A Puzzle with its Ups and Downs

Published on Sunday, November 04, 2012 in , , , , , ,

Konrad Jacobs' photo of Martin GarnderMartin Gardner made many puzzles and magic tricks popular over the years.

This post focuses on one particular bar bet whose popularity seems to come and go like the tide. It involves nothing more than 3 glasses and someone to challenge.

Martin Gardner first wrote up this puzzle in his December 1963 Scientific American column. It was later reprinted in the “Parity Checks” chapter of Martin Gardner's 6th Book of Mathematical Diversions from Scientific American.

A brief write-up of it is also found in his book Entertaining Science Experiments with Everyday Objects under the name “Topsy-Turvy Tumblers,” and Google Books has made the Topsy-Turvy Tumblers page available online for free.

Usually, it's described with the objective as getting all 3 cups mouth up, but it's easy enough to alter the goal to getting all 3 cups mouth down, as in the following video:

Whether you decide your challenge will be to get all the cups face-up or face-down, the process is the same. The spectator must follow your action exactly, do everything in 3 moves, and wind up with the cups facing the same way as you.

The three moves are a throw-off. When you look at the alternating set-up at the beginning, it's easy to see that you could achieve the goal by just flipping the two outer cups. Once you realize this, it becomes an easy way to check that you have the cups set in the right position for the correct goal when you do it, and in the wrong position for the wrong goal when the other person does it.

The pattern of moves they have to follow is easy enough. Turn the two rightmost cups, followed by the two outer cups, followed by the two rightmost cups again. When performing this, you really only need to think of this as right-outer-right.

Most people who do this puzzle stop with this once they win their money or drink. There is, however, a little-known sequel to this puzzle. Martin Gardner and Karl Fulves developed it together, but taught it with pennies instead of cups, so few have made the connection between the two routines.

In the sequel, you bet that you can get all 3 glasses facing the same way while blindfolded, and without even knowing the arrangement of the glasses!

You explain that you are going to be blindfolded, or otherwise prevented from seeing and touching the cups (this could be done over the phone, if desired). You mention that since you'll be blindfolded, you need a little leeway and will instruct the other person to flip the glasses one at a time.

The original write-up is a little hard to find, but thankfully, it was printed up in the American Scientist article, “Puzzles and tricks from Martin Gardner inspire math and science,” which is available for free online. It was also discussed further in the January 2012 issue of the College Mathematics Journal, which is also available in full online, in an article by Ian Stewart titled, “Cups and Downs.”

How is this possible? The method is simply this: First, you tell them to flip the leftmost glass. Next, you tell them to flip the middle glass. At this point, you ask them whether all the glasses are facing the same way yet. If so, you stop, of course, and if not, ask them to flip the leftmost glass one more time. At this point, the glasses are guaranteed to all be facing the same way!

After the second flip, the step where you flip the middle glass, you may get lucky and hear audible gasps, indicating that the people are amazed you reached your goal so quickly without looking.

If you don't hear any reactions after the second flip, you'll need to ask a question without appearing to do so. The most effective way to do this is simply to ask, "The cups aren't all facing the same way, are they?" Note that this starts with a negative statement, and then asks the question briefly.

If they reply that the cups are NOT facing the same way, you simply say, “I didn't think so,” and then make the last flip. This way, it sounds to the audience like you knew that wasn't the case all along.

If they reply that the cups ARE all facing the same way now, you say, “I thought so! Thank you!” When it happens this way, it simply seems like you're confirming your success, and knew your challenge was complete!

The Ian Stewart article linked above explains the mathematics behind this in a very clear manner, largely with a simple diagram of a cube. The American Scientist article also features a 4-object flipping sequence in which 2 objects are flipped at a time, and it still takes 3 moves or less without looking.

Play around with this bet, and better yet, take the time to examine the mathematics behind it. For such a seemingly simple bit of business, it has plenty to teach.

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