Question 63
There are $2018$ distinguishable boxes on the table. Starting Writer, Writer and Braker take turn writing a box pair to the table (each pair can be written at most once). They stop when there are $4032$ written pairs on the table. After that Braker numerates box pairs by numbers $1, 2, \dots, 4032$ and for each $k = 1, 2, \dots, 4032$ puts $k$ balls into each box belonging to pair numbered $k$. Can Braker guarantee that any two boxes will contain different number of balls?