that stage in order to perform well on the J.C.E.E.
長方體中切割正立方體之研究
在1940 年代，Bouwkamp 提出一系列有關如何將矩形切割成若干個正方形的研究報告，但是如何找出正方形個數最少的方法仍是長久以來懸而未決的問題。在本研究報告中，首先引進「四角切割」的方法，並結合輾轉相除法的概念，來研究矩形的切割問題。我們的方法能大幅度降低正方形的個數，也適合做為此問題的上界函數。有關如何在長方體中切割出正立方體的組合，我們也將輾轉相除法的概念延伸到三維空間，進而建立所切割出最少個正立體數的一個上界模式。此外，藉由四角切割概念的延伸，我們也發現這個上界亦可再予修正。In 1940’s, Bouwkamp proposed the study of dissecting squares from rectangles. Among the study, the problem of the least number of dissected squares has been open for decades. In this project, we first propose a corner dissection method, associated with the famous Euclidean algorithm. By reducing nearly three fourths of the number dissected by the primitive Euclidian algorithm, our method indeed establish a suitable upper bound of the minimal number of dissected squares from the given rectangles Meanwhile, the Euclidean algorithm has also been considered to dissect the cubes from cuboids. We analyze the fundamental properties of the method and establish a prototype of upper bound function for the minimal number of dissected cubes. Moreover, the method of corner dissection has also been implemented for some cuboids, which also exhibits the acceptable improvement being a suitable upper bound.