数学
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.
参见
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