Six drinking glasses are arranged in a row. The first three are filled with water; the next three are empty. Is it possible to get the full and empty glasses to alternate by moving only one glass? This classic brain teaser has been making the rounds in recreational mathematics circles for years and is the inspiration for Alternate Arrangements, the Puzzle Corner activity for this week. It is possible to alternate the glasses by moving only one, but you’ll need to think outside the box to do this. If you have never seen this brain teaser, try to solve it before reading on.
I remember encountering this puzzle long ago. At first it seemed impossible, but I persisted and finally made the leap of insight necessary to solve it. Doing so gave me a great sense of accomplishment. If I had given up and looked at the solution, I wouldn’t have gotten the satisfaction of having solved the puzzle. For this same reason, you should caution students not to share their solutions with others until everyone has had ample time to try the puzzle.
When using a puzzle like this one in the class room, my practice was to introduce it after the math period on Monday. I then asked students to work on it in their free time during the week. Students who solved the puzzle were asked to keep the solutions to themselves until the sharing session at the end of the week. This session was held during the math period on Friday. At this session, students who had solved the puzzle or found something interesting while working on it could share their discoveries with the rest of the class. The span from Monday to Friday gave ample time for students to work on the puzzle, and even those who had not solved it had at least had the chance to give it a good try. This format required patience on the part of students and was difficult for them at first; those who got the solution wanted to share it with their friends right away and others gave up before really trying to solve the puzzle. After a month or two, however, most students accepted the puzzle-solving cultural traits (not spoiling the joy of discovery for others and developing persistence) that I was trying to promote.
Look at the four glasses pictured above. Describe what you need to do so that the full and empty glasses alternate.
Can you alternate the glasses by moving only one glass? Describe how you would do this.
If you had six glasses in a row with the first three full and the next three empty, could you still get them to alternate by moving only one glass? How would you do this?
Click the arrow below to view the solution.
In this puzzle you were asked to rearrange 4 glasses of water by moving only one glass, so that the glasses alternated between those that had water and those that were empty. To solve this puzzle all you have to do is to take glass number two from the left and pour its contents into glass 3 from the left.