Math For Fun


Math For Fun is a series of math puzzles that appear in the PTA newsletter (without solutions). Solutions are posted here after about a month of publication. Send comments, questions, and corrections to Robbert van Renesse (rvr@cs.cornell.edu).


September 2009:

Puzzle 1a: You paint and you cut and what do you get?

You have a wooden cube with 4-inch edges, and paint it on all sides. Now you cut the cube into small cubes with 1-inch edges.

  • How many of the small cubes have paint on them?
    • 56: there are 4^3 = 64 blocks (small cubes) total, but the inside 2^3 = 8 blocks don't have paint on them. x^y means x to the power y.
  • What if the original cube had 2-inch edges?
    • 8: there are no blocks with no paint on them!
  • 3-inch?
    • 26: just 1 block in the middle with no paint on it
  • 5-inch?
    • 98: 5^3 = 125 blocks total, minus 3^3 = 27 blocks in the middle
  • Can you derive a formula for the number of 1-inch cubes with paint on them cut from an original cube with edges of length N inches, for all whole numbers N larger than 0? (hint: the highest exponent should be no more than 2)
    • Answer: To get the number of blocks that have paint on them, subtract the number that do not have paint on them (the "inside" cube with (N - 2)^3 blocks) from the total number of blocks (N^3). N^3 - (N - 2)^3 = N^3 - (N^3 - 6N^2 + 12N - 8) = 6N^2 - 12N + 8. However, this formula only works if N is at least 2. If N is 1, there is no cutting to be done, and the answer is 1, not 2 as the formula would compute.

Puzzle 1b: Logically speaking, a candle is brighter than the sun.

If X is brighter than Y, and Y is brighter than Z, then X would have to be brighter than Z, right? That is called transitivity. But now the brainy little kid next door asks you: "if a candle is brighter than nothing, and nothing is brighter than the sun, then why isn't a candle brighter than the sun?"

  • What is your logical answer?
    • English is not always a good language to express logic. Math is a better language for that. "A candle is brighter than nothing" would be expressed in math as "brightness(candle) > 0". "Nothing is brighter than the sun" would be expressed as "There does not exist an x such that brightness(x) > brightness(sun)". These two statements taken together do not imply that "brightness(candle) > brightness(sun)".


October 2009

Puzzle 2: Should I switch or not?

You are a contestant on a TV game show! The game host shows you three closed doors and tells you there's a big prize behind one and only one of those doors. You get to choose which door to open. Not knowing anything more, you pick door number 1. The host increases the suspense by opening door number 2 and showing that there is no prize behind it. The host then asks you if you would like to change your mind and pick door number 3 instead.

  • Would it be a good idea to switch, or should you stick with your first choice? Does it matter?
    • Switch, because it matters! The probability that the prize is behind door number 1 is 1/3, while the probability that it is behind either of the other two doors is 2/3. Now that you know that the prize is not behind door number 2, the probability that the prize is behind door number 3 is 2/3, while the probability that the prize is behind door number 1 is still 1/3. You double your chances by switching.
  • What is the probability that the prize is behind door number 1, and what is the probability that the prize is behind door number 3?
    • 1/3 and 2/3 respectivily.


December 2009

Puzzle 3: Stretch that rubber band!

Suppose the equator were covered by land, and you stretched a big rubber band around it. The equator is 40 million meters long. Now you invite 40 million of your Facebook friends to stand one meter apart on the equator and lift the rubber band by one meter.

  • How much longer does the rubber band become?
    • The circumference of a circle with radius r is 2*pi*r. If we increase the radius by 1, then the new circumference becomes 2*pi*(r+1), which is the same as 2*pi*r + 2*pi. Thus no matter what r was, the circumference grows by 2*pi meters, or approximately 6.28 meters. This might be a lot less than you expected, but that's what it is.
  • Now your repeat the experiment on the moon. (The moon is approximately 1/4 the size of the earth.) Does the rubber band stretch by about 1/4 as much as it does on earth?
    • No. As explained in the previous answer, the increase does not depend on the original radius. Indeed, if you repeat the experiment on a ping pong ball, the result would still be the same. For every meter that you lift up the rubber band, its length increases by about 6.28 meters. Until it snaps of course...


February 2010

Puzzle 4: Skating time!

It's winter and you go out skating because you are a phenomenal skater who can carve straight lines into the ice on one foot. Each time you switch feet you carve a new straight line. You invent a new game, trying to make as many intersections as possible. The first time you switch feet, you can make one intersection with the line you just finished carving. The second time you can add two more intersections, because now there are two finished lines, and so after switching feet twice you have three intersections.

  • How many intersections can you make after switching your feet 3 times?
    • Three more, so the total becomes six.
  • How many times do you have to switch your feet in order to create 100 or more intersections?
    • The series is 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, ... So you need to switch 14 times.
  • Can you come up with a formula that gives the number of intersections as a function of the number of times you switch feet?
    • Let N be the number of times you switch feet. Here is a graphic representation of the number of intersections in case N = 5 (just count the number of Xs):


X
XX
XXX
XXXX
XXXXX

Now we double up the triangle, using another triangle of the same size but consisting of 0s:

X00000
XX0000
XXX000
XXXX00
XXXXX0

This rectangle has N * (N + 1) Xs and 0s, so each triangle is of size N * (N + 1) / 2. Let's try this out for N = 14: 14 * 15 / 2 = 7 * 15 = 105, which is indeed the number of intersections you can make after switching your feet 14 times.



April 2010

Puzzle 5: Two-dimensional movers

Movers sometimes have a hard time figuring out if they can get a couch up the stairs, around the corner, through the door, and so on. Suppose we lived in a 2-dimensional world, and you had a 2-dimensional corridor, two meters wide, that makes a 90 degree turn like so:

  • What's the largest rectangle you could move through the corridor?
  • What's the largest circle you could get around the corner?
  • How about the largest triangle?
  • What's the largest object (the one with the largest area) you could move through the corridor?