# A conjecture on tiling

Here's the problem:

An $$L-tile$$ is the one which looks like this, and covers $$3$$ square units:
A A
A
Now we will define a term called an $$L_k-good$$ rectangles where $$k$$ is a nonnegative integer. First of all, for two positive integers $$m,n$$ let $$p_{mn}$$ be the remainder when $$mn$$ is divided by $$3$$. Now, we call an $$m*n$$ rectangle $$L_k-good$$ if the following statement is true: For any possible choice of $$p_{mn}+3k$$ mutually non-adjacent unit squares(non-adjacent means that it doesn't even share a vertex) from the $$mn$$ squares of the rectangle, if we remove those $$p_{mn}+3k$$ squares, the remaining grid becomes completely tilable with only using $$L-tiles$$. This property holds true for all choices satisfying the "mutual non-adjacency" condition, only then the $$m*n$$ rectangle is $$L_k-good$$. So, here are my conjectures:

1)For any $$k$$, there exist finite positive integers $$A,B$$ such that for any integers $$m,n$$ such that $$m>A,n>B$$, the rectangle $$m*n$$ is $$L_k-good$$. As an example, for $$k=0$$, we can show $$A=7,B=7$$ to work.

2)For any $$k$$, $$A=B=8+12k$$ is a pair for which the first part holds.

I tried this for quite a while, got no progress. This seems quite hard. Can anyone help me with this conjecture?

• Feels sad we cant bump this here :( – Shamim Akhtar Aug 31 at 20:15