Coke Versus Sprite: The Dan Meyer Problem

Last fall, I received a flood of texts from former colleagues to whom I had just said goodbye when we moved here to Raleigh. My math coach friends back in DC were attending a professional development opportunity and were given a problem that had them mad as a bunch of hornets. None of them could agree with one another and they all had compelling and convincing mathematical proofs that they were right. They were hoping I would settle the debate, which I (unsuccessfully) tried to do. I very confidently informed them of the answer, which turned out to be COMPLETELY WRONG.

Here was the problem that was causing such a stir:

There are two identical glasses and two full cans of soda: a Sprite and a Coke. The entire contents of the Sprite are emptied out into one glass and then the entire contents of the Coke are dumped out into the second glass.

A medicine dropper is used to remove 5 Mls of Sprite, which is added to the Coke. The Coke is stirred vigorously.

A medicine dropper is now used to remove 5 MLs of the Coke and added back to the Sprite.

The question is: WHICH OF THE TWO GLASSESS CONTAINS MORE OF ITS ORIGINAL DRINK? In other words, is the Coke "cokier" or is the Sprite "spritier?"

My initial reaction was that the Sprite was "spritier." I reasoned that the Coke that was added to the Sprite contained traces of Sprite in it.

When I finally figured out the precise mathematical calculations I was SHOCKED to discover that my math intuition was VERY, VERY wrong.

Recently, I challenged the staff here at Brassfield to answer the problem.

I LOVED watching and hearing the wonderful teachers at our school talking this out. The majority (about 60%) chose Sprite, while about 30% chose Coke. There were a few outliers (about 10% or so) who chose "neither." What do YOU think?

Watch this quick clip to help you decide: Coke V Sprite

Our very own Kindergarten teacher extraordinaire, Mrs. Dobner (formerly Miss McKinney) brought the problem to the attention of her husband that evening and engaged him in a delightful debate. (Have I mentioned that he is "mathy" and an engineer?) He didn't believe her when she told him the answer and set about writing a proof for his lovely bride on a napkin and, in the process, proved her right.

In fact, the two drinks are EQUAL. It defies common sense and intuition, but it's true. Here are a few ways to prove it:

1. If we started with 100 units of Coke and 100 units of Sprite, and now the Sprite glass has 98 units of Sprite and 2 units of Coke, then the Coke glass MUST have what’s left, which would be 98 units of Coke and 2 units of Sprite.

2. Mr. Dobner's response (on the napkin)

100 ml of sprite.
100 ml of coke.

Transfer 10 ml of sprite into the coke glass. Mix.
So now 10/110 = 9.09% is sprite, the remaining 100/110 = 91.81% is coke.

Taking 10 ml from this glass back into the coke. 9.1% of this 10 ml should be sprite, and 90.9% should be coke.

So final totals would be 90 ml sprite + 9.1% of 10 ml = 90.091 ml
and 100 ml coke – 90.9% of 10ml = 90.091 ml

3. 

• Start with 12 green tiles on the left and 12 red tiles on the right.
• Move 4 green tiles to the right. Now, 4/16, or 1/4, of the tiles on the left are green. 12/16, or 3/4, are red.
• 4 tiles are moved back to the left. To simulate the effect of stirring, 1 of these 4 are green. 3 of these 4 are red.
• The number of green tiles on the left is now 8 + 1 = 9.
• The number of red tiles on the right is now 12 – 3 = 9.