If we work backward, we can see that at each step there’s only one way in which the money can have been distributed:

After three rounds, they have $24, $24, and $24.

So after two rounds the amounts must have been $12, $12, and $48.

Hence after one round the amounts must have been $6, $42, and $24.

And thus originally the amounts were $39, $21, and $12.

Under the rules as described, with each man losing once, these must be the amounts that they started with. Note that the total amount held by the group is always $72.

From Edward Barbeau, Murray Klamkin, and William Moser, Five Hundred Mathematical Challenges, 1995.