Candy Bar Puzzle

Date: 10/28/96 at 23:3:34
From: Crystal Ratzlaff
Subject: Math puzzle

Dr. Math,

Raymond had a box of candy bars.

He gave Monique half of what he had, plus half a bar.
He gave Claude half of what he had left, plus half a bar.
He then gave Mei-Lin half of what he had left, plus half a bar.
Finally, he gave Laura half of what he had left, plus half a bar.

Raymond then had no bars left.

He didn't break any bars in half. How many candy bars did Raymond have 
to start?

Date: 11/12/96 at 11:53:11
From: Doctor Allen
Subject: Re: math puzzle

To find the answer, we call the number of candy bars Raymond had to 
start with x. We then perform operations on x as described in the 
puzzle.

He gave Monique half of what he had, plus half a bar:
Half of what he had is 1/2 of x or x/2. If Raymond gave Moniqui half 
of what he had plus half a bar, then Raymond has x/2-1/2 bars left.

He gave Claude half of what he had left, plus half a bar:
Remember, Raymond has x/2-1/2 bars left. Half of that is 1/2(x/2-1/2), 
which is x/4 - 1/4. If he gave Claude half of what he had plus 1/2 a 
bar, then Raymond has x/4-1/4 - 1/2 bars left. This can be simplfied, 
since -1/4 - 1/2 = -3/4, so Raymond has x/4-3/4 bars left.

He then gave Mei-Lin half of what he had left, plus half a bar:
Half of what Raymond has left is 1/2(x/4-3/4) = x/8 - 3/8. 
If Raymond gives Mei-Lin half of what he has plus half a bar, then he 
has x/8 - 3/8 - 1/2 left. Simplifying, this is equal to x/8 - 7/8.

Finally, he gave Laura half of what he had left, plus half a bar:
Half of what he had left is, 1/2(x/8-7/8) = x/16 - 7/16. Giving away 
half of what he has plus half a bar leaves Raymond with 
x/16 - 7/ 16 - 1/2 which simplifies to x/16 - 15/16.

Raymond then had no bars left:
The number of bars Raymond has left is zero.

x/16-15/16 =  0
    x - 15 =  0 (multiplying both sides by 16)
         x = 15 (adding 15 to both sides)

Raymond had 15 bars to start with.

Now we need to explain how Raymond managed not to have to break any 
candy bars in half.  Using 15 as a starting point, we can see how many 
candy bars he had after each step:

He gave half of what he had plus a half to Monique.
  1/2(15) + 1/2 = 71/2 + 1/2 = 8 to Monique (Raymond has 7 left)

He gave half of what he had plus a half to Claude.
   1/2(7) + 1/2 = 31/2 + 1/2 = 4 to Claude (Raymond has 3 left)

He gave half of what he had plus 1/2 to Mei-LIn.
   1/2(3) + 1/2 = 11/2 + 1/2 = 2 to Mei-Lin (Raymond has 1 left)

Finally, he gave half of what he had plus half to Laura

    1/2(1) + 1/2 = 1/2 + 1/2 = 1 (Raymond has no bars left)

Because Raymond starts with an odd number of candy bars and always 
ends up with an odd number of candy bars, dividing this number in half 
and then adding a half to it always produces a whole number.  This 
means Raymond never has to break a candy bar in half.   

-Doctor Allen,  The Math Forum
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