World's Hardest Easy Geometry Problem


Using only elementary geometry, determine angle x. Provide a step-by-step proof. You may only use elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use more advanced trigonomery, such as the law of sines, the law of cosines, etc. There is a review of elementary geometry below. This is the hardest problem I have ever seen that is, in a sense, easy. It really can be done using only elementary geometry. This is not a trick question. Here is a very small hint. Here is a small hint.

World's Second-Hardest Easy Geometry Problem


Using only elementary geometry, determine angle x. Provide a step-by-step proof. This is a variation of the problem above. This is also a very hard problem that is, in a sense, easy. Here is a very small hint. Here is a small hint.

Sorry, but I'm not giving the answer nor the proof here. You will just have to work on it until you either solve it or are driven insane. If you email me at k.enevoldsen@wlonk.com, I may give you a bigger hint (if I feel like it). If you think you have solved it, you can ask me if your answer is correct, but please also tell me how you got the answer. Tell me the key steps in your solution or send me your diagram. Try to persuade me that you are not just guessing. I have additional small, medium, and large hints, but you must first show your efforts to convince me that you have struggled valiantly.

I did not invent these problems. After I first read problem 1, I worked on it for many hours over several days before I eventually figured it out. A couple of years later I came back to the problem, but I had forgotten my proof. It took me many hours to figure it out again! Problem 2 also took me many hours to solve.

How hard are these problems? Any teenage student and some younger students can understand the proof, but very very few are able to discover the proof on their own. Of the hundreds of people that have emailed me, I'd estimate only one or two percent (mostly math professionals and college students) have solved it without significant hints. (The hints given above are not significant hints.) Most people who think they have found the solution are wrong.

These problems have been published in several places. Problem 2 first appeared in the 1920s and problem 1 first appeared in the 1970s. However, I will not name the sources here, because that will just encourage people to search the web for the answers, rather than busting their brains to solve them.

Elementary Geometry

Here is everything you need to know to solve the above problems.

Lines and Angles: When two lines intersect, opposite angles are equal and the sum of adjacent angles is 180 degrees. When two parallel lines are intersected by a third line, the corresponding angles of the two intersections are equal.

Triangles: The sum of the interior angles of a triangle is 180 degrees. An isosceles triangle has two equal sides and the two angles opposite those sides are equal. An equilateral triangle has all sides equal and all angles equal. A right triangle has one angle equal to 90 degrees. Two triangles are called similar if they have the same angles (same shape). Two triangles are called congruent if they have the same angles and the same sides (same shape and size).

  • Side-Angle-Side (SAS): Two triangles are congruent if a pair of corresponding sides and the included angle are equal.

  • Side-Side-Side (SSS): Two triangles are congruent if their corresponding sides are equal.

  • Angle-Side-Angle (ASA): Two triangles are congruent if a pair of corresponding angles and the included side are equal.

  • Angle-Angle (AA): Two triangles are similar if a pair of corresponding angles are equal.


The nature of nothingness|







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让数字也有语义|:数字是科学和工程的命脉,然而如果没有上下文背景,是难以体会到某个数字的含义的。一家叫True #的公司试图提供一种方式,让数字嵌入到常用文档的语义背景中,去改变这种状况。

Keith Enevoldsen's Think Zone

参见:  数学家经常被描述为狂热的着迷于严谨优雅的证明,其实数学也可创造出纯粹的美,很容易体会到的美。今年1月在圣迭哥进行的Joint Mathematics Meetings会议上,40位艺术家展示了他们用数学创造的艺术作品。 Michael Field,休 斯顿大学的数学教授,从他研究的动力系统上发现了艺术灵感。数学动力学系统只不过是几个规则,决定了点如何在平面上移动。Field用一个方程式,将一张 纸上的任意一点移动到另一个地点,Field 不断的重复这一过程——大约有50亿次——保留平面上每个像素大小的点经常性停留的轨迹,一个像素点停留的次数越多,Field就涂上更深的颜色。 数学家对动力系统着迷的原因是非常简单的方程式就能产生非常复杂的行为,Field发现这种复杂的行为能够创造美丽的图像。 数学艺术很流行,这里收集了几个相关的程序:屏幕保护软件electric sheepJenn3dContext Free ArtApophysis(都是开源的软件,主要使用了Fractal flames)。 数学分析需要的软件分类 English mengchuanjin? 文章分类: 学术动态 origin7 的插件很多,支持多种绘图以及基于图形的线性非线性拟合,一部分的统计工作完成,而且数据的共享支持excel,不过图形比excel好看不知道多少倍~ 作图软件:Origin7?.0, Sigma-plot10.0   优点:作图形式多样,图片漂亮 统计软件:Excel2003?, Spss13?.0   优点:Excel可以做简单的数据处理,Spss做进一步的检验和分析 绘图软件:Visio2003?, AutoCAD2006?   优点:前者做流程图,后者工程图 修图软件:系统自带的画图,Photoshop7?.5   优点:对图进行修改 文献管理软件:endnote9.0或reference manager   优点:这方面的介绍很多,去搜索一下吧 数模建立软件:lingo9.0或matlab lingo是数模运行软件,matlab擅长于矩阵的计算与编程

GNU Octave是一个开源数值计算高级语言,可以数字化地解决线性和非线性问题。它提供了一个简单的命令行界面,与MATLAB语言高度兼容。自2.0发布后,经历长达11年的开发,稳定版GNU Octave 3.0于12月21日正式发布。与其它免费的MATLAB竞争对手如Scilab不同,优先兼容MATLAB是GNU Octave的主要设计目标。新版的一些图形功能类似Matlab的图形和可视化系统Handle Graphics,以及与MATLAB的相近的语法,新增加的一些函数来自于子项目,Octave-Forge,它从功能上类似于MATLAB的工具箱。GUI开发正在进行中,但仍然不是Jit编译的执行方式。 一款叫Sage的开源软件正把常见的商业软件排挤出数学教育、政府实验室和以数学为基础的研究领域。这款开源软件的支持者称Sage能够完成任何事情,从12维物体到计算全球变暖效应数学模型中的降雨量。Sage是基于浏览器的开源工具,由华盛顿大学开发,学校声称软件是在全世界100多位数学家的帮助下开发的。数学副教授、程序的开发主管William Stein表示现在的商业数学软件MatlabMapleMathematicaMagma太昂贵了,而且它们不向外透露计算过程的代码,让使用者无法了解结果是如何获得的。有兴趣不妨下载试用下,程序支持Mac OS X,Windows和Linux。