HMMT February, 2024

Solution:
We claim the answer is all multiples of $3$ from $0$ to $2000+2 \cdot 2024 = 6048$.

First, we prove no other values are possible. Let $\ell(x, y)$ denote the label of cell $(x, y)$.

## The label is divisible by 3.

Observe that for any $x$ and $y$, $\ell(x, y)$, $\ell(x, y+1)$, and $\ell(x+1, y)$ are all distinct mod $3$. Thus, for any $a$ and $b$, $\ell(a+1, b+1)$ cannot match $\ell(a+1, b)$ or $\ell(a, b+1) \bmod 3$, so it must be equivalent to $\ell(a, b)$ modulo $3$.

Since $\ell(a, b+1)$, $\ell(a, b+2)$, $\ell(a+1, b+1)$ are all distinct $\bmod 3$, and $\ell(a+1, b+1)$ and $\ell(a, b)$ are equivalent mod $3$, then $\ell(a, b)$, $\ell(a, b+1)$, $\ell(a, b+2)$ are all distinct $\bmod 3$, and thus similarly $\ell(a, b+1)$, $\ell(a, b+2)$, $\ell(a, b+3)$ are all distinct mod $3$, which means that $\ell(a, b+3)$ must be neither $\ell(a, b+1)$ or $\ell(a, b+2) \bmod 3$, and thus must be equal to $\ell(a, b) \bmod 3$.

These together imply that
$$\ell(w, x) \equiv \ell(y, z) \bmod 3 \Longleftrightarrow w-x \equiv y-z \bmod 3$$
It follows that $\ell(2000,2024)$ must be equivalent to $\ell(0,0) \bmod 3$, which is a multiple of $3$.

## The label is at most 6048.

Note that since $\ell(x+1, y)$, $\ell(x, y+1)$, and $\ell(x, y)$ are $3$ consecutive numbers, $\ell(x+1, y)-\ell(x, y)$ and $\ell(x, y+1)-\ell(x, y)$ are both $\leq 2$. Moreover, since $\ell(x+1, y+1) \leq \ell(x, y)+4$, since it is also the same mod $3$, it must be at most $\ell(x, y)+3$. Thus, $\ell(2000,2000) \leq \ell(0,0)+3 \cdot 2000$, and $\ell(2000,2024) \leq \ell(2000,2000)+2 \cdot 24$, so $\ell(2000,2024) \leq 6048$.

## Construction.

Consider lines $\ell_{n}$ of the form $x+2y=n$ (so $(2000,2024)$ lies on $\ell_{6048}$). Then any three points of the form $(x, y)$, $(x, y+1)$, and $(x+1, y)$ lie on three consecutive lines $\ell_{n}$, $\ell_{n+1}$, $\ell_{n+2}$ in some order. Thus, for any $k$ which is a multiple of $3$, if we label every point on line $\ell_{i}$ with $\max(i \bmod 3, i-k)$, any three consecutive lines $\ell_{n}$, $\ell_{n+1}$, $\ell_{n+2}$ will either be labelled $0,1,2$ in some order, or $n-k, n-k+1, n-k+2$, both of which consist of three consecutive numbers. Below is an example with $k=6$.

Hence the possible labels are precisely the multiples of $3$ from $0$ to $6048$.