The Value of 'Impossible' ProblemsLesson Plans > Mathematics > Problem Solving
The Value of 'Impossible' Problems
One day, back when I was a high school student, a friend handed me a puzzle book. "Doug," he said, "I can't figure this puzzle out. I've been working on it forever. You want to take a look at it?"
So I took the book from him and read the problem. I tinkered with it for quite a while, then started laughing. "What?" he asked.
"Jeff, the reason you can't solve the problem is because it can't be solved." Of course, Jeff was very annoyed. I proceeded to write a proof that the problem could not be solved. Below is the problem, as it was stated in the puzzle book. I am not providing my 'solution'--I will leave that for you to figure out.
Change the string of letters 'MI' to the string 'MU', using the following rules.
- If a string ends with 'I', 'U' can be added ('MI' can be changed to 'MIU')
- Three 'I's in succession can be changed to a 'U' ('MUIII' can be changed to 'MUU')
- The string 'Mx' (where x is any sequence of letters) can be changed to 'Mxx' ('MUIU' can be changed to 'MUIUUIU')
- Two 'U's in succession can be deleted ('MIUU' can be changed to 'MI')
Okay, so it's a cruel thing to give a student an unsolvable problem, without warning him it's unsolvable. But at the same time, isn't that how real life is?
When Orville and Wilber set out to build an airplane, did they know that it could be done? How advanced would we be technologically if everyone said "I don't know if it can be done, so I'm not going to waste my time trying."
Curiosity and a willingness to experiment are traits that we need to instill in our students. Tomorrow's scientists and researchers are in our classrooms today. Some of the best words for them to hear from their teachers are "I don't know if this can be done, but why don't you see if..."
I still remember the day my high school physics teacher handed me and my friend Mike a thermometer and a computer, and said, "See if you can make the computer read the thermometer." I immediately asked if he knew how to do it. (I was suspicious, because my physics teacher was famous for just making things up on the spot) He said, "Oh, I'm sure it can be done, I just don't know how."
Here are a few 'impossible' problems (only a few of them are truly impossible) to set your advanced students working on. Some of these I know can be done, because I've done them myself. Others I've either tried and failed, or never tried at all. Some, like the first problem above, I know can't be done. But if you have students who are interested in trying, think of the mathematics they might learn in the process of attempting these!
- Can you create a cubic formula? (Like the quadratic, only for solving cubic equations)
- What about a quartic formula (fourth degree)?
- Can you trisect an angle using just a compass and straight edge?
- Can you generate a method for finding the inverse of an NxN matrix?
- Can you draw a map in which each bordering region is a different color, and the map requires more than 4 colors?
- Can you determine whether the series 1/2 + 1/3 + 1/4 + ... converges or diverges?
- Can you find two irrational square roots, the sum of which is an integer?
- Can you generate a formula which will give you all the pythagorean triples?
- Can you create a formula for the sum of the first N perfect squares? What about perfect cubes?
- Does the ratio of (primes less than N) to (composites less than N) approach a limit as N goes to infinity?