Albion College

Albion, Michigan 49224-5013

e-mail: ram@albion.edu

- To solve the equation
*x*= 2, notice that the exponent of the base^{xx ···}*x*is the same as the entire left-hand side of the equation. Hence the equation can be rewritten*x*^{2}= 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*x*= 4 similarly leads to^{xx ···}*x*^{4}= 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? - What is wrong with the following implication?
If + 3 = 1, then (by subtracting 3 from both sides and squaring)

*x*= 4. - Suppose
**v**is a nonzero vector. Is the set {**v**,**v**} linearly independent? - 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?
- What is nonorientable and lives in the sea?
- 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".

Favorite links:

- Topology Now!
- Barnes Three Dance Database
- Michigan dance calendar
- Country Dancing in Kalamazoo
- Lansing English Country Dancers
- Michigan Dance Heritage
- Kalamazoo College Guild of Change Ringers
- North American Guild of Change Ringers
- Southern Michigan Orienteering Club
- Morris the Cat, with Socks
- Michigan Mathematics Prize Competition
- Knots on the Web
- Car Talk
- The Set Game Company Homepage
- Mysterious Rabbit
- Business Card Menger Sponge Project
- The Elements
- Clone of the Attacks
- Recipe for apple puff pancake.
- Recipe for carrot cake.
- Recipe for cherry-almond biscotti
- Sudoku spreadsheet
- Michigan licnese plate lettters

Modified September 7, 2023, by Robert Messer