Pigeonhole principle

[swastik pramanik]

A point \((a_1, a_2)\) in the \(x-y\) plane is called a lattice point if both \(a_1\) and \(a_2\) are integers. Given any set \(L_2\) of \(5\) lattice points in the \(x-y\) plane, show that there exist \(2\) distinct members in \(L_2\) whose midpoint is also a lattice point (not necessarily in \(L_2\))

[Anu Radha]

L_2 has five lattice points.

Since the coordinates are integers, we can talk about whether they're odd or even.

Now, the coordinates of the midpoint of two points (x,y) and (z,u) are [(x+z)/2, (y+u)/2]

So, for the midpoint also to be a lattice point, x+z and y+u should be even.

Also notice that: odd+odd and even+even both give an even number

But, we can always find two distinct numbers x and z belonging to points in L_2 such that their sum is even (all odd, 1 even 2 odd, 2 even 3 odd, 3 even 2 odd, 4 even 1 odd or all even), so that their sum is even, and the same is the case for y and u.

(Note: this is true because all x,y,z,u are all integers)

So we can say that there are always two distinct points belonging to L_2 such that their midpoint is also a lattice point, not necessarily belonging to L_2.

[Ananya Bhattacharya]

But, we can always find two distinct numbers x and z belonging to points in L_2 such that their sum is even (all odd, 1 even 2 odd, 2 even 3 odd, 3 even 2 odd, 4 even 1 odd or all even), so that their sum is even, and the same is the case for y and u.

Dear @Anuradha, I have two questions for you.
1. Can you please rethink on the cases you have mentioned here?
there the cases you have mentioned, 3 of those cases add up to 5 entities.
can you please think again that what these entities are and state here?
To remind you, there can be two types of entities here. i) points, ii) Integer coordinates.

2. Secondly, are you sure all the cases suggest Characteristics of one specific set? then why some of them add up to \(5\) while one adds up to \(3\). Also i am not sure about what you denote by saying "all", in first and last case.

If you are interested, i can let you know about what other aspects , thoughts i have on this problem. Please Let me know.

[Ananya Bhattacharya]

@Swastik Pramanik,
You can try to pick a Point and see what kind of integer coordinates they may have, ood or even or both, (with positions)
then u can think of picking another point that may take any of these and what will happen then to their mid point.

Try this, it is an interesting thought you may find.

[Furious Fthefics]

It is quite esay to say there exist exactly $$2^2$$  kind of pairty .I mean to say that the pairty are $$(o,o),(o,e),(e,o),(e,e)$$ .here e denote even and o denote odd.so there is atleast two points having same pairty.

 

So the mid points of the straight line connecting those two points are also a lattice point.

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