Let's think using the "Pigeonhole Principle"
[Problem] Place several points on or inside an equilateral triangle with a side length of 2.
However, ensure that the distance between any two points is greater than 1.
What is the maximum number of points that can be placed?
By the way, to say the answer is "n points",
・ Show an example where n points work
・ Show that it is impossible with n+1 points
is necessary.
Also, as stated in the sentence above, please make the distance between two points "greater than 1". "Exactly 1 is not allowed".
Here, let's introduce the concept called the "Pigeonhole Principle". This is also known as the "Dirichlet's box principle".
・ If n+1 pigeons enter n nests, at least one nest will contain two or more pigeons
・ If n+1 people enter n rooms, at least one room will contain two or more people
This is the principle. It states something that seems completely obvious, but to answer the [Problem] above, that way of thinking can be used effectively. Or rather, it is difficult to answer without using it.
Now, let's think about it. First, 4 points can be placed. For example, as shown in (Figure 1), if you place 3 points at the vertices of the equilateral triangle and the 4th point at the center of gravity of the equilateral triangle, the condition is satisfied. At this time, the distance between any two points is 2 or 2/√3, so "the distance between any two points is greater than 1".
However, if you try to place 5 points, it will be difficult to place them. No matter how you move the points, the distance between some two points will likely be 1 or less. Therefore, the answer seems to be "4 points".
So, how can we properly show that "5 points cannot be placed"? Yes, this is where the "Pigeonhole Principle" comes in. Let's divide the original "equilateral triangle with a side length of 2" into "four equilateral triangles with a side length of 1" as shown in (Figure 2). If you place 5 points, you will inevitably place 2 points on or inside one of the small triangles. Since the distance between those two points will necessarily be 1 or less, we can see that "5 points cannot be placed".
This way of thinking is the "Pigeonhole Principle". The four small triangles divided correspond to the "nests". The 5 points are the "pigeons". To make the distance between two points greater than 1, you must put the points (pigeons) into separate small triangles (separate nests), but it is impossible to put 5 points (5 pigeons) into 4 separate small triangles (4 nests).
The "Pigeonhole Principle" seems so obvious that it doesn't even seem worth naming. But for this problem, I can't find any other way to show that "5 points cannot be placed". It's modest, but you could say it has great power.
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~ Math Problem Collection for Thinking ~
▷ Instant kill if you understand, obvious at a glance if you see the answer
▷ Pythagorean triples consist of multiples of 3, 4, 5
▷ Similar figures
▷ Let's think using the "Pigeonhole Principle"
▷ Proof problems that do not state the conclusion