How to Solve the Famous Coffee and Cream Puzzle
Summer is brain-teaser season in my house. And since summer has now officially arrived, it’s officially brain-teaser season for The Math Dude. What’s the puzzle for today? Keep on reading to find out!
Jason Marshall, PhD
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How to Solve the Famous Coffee and Cream Puzzle
Over the past few weeks, summer decided to make its grand entrance for the year here in Los Angeles. By which I mean to say that it’s been hot. And I don’t just mean kind of mildly toasty hot, I mean super duper unpleasantly hot—so hot that you don’t want to step outside during the day (or even night) hot.
While this forced indoor isolation is kind of a bummer, I’m a glass-half-full kind of guy so I prefer to see it as an opportunity to do one of my favorite things—sit indoors in an air-conditioned coffee shop and think about math. In particular, I’ve been digging around the internet looking for great new brain-teaser puzzles. And I’ve found a bunch—one of which, I’d like to share with you today.
So, without further ado, let’s begin teasing our brains.
The Coffee and Cream Problem
Imagine that you and I are sitting together at a table in a nice and cool coffee shop escaping the summertime heat. A waiter approaches the table and carefully sets a single cup of coffee and a pitcher of cream down. He says that each container is filled with exactly 10 ounces of liquid. At this point in such a story, most normal people living in a normal world filled with normal situations would pour a dabble of cream in their coffee and get on with the show.
Here’s a puzzle for you to ponder …
But this is not a normal coffee shop and the waiter doesn’t walk away. Instead, he sets down a spoon which holds exactly 1 ounce of liquid, and says:
“Here’s a puzzle for you to ponder. If you put a spoonful of cream in your coffee and then put a spoonful of creamy coffee back in your cream, will there be more cream in your coffee or more coffee in your cream?”
At this point, we both look up at the waiter with utter confusion on our faces. And then we notice who he is, and it all makes sense. The waiter is none other than Charles Dodgson, the English mathematician and writer that you probably know as Lewis Carroll (of Alice in Wonderland fame).
Many people don’t know it, but Carroll was actually first-and-foremost a mathematician and logician, and this is one of the puzzles with which he loved to tease his friends’ brains.
So, my friends, I now use it to tease your brains. Take a few minutes, think about it, and then when you’re ready, continue on for the answer.
How to Solve the Puzzle
As with every good puzzle, there are multiple ways to arrive at an answer to today’s brain-teaser. Of course, there’s only one actual answer, but there are multiple roads you may travel along to arrive at it. To show you what I mean, we’re going to think about today’s puzzle in two different ways.
First up, let’s think about the volumes of the various liquids we’re dealing with as we go through the motions of moving cream to coffee and then vice versa as outlined in the story. In particular, imagine we move 1 ounce of cream from the 10 ounce cream container and pour it into the 10 ounces of coffee. At this point, our coffee cup would container 10 ounces of coffee and 1 ounce of cream, and our creamer would contain 9 ounces of cream.
There’s only one actual answer, but there are multiple roads you may travel along to arrive at it.
The next step in the story is to take 1 ounce of the creamy coffee mixture and move it back into the creamer. How much liquid will be in each cup? There will be 10 ounces in each … right back where we started from. Of course, the difference is that there is now cream in the coffee and coffee in the cream. But the question is how much of each?
Well, after pouring the initial 1 ounce of cream into the coffee, 10/11 of the cup will be coffee and 1/11 will be cream (since there are now 11 total ounces of liquid in the cup: 10 ounces of coffee + 1 ounce of cream). If we assume that the coffee and cream mixture is stirred up evenly, the 1 ounce spoonful of creamy coffee mixture that we pour back into the creamer will contain 10/11 ounces of coffee and 1/11 ounces of cream.
So after all of the liquids are moved around, the creamer must contain 9 ounces of cream (that was never removed) + 1/11 ounces of cream from the spoon + 10/11 ounces of coffee. And the coffee cup must contain 10 ounces of mixture, 10/11 of which will be coffee, and 1/11 of which will be cream. So, putting this all together, the creamer will contain 10/11 ounces of coffee, and the coffee cup will contain 1/11 x 10 ounces = 10/11 ounces of cream.
It’s the exact same amount!
An Easier Way to Solve the Puzzle
It turns out there’s actually an easier way to think about all of this. Instead of worrying about the specific numbers for a given situation, we can just think about what has to happen to any cream or coffee that you move around. In particular, notice that since both cups begin and end with the same amount of liquid, the amount of cream that ends up in the coffee cup after all is said and done must be equal to the amount of coffee that is no longer in the cup. Where did that coffee go? Of course, it’s in the creamer. Which means the amount of cream in the coffee has to be equal to the amount of coffee in the cream.
Done!
If you think about it, you’ll see that this logical way of thinking about the problem is quite powerful since it tells us that the number of ounces in the cups is irrelevant—the answer holds no matter how large or small the cups. And if you really think about it, you’ll see that it doesn’t even matter if you stir the coffee and cream before removing a spoonful since any amount you take from one cup (no matter which substance) will always be equal to the amount that’s dumped in the other cup. No matter what, you always end up with the exact same amount of each liquid in the other cup. Pretty cool, right?
Wrap Up
Okay, that’s all the brain-teasing math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Coffee and cream image from Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.