In the card game “Match,” the loser has to double the money of the other players. Juan, Jamal and Joey are playing. First Juan loses and doubles the money of Jamal and Joey. Joey loses the next hand and doubles the money of Jamal and Juan. Jamal loses the next hand and doubles the money of Juan and Joey. At this point they quit and each has $10. How much did each start with and how much did each win or lose relative to the money they started with?
If the start with amounts x,y,z
after the 3 rounds, the holdings are
x-y-z, 2y, 2z
2(x-(y+z)),4y,2z-(x-(y+z))-2y
= 2x-2y-2z, 4y, 3z-x-y
2(2x-2y-2z), 4y-(2x-2y-2z)-(3z-x-y), 2(3z-x-y)
= 4x-4y-4z, 7y-x-z, 6z-2x-2y
So, now we have three equations:
4x-4y-4z = 10
-x+7y-z = 10
-2x-2y+6z = 10
x = 16.25
y = 5.00
z = 8.75
Note: total=30 as desired.
check the holdings after each round.
round 1: 2.50, 10.00, 17.50
round 2: 5.00, 20.00, 5.00
round 3: 10.00, 10.00, 10.00
In retrospect, it would probably have been easier to work this one backwards, with no algebra:
After round 3: 10,10,10
After round 2: 5,20,5
and so on...
To solve this problem, let's work backwards from the final situation where each player has $10.
At the end, each player has $10, so the total amount of money is $30 ($10 each for Juan, Jamal, and Joey). We know that Jamal lost the last hand and doubled the money of Juan and Joey. Therefore, before Jamal lost, each of them had $5.
Now let's go back to the previous hand, where Joey lost and doubled the money of Jamal and Juan. Joey's loss doubled the money of Juan and Jamal, so they each received $10 from him. Before Joey lost, each of them had $0.
Finally, in the first hand, Juan lost and doubled the money of Jamal and Joey. He had to give $10 each to Jamal and Joey, so Juan's initial amount must have been $20.
To summarize:
- Juan started with $20 and ended with $10.
- Jamal and Joey both started with $0 and ended with $10.
Relative to the money they started with:
- Juan lost $10.
- Jamal and Joey both won $10.