# problems 21-30 (problems only)

dan's note:  These problems were posted as an ongoing contest on www.dansmath.com from late 1997 to late 2008, divided into 12 seasons.

# Men and/or Women?

In a certain algebra lecture class, Chris and Pat count the students and compare notes.

"Hmm, 12/17 of my classmates in here are women," notes Pat.
"Funny," recounts Chris, "5/7 of my classmates are women."

They were both right. How many students, men and women, were in the class, and what were the genders of Chris and Pat?

Men and women are mutually exclusive sets for the purposes of this problem. Explain your reasoning.

# How Much Space?

A wire belt is wound tightly around the equator of the earth (circumference 25,000 miles), then 25 feet of wire is added and the belt is propped up at an equal height all the way around the planet.
How much space will there be under the wire?   (Please explain your reasoning.)

a)  Not enough for an ant to crawl under            b)  Enough room for an ant, but not a mouse,

c)  A Siamese cat can just squeeze under it       d)  Dan the math teacher could limbo under it!

# Fermat's First Theorem?

You've heard of Fermat's Last Theorem? As stated in 1637, and proved (by Andrew Wiles) in 1995:
a^n + b^n = c^n has no integer solutions for n > 2.
But 6^3 + 8^3 is very close to 9^3 . . . 216 + 512 = 728, not 729.)

What is the smallest number that equals the sum of two (positive) perfect cubes in two different ways?  (For example, 65 is the sum of two squares in two different ways: 65 = 8^2 + 1^2 = 7^2 + 4^2.)

# That's Sum Product!

Pat and Chris are gambling in Las Vegas. I asked them to 'put their winnings together.'

"Nine factorial," said Pat product-ively."

How much money did each person win?

(Give exact reasons and explain as well as you can.)

# To Seven-Eleven?

Solve this system of equations for x and y :

x + -/y = 7
y + -/x = 11

That is, find a pair of real numbers x and y making both equations true.

(Here -/n means square root of n)

# The Two Elephants: Tons of Tens, or Tens of Tons?

Tenny and Tonny are two elephants. Every winter, Tenny gains ten percent of his body weight, and Tonny loses ten percent. Every summer, the opposite happens: Tenny loses ten percent and Tonny gains ten percent. This has gone on for ten years, and now they each weigh ten tons.

How many tons did Tenny and Tonny weigh ten years ago?

(Explain steps, round to nearest pound.)

# How Many Tiles?

A rectangular floor is composed of whole square tiles. A diagonal line is drawn and ruins some of the tiles. (See that on a 2 x 5 floor, 6 tiles are ruined, on a 2 x 4, only 4 are ruined.)

a)  How many tiles are ruined on a 4 by 6 floor?

b)  How about a 63 by 81 floor?

c)  Generalize to an m by n floor.

# Where On Earth?

I start running at 12:00, go 2 miles south by 12:15, 2 miles west by 12:30, and 2 miles north by 12:45.

After 45 minutes I'm right back where I started! Where on Earth am I?

(There's more than one possibility)

# Fridays the Thirteenth

What is the maximum number of Friday-the-13ths that there can be in a single year?

What is the minimum number?

# Turk-o-nacci Sequence!

Like the Fibonacci sequence  1, 1, 2, 3, 5, 8, . . .  a certain turkey flock has as many turkeys on a given day as the sum of the number of turkeys on the previous two days.

If there were 79 turkeys on November 7-th, and 542 turkeys on November 11-th, how many turkeys were there on November 18th?