# Robert Messer

Department of Mathematics and Computer Science
Albion College
Albion, Michigan 49224-5013

office: Mudd 109
e-mail: ram@albion.edu

Surprising results:
1. To solve the equation  xxx ···= 2,  notice that the exponent of the base  x  is the same as the entire left-hand side of the equation. Hence the equation can be rewritten  x2 = 2.  It follows that  x = is a solution to the equation.  In other words, when is substituted for  x  in the tower of exponents, the value of the expression is  2.
Now the equation  xxx ···= 4  similarly leads to  x4 = 4.  So  x = = is a solution to this equation.  In other words, when is substituted for  x  in the tower of exponents, the value of the expression is  4.

It appears that for  x = the tower of exponents has the value of both  2  and  4.  Does this mean that  2 = 4?

2. What is wrong with the following implication?

If + 3 = 1, then (by subtracting 3 from both sides and squaring) x = 4.

3. Suppose v is a nonzero vector. Is the set {v,v} linearly independent?

4. Here are two dogbone toys made of completely malleable rubber. Can the object on the left be unlinked (without cutting it apart or merging anything together) so it looks like the one on the right? 5. What is nonorientable and lives in the sea? 6. You have probably seen or made a paper model of a Möbius band and observed that is has only one side. Is it possible for a Möbius band to have two sides? Here are the notes for a talk I gave titled "The No-Sided Möbius Band".