Math 492/592 W05
Problem Set #1
Directions: Try to solve each of the following problems. Keep track of the strategies that
you use so that you can include a description of your progress in your write-up. Note that
I am not looking for a specific approach to the problems. Some of the problems can be
solved using algebra or calculus, but they can also be solved other ways and it is these
other ways that are often the most interesting. Try to use middle-school mathematics
whenever possible and try to justify any assumptions, facts, or claims/conjectures that
you use. Try to prove that your solution is correct.
1. Boxing a triangle. Suppose you had a piece of cardboard in the shape of triangle
where each side is 20 cm long. What is the largest open-top (triangular) box you
could make by cutting sections out of each corner and folding up the sides?
2. The Shriveled Cucumber. A 100-gram cucumber fell out of a truck and rolled
out into the hot sun. When this first happened the cucumber was 99 percent water
(by weight). Sometime later (due to evaporation), the cucumber was only 98
percent water (by weight). How much did it weigh at that time?
3. Saved by Zero. How many zeros occur at the end of the expanded numeral 999! ?
Recall that “!” is the “factorial” of a number. For example 4! = 4x3x2x1 = 24.
4. Let My Love Open The Door. There are 1000 doors numbered 1 to 1000.
Suppose you open all of the doors and then close every other door. Then, for
every third door, you close each opened door and open each closed door. You
follow the same pattern for every fourth door, every fifth door, and so on up to
every thousandth door. Which doors will be open when the process is complete?
5. The Last Straw. Two piles of straws are on a table. A player can remove a straw
from either pile, or a straw from both piles. The player who takes the last straw
loses. If there are two players how should you play?
What counts as progress?
Some things you could do to make (and document) progress on a problem are…
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Draw sketches, diagrams, or tables
Try to solve an easier or slightly different version of the problem
Guess and check your guess
Make an assumption and then try to solve the problem … and then go back and
see if you still need the assumption or if you can prove it is true.
Do a special case or example. Plug in numbers
Try to figure out what is standing in your way and write it down. “I don’t know
…” or “… doesn’t make sense”