# problems 31-40 (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.

# Paint My House!

Aaron, Brenda, Charlie, and Danita are house painters. The team of Aaron, Brenda, and
Charlie can paint my house in 3 days; if Aaron and Danita work as a pair it takes 6 days;
the pair of Charlie and Danita takes 8 days, and the team of Brenda, Charlie, and Danita
can finish in 5 days. How long would it take the whole 4-person crew to paint my house?
(answer to the nearest minute if necessary.)

# Squares in a Box

Suppose you have one square tile of each of these side lengths: 1, 4, 7, 8, 9, 10, 14, 15, and 18 feet.

How can you fit them together into a rectangle? (With no cutting or overlapping, of course.)

To answer, give the coordinates of the lower left corner of each square, starting with the square at (0,0).

# Disk-O-Mania!

a)  How many circular disks of diameter 1 inch can you fit (without overlapping) inside an 8 by 8 square ?

b)  What is the smallest n for which you can fit more than n^2 disks of diameter 1 inside an n x n square ?

# Party Like It's 1999

a)  The number 1999: is it prime or composite? (Prove it's prime or find its prime factorization.)

b)  Is 1999 the sum of two perfect squares? Three? How many does it take, what are they?

c)  What's the smallest number of perfect cubes that add up to 1999 and what are they?

# The Fifth is Half Asleep!

In a 3:00 presentation Mr. Forth gave to less than 104 fifth-graders, there were three boys for every
five girls. At 3:04 some were asleep and some were awake; by 3:05 one-fifth of those awake fell asleep,
and one-fourth of those that had been asleep woke up. There were then as many kids awake as asleep.

How many fifth-graders were there; how many boys and how many girls?

# Under the Salary Cap!

An NBA basketball team has 12 players and the total team salary must be sixty million dollars.
If players can earn only 2, 4, 6, or 8 million, in how many "ways" can the 60 million be paid out?
List all the ways!

(give the lists of possible salaries, from largest to smallest. for example, one "way" would be
60 = 8+8+8+8+8+8 + 2+2+2+2+2+2.)

### Problem #37 - Posted Saturday, February 6, 1999

A former student of mine mistakenly canceled the 6's and got lucky:
1 6       1
----  =  ---
64       4   . . .   (This says 16/64 = 1/4.)

Find three other two-digit examples like this and at least two three-digit examples of lucky cancelling without using the digit 0.

(the numbers must have different digits; i.e. 33/33 = 3/3 doesn't count.)

# The Same Birthday?

a)  Choosing four people at random, what is the probability that none of them have their birthdays on
the same day of the week?

b)  What is the smallest number of (random) people such that there's a better-than-50% probability of
at least two of them having the same birthday?  (such as feb. 14 ; not nec. the same year!)

# Holy Triangle!

If I rearrange the four pieces on the left (base = 5, height = 13) to make the picture on the right (same base, same ht), there's one empty square! What the hole is going on here?

Explain how the same pieces seem to occupy different areas.

# More On Twins

### (not 'moron twins'!)

"Twin primes" are prime numbers that are 2 apart (such as 17 and 19).
Consider the set of all pairs of twin primes, including (17, 19).
a) If we pick a pair of twin primes, each less than 100, what is the probability that their
sum is divisible by 12? b) If we pick a pair of twin primes between 10 and 1000, then
what's the prob. their sum is div. by 12?