Chapter 1: A Unique Challenge

I recently encountered an intriguing problem from an old Cambridge University entrance examination. What struck me as remarkable is that this question doesn't necessitate any advanced knowledge; rather, it hinges on sound reasoning and a systematic approach to problem-solving.

The problem involves six points—A, B, C, D, E, and F—situated in three-dimensional space. While no diagram accompanies the question, sketching one could aid in visualizing the situation, though it's not essential. Here’s the challenge:

The Problem Six points A, B, C, D, E, and F are positioned in three-dimensional space in such a way that no three points are collinear and no four points lie on the same plane. Lines are drawn between each pair of points, colored either gold or silver. Your task is to demonstrate that there exists at least one triangle where all sides are the same color. Additionally, prove that this assertion does not hold true when only five points are present.

Solution for Six Points Don’t let the term "general position" confuse you; it simply ensures that any three selected points will indeed form a triangle rather than a straight line. Thus, we can immediately ascertain that any trio of points creates a triangle with three distinct edges.

Now, let’s choose a point—say, point A (the choice is arbitrary). Examining the lines connecting point A to the others—AB, AC, AD, AE, and AF—we find five lines in total. With only two color options, it follows that at least three of these lines must share the same color.

Assuming the three lines AB, AC, and AD are gold, we can analyze the lines BC, BD, and CD that connect the endpoints of the gold lines. If any of these three lines is gold, then at least one of the triangles ABC, ABD, or ACD will have all gold sides. Conversely, if BC, BD, and CD are all silver, then the triangle BCD will consist entirely of silver sides.

Thus, we have established that at least one triangle exists where all sides share the same color.

Solution for Five Points To illustrate that the previous conclusion does not apply when there are only five points A, B, C, D, and E, we can create a counterexample where no triangle exhibits sides of the same color.

A straightforward approach is to connect the points A-B-C-D-E-A with one color while connecting all remaining combinations (A-C, A-D, A-E, B-D, B-E, C-E) with the other color. This configuration shows that no triangle can have all sides in the same color.

What are your thoughts on this question? I’d love to hear your comments!

Chapter 2: Video Insights

In this video, titled "Can you solve this Cambridge Entrance Exam Question?" we delve deeper into the thought processes required for this intriguing problem.

Another insightful video, "TMUA 2022 Paper 1," explores similar mathematical reasoning challenges that enhance our understanding and skills in logical problem-solving.