Question 178
The number $2022$ was initially written on the board. Sophia and Isabella take turns adding numbers to the board. Sophia can add a number that is $2$ less or $2$ more than the last number written, while Isabella can add a number that is $3$ less or $3$ more. Sophia takes the first turn, and they continue alternating their turns. After $2022$ turns, is it possible that the numbers $1, 2, 3, \ldots,$ and $2023$ to be written exactly once on the board?