Since my last post, I actually dug up one of my books with Carroll problems. I’ll present this one in Carroll’s own words and add a few notes:
Some men sat in a circle, so that each had 2 neighbours; and each had a certain number of shillings. The first had I/ more than the second, who had I/ more than the third, and so on. The first gave I/ to the second, who gave 2/ to the third, and so on, each giving I/ more than he received, as long as possible. There were then 2 neighbors, one of whom had 4 times as much as the other. How many men were there? And how much had the poorest man at first?
A ‘/’ is clearly to be read as a shilling and the ‘I’ is to be read as 1. With that, I think the operations is clear. It is also clear that eventually someone will not be able to pass along 1 more shilling than he was passed, given the finite number of shillings in the game. When that state occurs, instead of passing that person retains the shillings he was just passed. We are then told that it is true that someone now holds 4 times as many shillings as one of his neighbors and are asked how many men there are and how many shillings the poorest of the group must have had to start.