# Jo is sitting at a round table with a bunch of other shmucks. What if.........?

Jo is sitting at a round table with a bunch of other shmucks. He has one more dollar than the person to his right and that person in turn has one more dollar than the person to his right and so on around the table. Then, Jo gives one dollar to the person to his right and he in turn gives 2 dollars to the person to his right and that person gives 3 dollars to the person to his right and so on. This process continues around the table as many times as is necessary until someone has no money left. At that time, Jo has 9 times the money as the person to his right. How many men are there along with Jo, how much money did he start with, and how much money does he have at the end?

Relevance

let Jo have \$J with him initially, and there are p other people around the table with him. so we know the person on his left has \$(J-p) with them, the least among all of them.

the process is decreasing \$1 of their money at each round, and the "accumulated" donation comes around back to Jo. p+1 people donated, so Jo lose \$1 and gets \$(p+1) per round. the continuation of the process is until someone has no money left. that means the person on Jo's left is that person, with them possessing the smallest amount of money making them the first to bankrupt, after (J-p) rounds.

so, Jo having 9 times the money as the person to his right :

Jo's money after (J-p) rounds

= J + (J-p)*(p+1 - 1)

= J + (J-p)*p

= J + Jp - p^2

the person on Jo's right's money after (J-p) rounds

= (J-1) - (J-p)*1

= p - 1

J + Jp - p^2 = 9(p - 1)

J(p + 1) = p^2 + 9p - 9

J = (p^2 + 9p - 9) / (p + 1)

= [(p + 1)(p + 8) - 17] / (p + 1)

= p + 8 - [17/(p+1)]

since J is a whole number, p+1 must divide 17, a prime. so p = 16, and J = 16 + 8 - 1 = 23.

How many men are there along with Jo = p+1 = 17

and how much money does he have at the end

= J + p(J - p)

= 23 + 16*7

= 23 + 112

= \$135

If Jo had \$23 and there were 17 men at the table, the amounts would be:

23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7

After 1 round:

22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6 --> \$17

After 2 rounds:

21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5 --> \$34

After 3 rounds:

20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4 --> \$51

...

After 7 rounds:

16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 --> \$119

At that point Jo has \$16 + \$119 = \$135

The person to his right has \$15

\$15 x 9 = \$135