dan's math@home - problem of the week - archives
Problem Archives page 12
Problems Only. For answers & winners click here.
1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151+ . prob index
111 -- Wedding Cake
112 - Hands On Time
113 - The Two Cubes
114 - Price of Stamps
115 - Thinking Caps!
116 -- Hoop It Up ! !
117 - Kissing Circles
118 -Cubes & Squares
119 Better than Perfect
120 -- Rotating Tires!
 Problem #111 - Posted Tuesday, February 20, 2001
The Wedding Cake (back to top)
"Sorry, I'm all out of two-tiered cakes," said the pastry chef. "What about
this square one; could you make two layers of it?" "Sure, I could make two
square tiers-- the bottom with a side that's twice as long as that of the top.
And to avoid crumbling, I'll use the fewest number of straight cuts possible."
What is that smallest number of cuts, and exactly how is it done?
Problem #112 - Posted Friday, March 2, 2001
Hands On Time! (back to top)
"What time is it, Rory?" asked Cory one lazy day. "When I checked my watch this morning, the
hour hand was where the minute hand is now, and the minute hand was one minute before where
the hour hand now sits. I notice both hands are now at exact minute divisions."
What is the time now? When did Rory check this morning?
Problem #113 - Posted Sunday, March 11, 2001
The Two Cubes (back to top) . . . graphic by dan bach ->
Woody had two solid cubes; one was 10 cm on a side and
the other was slightly larger. He showed me how he'd cut
a square tunnel completely through one cube(which remained
intact) so the other could pass through it. The amazing thing
was that it was the smaller cube that had the hole in it !!
How was this possible, and what is the biggest theoretical
side of the larger cube? (Hint: Look at the smaller cube from a corner.)
Problem #114 - Posted Wednesday, April 4, 2001
The Price of Stamps (back to top)
Philo just spent a total of about 32 dollars on two types of stamps.
In the first batch, he bought as many stamps as the average cent value of each stamp.
For the second batch the average price per stamp was five times the number of stamps.
If he spent a total of $3.19 less on the second set than on the first, what was the value
(and thenumber bought) of each stamp, and what was the exact total spent?
Problem #115 - Posted Thursday, April 19, 2001
Thinking Caps! (back to top)
Three subjects of an experiment are to be let into a room. They each recieve, at random,
either a red or blue cap. After entering the room each person must simultaneously either
guess their own cap color or say nothing; they can see the other two caps but not their own.
If anybody guesses right and nobody guesses wrong, the three will split six million dollars.
But if there is no guess or a wrong guess, they get nothing. They cannot communicate in
any way once they enter the room but can agree on a strategy in advance.
What strategy will maximize their chance of winning?
Problem #116 - Posted Sunday, April 29, 2001
Hoop It Up ! (back to top)
In the eternal battle to excel, two basketball players (we'll call them
Chris and Pat, to be gender-neutral) compared their stats after a tough game.
"I had a better game," said Chris. "I scored more points than you did tonight."
"No, I had the better game," countered Pat. "I had a better percentage on my two-point
shots, a better percentage on my three-point shots, and I took more total shots than you."
"That's bizarre," said Chris. "Maybe we figured this wrong; we'd better check our math."
(For you rabid or at least avid basketball fans, we are assuming neither player attempted any free throws.)
Was this really possible? Give an example, or else prove that it's impossible.
Problem #117 - Posted Thursday, May 10, 2001 #
The Kissing Circles (back to top)
Inside the (orange) unit circle we fit two blue circles of radius 1/2.
a) The yellow circle is tangent to the two blue circles and the
inner edge of the orange circle. What is its radius, a?
b*) The green circle is tangent to one of the blue circles, the
yellow circle, and the orange circle. What's its radius, b?
c*) If the orange circle is centered at (0, 0) what are the coords
of the centers of the four inner circles? (The blue ones are easy.)
* You can enter even if you just answer part a); keep resubmissions to a minimum
# You can still earn a point for answering this while this sentence is here!

Problem #118 - Posted Sunday, May 20, 2001
More Cubes & Squares ! (back to top)
Here are three nice puzzles about perfect squares and cubes.
a) The sum of the cubes of the digits of 407 is :4^3 + 0^3 + 7^3
= 64 + 0 + 343 = 407 itself. What's the smallest number, greater
than 1, not containing any zeroes, with this cubical property?
b) Find two whole numbers > 1 such that the difference of their
cubes is a square, and the difference of their squares is a cube.
c) What cube has volume numerically equal to its surface area?
Can a cube have the cube of its area equal the square of its volume?
Problem #119 - Posted Wednesday, May 30, 2001
Better Than Perfect ! (back to top)
A "perfect number" is the sum of its proper divisors, like 1 + 2 + 3 = 6.
The proper divisors of a "doubly perfect number" add up to twice the number, and so on.
a) What are the next two perfect numbers, after 6? b) What is the sum of the reciprocals
of the proper divisors of a perfect number? c) Find the first doubly perfect number, and
figure out the sum of reciprocals of its proper divisors. d) Find the second doubly perfect
or the first triply perfect number. (One bonus point for both!)
Problem #120 - Posted Thursday, June 21, 2001
Rotating Tires ! (back to top)
Tires placed on the rear of your car will wear out after 21000 miles, while tires on the front
of your car will last for 29000 miles.
Suppose you have a new car and five identical new tires (four installed and one spare).
a) What is the maximum distance you can drive, assuming you can easily change the tires
any time you want? b) Describe a rotation schedule that allows you to drive this distance.
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Problem Archive Index
Probs & answers . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90
Problems only . . . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90
Probs & answers . . . 91-100 . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151+
Problems only . . . . . 91-100 . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151+
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