Question 12
Positive integers $K,B,W$ satisfy $B\gt W\gt 1$. Initially one pile has $B$ black balls and another has $W$ white balls. At each operation sort all piles in nonincreasing size, putting a white pile before a black pile of equal size. Select the first $K$ piles, or all piles if fewer, and split each selected pile into two whose sizes differ by at most 1.
For example, the source takes $K=4$, black piles 5,4,4,2 and white piles 8,4,2, then lists the sorted order as $(w8,b5,w4,b4,w2,b2)$. The first four split into $(4,4),(3,2),(2,2),(2,2)$, and the stated next order is $(w4,w4,b4,b3,w2,w2,w2,b2,b2,b2)$.
Repeat until a white singleton first appears at the end of an operation. Prove there is still a black pile of at least 2 balls.
For example, the source takes $K=4$, black piles 5,4,4,2 and white piles 8,4,2, then lists the sorted order as $(w8,b5,w4,b4,w2,b2)$. The first four split into $(4,4),(3,2),(2,2),(2,2)$, and the stated next order is $(w4,w4,b4,b3,w2,w2,w2,b2,b2,b2)$.
Repeat until a white singleton first appears at the end of an operation. Prove there is still a black pile of at least 2 balls.