Cups and Downs

This week’s Puzzle Corner activity is a magic trick with a mathematical, as well as a slight-of-hand, component. I first came across this trick in one of Martin Gardner’s many books on recreational mathematics. I liked it so much that I have been stumping students, friends, and family members with it ever since. In order to make this trick work, you will need to practice it by yourself until the moves (illustrated at bottom) become automatic, before trying it out on someone else. Its success, like the success of many magic tricks, depends on diverting the audience’s in this case, your students’ attention. You can’t do this if you are uncertain of all the moves and take too much time making them. We’ll start by looking at how the trick is performed and then look at the mathematics involved.

You will need at least three cups to perform this trick. Plastic cups that are light and easy to turn over work best. When I do this trick in a classroom with students, I have as many sets of cups as I have groups. I quickly move from group to group doing the trick and leaving the cups there for students to ponder how it works. This works better than doing it once for the whole class.

Begin this trick by explaining that the goal is to get all three cups facing up after making exactly three moves. In each move two of the three cups must be flipped simultaneously. After this explanation, place the cups on a table in a row so that the center cup is facing up, and the two outside cups are facing down. This initial setup is critical for successfully performing the trick, and understanding how it works, but you should not draw attention to it. In fact, as soon as you set them up you need to start flipping the cups as practiced. The trick works because the audience concentrates so hard on your moves, that they don’t remember the initial setup. At the end of the three moves when you have all the cups facing up the solution quickly flip the middle cup over so that it faces down.

Next, invite one of the students who was watching to get all the cups facing up in three moves. This is impossible, because the cups are now in a different starting position, but it is a rare student who realizes this. In fact, the student usually begins to flip the same cups you did (students were watching the moves not the initial conditions). The student will either get all the cups facing down or stop in puzzlement before the third move as the realization strikes that there is no way to get all cups facing up in one more move. If the three cups are all facing down, I tell the student that the trick has not been done correctly since the cups were supposed to be facing up. I then flip over the middle cup so that it now faces up the original setup and quickly do the three moves once more to prove that it can be done. After getting all the cups facing up, I flip over the middle cup, once more creating the impossible starting condition, and ask for someone else to try the trick. If the cups are not all facing down at the end of the student’s moves, I quickly set them up correctly and do the trick again to show that it can be done and then mischievously flip over the middle cup creating the impossible starting position once more. I continue to perform the trick until some students catch on that it is the initial starting position that allows the cups to all be facing up, not the series of moves.

What students realize at this point is that when they try the trick, the cups are set up differently than they are when you do the trick. This means that they will never be able to get all three cups facing up as long as they abide by the rule of flipping two at a time. With their starting position they will be able to get all of the cups facing down, but not facing up.

The reason for this is quite simple once you think about it. Consider the original orientation of cups. It is possible to get all three cups facing up in just one move by flipping the outside two cups; doing it in three moves is another deliberate distraction. Because you are always flipping two at a time, starting with two cups facing up or two cups facing down will determine the final possibilities. Whenever two cups are facing down at the beginning, you will always be able to get all three facing up, and never all three facing down. Likewise, whenever you begin with two cups facing up at the beginning, you will always be able to get all three facing down, and never all three facing up.

At this point, students are ready to talk about the mathematics involved in the trick. This trick is related to odd and even numbers. Since the goal is to get all three cups facing up (an odd number) and you start with one cup facing up (also an odd number) you must turn up an even number (two) of cups, because an odd plus an even is an odd. Likewise, in the students’ impossible starting position with one cup down and two cups up, no number of even flips will get all three cups facing up. This is because you start with an even number of cups facing up and you must flip an even number of cups over on each move and an even plus an even is an even, not an odd.

Below are questions about why the trick works and how it can be explained. You may find it easier to facilitate a class discussion asking the same questions rather than having each student answer them.

Performing the trick:

  1. Step One: Before placing the cups on the table, tell students that the challenge in this puzzle is to get all three cups facing up in exactly three moves. In each of the three moves, two cups must be flipped simultaneously.
  2. Step Two: Quickly set up the cups so that the center cup is facing up and the outside cups are facing down. Tell students to watch carefully as you immediately begin the three moves shown. This draws attention away from the initial setup which is critical to the moves, which are irrelevant.
  3. Step Three: Having demonstrated that the cups can all be made to face up after three moves, flip the middle cup over so that there is a different starting setup. Try not to draw attention to this different starting condition by nonchalantly asking a student to get all the cups facing up an impossibility.
  4. Step Four: After the student fails, quickly set up the correct starting conditions and demonstrate again that the trick does work. Set up the impossible condition once more by flipping the middle cup down and ask another student to try. Do this as many times as needed.

CupsandDowns1Move One: 
Flip right and 
center cups

CupsandDowns2Move Two:
Flip right and
left cups

CupsandDowns3Move Three:
Flip right and
center cups

CupsandDowns4All three cups
are now facing up;
the puzzle is solved.

1. Were you ever able to get all three cups facing up after your teacher did the trick? Why or why not?

2. Was it possible for you to get all three cups facing down? Why or why not?

3. What can you say about how the cups start out in relation to how they can end UP?

4. Why does this trick work the way it does?

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