[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]

Will try. But need to think about how recursion can play a role here.
Thank You Sir.

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