A Game in Three Rounds

Date: 04/19/99 at 22:54:28
From: Razelene
Subject: Can't figure out this problem

I can't seem to answer this math question. It's sort of long.

Alice, Bonita, and Carmen have just finished playing three rounds of a 
game. In each round there was only one loser. Alice lost the first 
round. Bonita lost the second. Carmen lost the third. After each round 
the loser was required to double the chips of each of the other by 
giving away some of her own chips. After three rounds each of the 
girls had 8 chips. How many chips did they have at the start?

I tried letting them have the same number at the start but it didn't 
work. But I tried the numbers 14, 4, and 6. Is that right?

Sincerely,
Razelene

Date: 04/20/99 at 09:03:53
From: Doctor Peterson
Subject: Re: Can't figure out this problem

Hi, Razelene.

This sounds like a good place to use unknown variables. Let's say 
Alice, Bonita, and Carmen start out with A, B, and C chips 
respectively. (I think they deliberately made that easy for us in the 
way they named the girls.) Now we can follow through the game and see 
what they end up with.

First round: Alice loses, and gives Bonita and Carmen B and C chips 
respectively, in order to double what each of them has by giving them 
as much as they already have:

    Alice:  A - B - C
    Bonita: B + B = 2B
    Carmen: C + C = 2C

Second round: Bonita loses, and gives Alice (A - B - C) chips and 
Carmen 2C chips:

    Alice:  (A - B - C) + (A - B - C) = 2A - 2B - 2C
    Bonita: 2B - (A - B - C) - 2C = -A + 3B - C
    Carmen: 2C + 2C = 4C

(There's some tricky work with negatives there; just take it slowly if 
you have trouble.)

Now do the same with the third round, and you'll have three equations.
Since you know the resulting expressions are all equal to 8, solve 
those equations, and you'll know what they started with.

The answer you guessed is not correct. This is not a problem to solve 
by guessing! It's not an easy problem, but it can be done by working 
through it carefully.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 04/21/99 at 20:51:20
From: Doctor Peterson
Subject: Re: Can't figure out this problem

Hi, Razelene. I was looking at my answer to this problem, noticed that 
you are probably too young to be doing algebra, and then realized that 
it can be done MUCH more easily without algebra than with it.

Let's try again.

You know what happens at each step, and you know how things ended up. 
So let's just work backwards:

End:  A has 8
      B has 8
      C has 8

Before the third round, when A and B were doubled and C lost what A 
and B gained:

      A had 8 / 2 = 4
      B had 8 / 2 = 4
      C had 8 + 4 + 4 = 16

Before the second round, when A and C were doubled and B lost what A 
and C gained:

      A had 4 / 2 = 2
      C had 16 / 2 = 8
      B had 4 + 2 + 8 = 14

Go back one more round and you're done! Wasn't that easier?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/