I need help to answer the following question:

Is it possible to place 26 points inside a rectangle that is 20cm by

15cm so that the distance between every pair of points is greater

than 5cm?I haven’t learned any mathematical ways to find a solution; whether it maybe yes or no, to a problem like this so it would be very helpful if you could help me with this question.

**Answer**

No, it is not. If we assume that P1,P2,…,P26 are 26 distinct points inside the given rectangle, such that d(Pi,Pj)≥5cm for any i≠j, we may consider Γ1,Γ2,…,Γ26 as the circles centered at P1,P2,…,P26 with radius 2.5cm. We have that such circles are disjoint and fit inside a 25cm×20cm rectangle. That is impossible, since the total area of Γ1,Γ2,…,Γ26 exceeds 500cm2.

Highly non-trivial improvement: it is impossible to fit 25 points inside a 20cm×15cm in such a way that distinct points are separated by a distance ≥5cm.

Proof: the original rectangle can be covered by 24 hexagons with diameter (5−ε)cm. Assuming is it possible to place 25 points according to the given constraints, by the pigeonhole principle / Dirichlet’s box principle at least two distinct points inside the rectangle lie in the same hexagon, so they have a distance ≤(5−ε)cm, contradiction.

the depicted partitioning of a 15cm×20cm rectangle R in 22 parts with diameter (5−ε)cm proves that we may fit at most 22 points in R in such a way that they are ≥5cm from each other.

**Attribution***Source : Link , Question Author : Aryan Patel , Answer Author : leo*