Here's the problem:

An $L-tile$ is the one which looks like this, and covers $3$ square units:
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?

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

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