Peter's homepage

**IN PROCESS**

**Organization of problems**

**Geometry**

**Two ants**
What's the shortest distance between two ants inside a room if they have to crawl along the inside surface?

**Polygon** Can you build a regular n-gon in 3 space with all angles 90? Get out the straws
and pipe cleaners.

**Arches** The arches in the photo are all the same size but they seem to get smaller as they get farther
from the camera. What sort of function describes this apparent size decrease? [This article has appeared in the Mathematical lens section
of the Oct. 2007 issue of Mathematics Teacher, Vol 101.]

**3-D view** You know those pictures made up of dots that give you a 3-D vision if you relax your eyes?
How do they work?

**Cutting planes** How many regions are created by the intersection of 5 planes?

**Regions in the circle** Take n points on the circumference and make all connecting lines.
How many regions?

**Cutcube** Take a 4-dimensional hypercube, hang it by a vertex, and slice it in half with a
3-D hyperplane. What does the cross section look like?

**Counting**

**Handshake**How long does it take for everyone to shake hands with everyone else?

**Stickers** Every time you buy a package of gum you get a new sticker. How long till you fill the sticker
book?

**Sum of Cubes** Variations on the formula for the sum of cubes.

**Pascal** Explorations with Pascal's triangle.

**Fibonacci ** Playing with Fibonacci numbers.

**Fibonacci modk** Now we write the Fibonacci numbers mod k.

**Fibonacci flowers and cones** How do we account for the extraordinary appearance of Fibonacci numbers in nature?

**Counting trains** How many trains of length 10 can I make out of cars of length 1 and 2?

**Even-odd** Ten coins are tossed. Eyeore wins if the number of heads is even and
Owl wins if the number of heads is odd. Is the game fair?

**Even-odder** Same game, but this time heads comes up 2/3 of the time.

**Probability**

**First hit** If A happens with probability p and B happens with probability q,
what's the probability that A will happen befopre B?

**Darts** If on each throw I hit the bullseye with probability p, how many throws does it take on average
to hit the bullseye?

**Boys and girls** If I know he has two children and I run into him downtown with a daughter,
what's the probability that his other child is a boy?

**Blood test** If I test positive for a disease and the test is 98% reliable, what's the
probability I have the disease?

**Car and two goats** The Monty Hall story complete with Marilyn's correspondence.

**Horse race** This is a game played at the Ex with 6 spinners. What's your expected net payoff?

**Hitting 10** If I flip a coin again and again and get 1 for a head and 2 for a tail, what's
the probability I'll hit 10?

**Snap** What's the probability of getting a snap with a standard 52-card deck?

**Bridge hands** What the probability of getting all four aces in a bridge hand?

**Strategy**

**Skunk** As long as you stay in the game your payoff can increase, but there's
also a chance you'll get knocked down to zero. Should you take what you've already got and run?

**Card trick** You do a bunch of things with the cards and and come up with some number, say 16.
Then without looking, I tell you the 16th card in the deck. How do I do it?

**Coffee and cream** You want the coffee to be as hot as possible when you get around to drinkintg it.
Do you put the cream in now or later?

**Four wells** In each well there is a glass either up or down. You get to reach in and alter the
state of one or both. The table spins. Can you get all four up or down?

**Prisoners and boxes** To be spared execution, they must all find their names.
It seems an impossible task. The solution is nothing short of extraordinary.