**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 index111 -- 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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