MathDB

Define a domino to be a

1\times 2

rectangular block. A

2023\times 2023

square grid is filled with non-overlapping dominoes, leaving a single

1\times 1

gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the

3\times 3

grid shown below, John could move 2 dominoes:

D

, followed by

A

.) [asy] unitsize(18); draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle); draw((0,1)--(3,1)); draw((2,0)--(2,3)); draw((1,1)--(1,3)); label("A",(0.5,2)); label("B",(1.5,2)); label("C",(2.5,2)); label("D",(1,0.5)); [/asy]

Problem Statement