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Everything posted by bonanova

  1. At an ever increasing pace Al, Bert and Charlie have been receiving into three identical boxes of limitless capacity identical pairs of silver coins engraved with the integers 1, 2, 3, 4, ... etc. These events occur on a precise schedule, each box receiving Coins marked 1 and 2 at 1 minute before midnight Coins marked 3 and 4 at 1/2 minute before midnight Coins marked 5 and 6 at 1/4 minute before midnight Coins marked 7 and 8 at 1/8 minute before midnight etc. But they were instructed at each event to remove a coin from their respective boxes and discard it. After some thought, Al decided each time to discard his lowest-numbered coin; Bert discarded an even-numbered coin; and Charlie thought what the heck and discarded a coin selected at random. Regardless of strategy, at each event the number of coins in each box grew by unity, so that after N events each box held N coins. Needless to say when midnight struck their arms were infinitely tired, but it was a small price to pay for infinite riches. But tell us, now, whether their expectations were met. Describe the contents of each box at midnight.
  2. Not bad. I like the fact you can throw away the connections you find without fear of removing a potential crossing with a future connection. It’s not the solution I had in mind but I think it works. Nice.
  3. Just an observation, that may already have been uncovered, but don't know where to go with it:
  4. Final clue Repeating an earlier comment,
  5. Belt "segment" just refers to portions of belt between adjacent bags. Top or bottom is not an issue. Consider bags are randomly placed points on a line if you like. The fraction of length that is not a near-neighbor segment changes with time. To imply a limit, the OP asks "on average" over time.
  6. OK a clue ( and it was plainglazed third answer that's close )
  7. plainglazed is closest so far (but I can’t say which of his guesses). Remember it’s a physics question.
  8. This puzzle has a nice "Aha!" proof.
  9. My freshman physics prof drew these on the board one morning. He asked, if the thing on the left is a centimeter, what is the thing on the right? Your turn (spoilers please.)
  10. At a busy airport a conveyor belt stretches from the runway, where all the planes land, to the baggage claim area inside the terminal. At any given time it may contain hundreds of pieces of luggage, placed there at what we may consider to be random time intervals. Each bag has two neighbors, one of which is nearer to it than the other. Each segment of the belt is bounded by two bags, which may or may not be near neighbors (to each other.) On average, what fraction of the conveyor belt is not bounded by near-neighbors? Example: ----- belt segment bounded by near neighbors ===== belt segment not bounded by near neighbors ... --A-----B==========C---D--------E=============F---G-H---I-- ...
  11. At 5-second intervals, 1024 automobiles enter a straight (and otherwise empty) single-lane highway traveling at initial speeds chosen at random from the interval [50, 70] miles per hour. Cars may not pass nor collide with other cars. When a slower car is encountered, a car must simply reduce its speed, and for the purposes of this puzzle we may consider the cars in such a case become permanently attached, traveling at the slower car's speed. Eventually there will be N clusters of cars. What is the expected value of N? (Equivalently, what is the expected cluster size?)
  12. I'm not sure I recognize your solution 6 from its description of how it's constructed, but your final description of it seems right. Nice solve.
  13. One might start by constraining the points to be on the circle's perimeter.
  14. Did OP leave out the part about starting from a random distribution of the (total number of) b = 3x2n beans on the plates? Apologies.
  15. Close, check the second case. Brain fart. Nice solution.
  16. You're correct. It can be a little bit shorter than that.
  17. Ten years ago I called attention to a number that when divided by a single integer p it left a remainder of p-1. (Help, a remainder is chasing me) Here is a chance to construct a nine-digit number, a permutation of { 1 2 3 4 5 6 7 8 9 } that has no remainders, sort of. The task is to permute { 1 2 3 4 5 6 7 8 9 } to create a number whose first n digits is a multiple of n for any single-digit n. For example, consider 123654987. Its first 2 digits (12) are divisible by 2. It's first 5 digits (12365) are divisible by 5. However this is not a solution, since 1236549 is not a multiple of 7.
  18. Arrange the Jacks, Queens, Kings and Aces of the four suits { Spades, Hearts, Diamonds, Clubs } in a 4x4 array, in such a manner that: Each row has exactly one card of each rank and one card of each suit. Each column has exactly one card of each rank and one card of each suit. Both major diagonals have exactly one card of each rank and one card of each suit. Euler, the great mathematician, proposed the task of constructing a similar 6x6 array, but instead it was proven to be impossible. Does a 5x5 array exist?
  19. Good start. Is it always possible to achieve a distribution ratio of 1:2:3?
  20. Four towns, A, B, C and D, are located such that their centers form the vertices of a square 1 mile on a side. Town planners want to build a set of roads that connect the four town centers while minimizing the cost, which can be considered to increase linearly with road length. What set of roads minimizes that cost? A B C D
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