Answers to Last Week’s CPCP Challenge

Last week I posted some math and logic problems. Here are the answers (I have chosen some of the succinctly-expressed answers submitted, rather than write them all out again myself):

  1. Light both ends of the first rope, and one end of the second.  When the first rope is completely burnt, light the other end of the second rope.  The 45 minute mark is when the second rope is completely burnt.
  2. Take X coins and flip them.  These form one pile, the rest of the coins form the other.
  3. Take the chicken and drop it off at the finish. Come back and get the dog, drop it off at the finish and grab the chicken again. Come back with the chicken, drop it off at the start and grab the corn. Drop off the corn with the dog. Head back to grab the chicken, and return to the finish.
  4. 3 cuts.  Cut each link in one chain.  Separate them, and use the links to join the ends of the 3 intact chains.
  5. She has two children, one of which is a daughter. Here are the possibilities: Boy/Boy — this is impossible. Boy/Girl, Girl/Girl, or Girl/Boy. So it’s a 1/3 chance that both children are girls.(NOTE: This answer is wrong, but I am leaving it here to explain the extensive debate in the comments below. The correct answer is 1/2.)
  6. Pick a jellybean from the box labeled blue&red. You can be sure that the all the jellybeans in there are the same color. The box labeled with the other color actually contains both blue and red. The box labeled with the color of the picked jellybean actually contains jellybeans of the opposite color.
  7. Unchanged. The floating cube displaces its own weight in water.
  8. First weight three coins against three others. If the weights are equal, weigh the remaining two against each other. The heavier one is the counterfeit. If one of the groups of three is heavier, weigh two of those coins against each other. If one is heavier, it’s the counterfeit.  If they have equal weight, the third coin is the counterfeit.
  9. Same amount of water in wine as wine in water. Think about it: however much water is missing in the one gallon jug of wine has to be in the other container and vice versa.
  10. 1&2 cross in 2 minutes. 1 returns in 1 minute. 5&10 cross in 10 minutes. 2 returns in 2 minutes. 1&2 cross in 2 minutes. Total: 17 minutes.
  11. Anywhere 1 mile north of the line of latitude near the South pole which is 1 mile in circumference will do, as will an infinite number of points below that point, all around the earth.
  12. 3, 3, and 8. The only groups of 3 factors of 72 to have non-unique sums are 2, 6, 6 and 3, 3, 8 (both add to 14). The presence of a single oldest child eliminates 2,6,6.
  13. Let’s say it takes 24 hours to circle the planet. So each plane can carry
    12 hours of fuel. At midnight THREE planes set out with full tanks. By 3 AM they have gotten 1/8 of the way around. Each has 9 hours of fuel remaining. Plane 1 gives 1/4 tank to each of 2 and 3, filling them up; it has 1/4 left and turns around. By 6 AM #2 and #3 have gotten 1/4 of the way around; each has 3/4 tank (9 hours )remaining.  2 gives 3 3 hours of fuel, filling him up and leaving himself with 6. He heads for home. 1 arrives home and refuels. At noon 3 is half way around. He has 6 hours of fuel remaining.  2 arrives home and refuels. 1 and 2 set out in the other direction. At 3 PM 3 is 5/8 of the way around, with 3 hours remaining. 1 and 2 are 7/8 of the way around; 1 fills 2 up and heads for home with 6 hours remaining. At 6 PM 3 is 3/4 of the way around and running on fumes.  He meets 2, who immediately gives him 3 hours of fuel, leaving himself with 6.  1 arrives home, refuels, and sets out again. At 9 PM 3 is 7/8 of the way around and running on fumes again, while 2 is down to 3 hours.  Luckily here comes 1 with 9 hours of fuel; he gives 3 3 hours. At midnight they all arrive safely; plane #1 even has 3 hours of fuel left.
  14. Flip the first switch and leave it on for ten minutes.  Turn it off, turn on the second switch and go upstairs.  Look at the lamp and feel its bulb if it’s off. If it’s on, the second switch controls the light. If it’s off and warm, the first switch controls the light. If it’s off and cool, the third switch controls the light.
  15. I repeated question number 9 here by mistake.

Oh, and I had promised a harder problem. Here it is (Jesse Mazer mentioned it in the comments to the original post as well, I believe):

You have 12 balls. One of them is either lighter or heavier than the others. You have a scale and can only use it three times to find out which ball is different, AND whether it is lighter or heavier. How will you do it? Good luck.

Don’t post answers in the comments. Email them to me at s.abbas.raza [at]