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| Math Central | Quandaries & Queries |
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Dear sir, pls answer for following questn: five players agreed that after every game, the loser doubles every body's money. they played 5 games and everybody lost once. after the 5 games, everybody had 4128. How much money did each of the players have before the game? tks Safdar |
Safdar,
After 5 games does each player have 4128 or is it the combined amount of money between all 5 players? My suggestion would be to be to try to work backwards in the problem.
Janice
Safdar wrote back
Dear Janice,
AFTER 5 GAMESEACH PLAYER HAS $128.
TKS N BST RGRDS
SAFDAR
Safdar,
The suggestion by Janice that you try working backwards is an excellent strategy. Suppose I number the players, player1, player 2 and so on and that player 1 lost the first game, player 2 lost the second game up to the fifth game that player 5 lost. I am going to use a table to keep track of the amount of money each player has. After the fifth game my table is
| Player 1 | Player 2 | Player 3 | Player 4 | Player 5 |
|---|---|---|---|---|
| $128 | $128 | $128 | $128 | $128 |
Player 5 lost the fifth game and doubled each other player's money so players 1 to 4 had $64 at the beginning of the fifth game. Player 5 gave each of them $64 so he gave a total of 4 × $64 = $256 and hence at the beginning of the fifth game he had $256 + $128 = $384. Thus my table after the fourth game is
| Player 1 | Player 2 | Player 3 | Player 4 | Player 5 |
|---|---|---|---|---|
| $128 | $128 | $128 | $128 | $128 |
| $64 | $64 | $64 | $64 | $384 |
Player 4 lost the fourth game and doubled the other player's money so at the beginning of the fourth game the table is
| Player 1 | Player 2 | Player 3 | Player 4 | Player 5 |
|---|---|---|---|---|
| $128 | $128 | $128 | $128 | $128 |
| $64 | $64 | $64 | $64 | $384 |
| $32 | $32 | $32 | $???? | $192 |
How much money did player 4 have before the fourth game?
Continue the table back to the beginning of the five games.
Notice that there is a check on your arithmetic. After the last game the players have a total of 5 × $128 = $640. Each game just moves the money around so the sum of each row in the table must be $640.
Harley
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