[Guest]

100 people ( Player 1, Player 2, ... Player 100) are playing a game.

In the first round Player 1 wins.

In the second round Player 2 wins

In the third round Player 3 wins and so on till 100

so in total 100 rounds of the game are played.

If a person wins a round he has to give some money to the other 99 players. The winner must double the money of each of the other players (that they had before the start of the round).

Example; if when Player 5 won the 5th round, he doubled the money all other players at the start of round 5. In the 6th round, player 6 doubled the money other players had at the start of round 6 (same as end of round 5 after player 5 had distributed the money to other players).

At last after the 100 th player wins the amount of money which everone has is same and is "X"

Questions:

1) Which player(s) ended up richest? (i.e. richer meaning that the player had more money at the end of the game than he had at the beginning of the game)

2) Which player(s) lost the most money?

For math lovers:

3) How much money did the 63rd player had in beginning of the game in terms of "X"?

Edited July 2, 2009 by DeeGee

[Guest]

Sparx1 and ZinY, you both have the first two questions correct. The third part is however not correct for either of you.

[Guest]

Sparx1 and ZinY, you both have the first two questions correct. The third part is however not correct for either of you.

Hi DeeGee!

Do you mean

Player 100 started with - (50/2⁹⁹+1/2¹⁰⁰) X is correct.

Player 1 started with ----- (50/2⁰+1/2¹⁰⁰) X is correct.

But, Player 63 started with - (50/2⁶²+1/2¹⁰⁰) X is 'not' correct?

I could not understand.

[Guest]

Player 63 started with - (50/2⁶²+1/2¹⁰⁰) X.

For the first 62 rounds, he did not 'win'. So, his amount doubled for 62 times.

(50/2⁶²+1/2¹⁰⁰) X times 2⁶²= (50+1/2³⁸) X.

On 63^rd round, he did 'win'.

After 63^rd round, he was left with (1/2³⁷) X only, after payout to others.

His amount doubled for the remaining 37 rounds.

(1/2³⁷) X times 2³⁷ = X.

[Guest]

...

by the way, was my answer on page 2 right?

Hi sparx1!

I realized that you already had exactly the same answer as mine for the 63^th player in your earlier post #13.

No wonder DeeGee is saying both of us are wrong!

Thanks for pointing out my mistake as well.

[Guest]

I said what was correct was that player 1 lost most money and player 100 won most money.

As for the calculations, for any player i (1 to 100), the starting amount should be:

x[ 100(2^(100-i)) + 1]/2¹⁰⁰

so for player 100, the starting amount was 101x/2¹⁰⁰

player 99, it was 201x/2¹⁰⁰

player 98, it was 401x/2¹⁰⁰

and so on

Edited July 6, 2009 by DeeGee

[Guest]

In fact, the answers are the same.

I said, player 100 started with - (50/2⁹⁹+1/2¹⁰⁰) X

If we use the common denominator 2¹⁰⁰, it becomes (100/2¹⁰⁰+1/2¹⁰⁰) X,

which is equal to your answer 101x/2¹⁰⁰.

Player 63 started with (50/2⁶²+1/2¹⁰⁰)X.

So, we would say he started with 13743895347201 X / 2¹⁰⁰