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bonanova

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Everything posted by bonanova

  1. Staying dry in the rain

    For the purposes of this puzzle, consider our old friend Albert to have the shape of a rectangular paralellepiped (when he was born the doctor remarked to his mother, I don't explain them, ma'am I just deliver them) just meaning a solid having (six) rectangular sides. At a recent physical exam, Albert was found to be 2 meters tall, 1 meter wide and .20 meters thick (front to back.) He maintains his geometric rectitude by never leaning forward when he walks or runs. So anyway, Albert, alas, has found himself caught in a rainstorm that has 1000 raindrops / cubic meter that are falling at a constant speed of 10 meters / second, and he is 100 meters from his house. Just how fast should Albert run to his house so as to encounter as few raindrops as possible?
  2. 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-- ...
  3. Baggage on a conveyor belt

    @plasmid Nice solve. The puzzle itself was more a math exercise than a puzzle -- that part was thinking it through and setting up the calculation. I thought is was interesting in that you can sort of envision the setup and know that it had to be e.g. greater than 1/3, but maybe not as great as 2/3, so the question was would it be greater than 1/2?
  4. Baggage on a conveyor belt

    Yes there is a rule, L'Hopital's rule. Basically you can just replace functions by their derivatives to resolve indeterminate values. Or, you can just evaluate expressions and see how they behave: It's intuitive that the exponential dominates for large x if you think of say x / e3x, instead of x e-3x.
  5. Squirrel

    I guess we can compute expectation value as well:
  6. Squirrel

  7. Grabbing marbles

    You are wearing gloves while trying to retrieve the marbles contained in a bag. Because of the gloves, you're doing a pretty poor job of it. With each grab, you are only able to retrieve a random number of marbles, evenly distributed between 1 and n, the number of marbles currently in the bag. With 30 marbles initially in the bag, how many grabs do you expect it will take to retrieve them all?
  8. Squirrel

    There is survival in numbers.
  9. Three matches

    If we place four matches in the form of a square, they form 4 right angles. If we place them like a hash-tag (#) they form 16 right angles. If someone removes one match, can we still form 12 right angles? (No bending or breaking of the matches is allowed.)
  10. Grabbing marbles

    The cars in front of the slowest car are the remaining cars. (Each slowest car captures those behind it.)
  11. Grabbing marbles

    I think this puzzle is exactly the same as the traffic jam puzzle I posted recently, although that's not obvious at first glance. @plainglazed found a solution in which he formed clusters of cars by recursively locating the slowest of a group of cars, assuming on average it was in the center of the remaining cars. This corresponds to "grabbing" on average one-half of the remaining marbles from the bag. Picture the marbles in a line and, grabbing a random percentage of them starting from one end. This has to end up having the same number of marble grabs and car clusters. It leads to a logarithmic answer, but to the wrong base -- it should be the natural loge, not log2, which gives too large an answer. Instead of decreasing the number by 1/2 each grab, the remaining number is decreased by 1/e each grab. Same must go for locating the slowest of the remaining cars. @plasmid found a solution that leads to Sum { 1/k }, which as you point out is ln { n } + gamma, and is here confirmed by simulation. What I can't find is a corresponding analysis for the marbles problem (although plainglazed's approach is applicable to both problems) that will lead to that same sum, instead of yours, where both sums appear to be correct. This has been fun to think about. .
  12. Take a guess

    If you chose to answer this question completely at random, what is the probability you will be correct? 25% 50% 0% 25%
  13. Grabbing marbles

    That agrees with simulations I ran, which gave, approximately, Nice solve.
  14. Staying dry in the rain

    You're right. And it's not much of a puzzle after all. You have to allow Albert to lean forward as he runs to make this puzzle at all interesting. But even then, the obvious solution is to have Albert lie horizontally and crawl at infinite speed. Only the top of his head gets wet then. I ran into this puzzle a few years back and "solved" it, more interestingly but also more incorrectly, by multiplying his front and top areas respectively by sin theta and cos theta where theta was determined by his speed compared to the speed of the rain, and some other stuff. It was nonsense.
  15. Here are the placeholders for a long division, solvable, even with none of the digits filled in. The quotient has been placed to the side. It has a decimal point, not shown, and its last nine digits are repeating. Meaning, of course, the last row of X's replicates a previous row. Can you piece together the dividend? -------------- _________________ x x x x x x / x x x x x x x ( x x x x x x x x x x x x x x x x x x ----------- x x x x x x x x x x x x x ----------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x ----------- x x x x x x
  16. Dividend please (on steroids)

    More clues...
  17. New math

    If 5+3+2 = 151022 9+2+4 = 183652 8+6+3 = 482466 5+4+5 = 202541 Then 7+5+2 = ______ ?
  18. Can you write down a 9-letter word that permits you to erase it, one letter at a time, such that after each erasure a valid (English) word remains? (As implied, the letter order remains the same throughout.) Clue: Example:
  19. Same numbers of heads

    @Thalia You are so right. Thanks. Locking this thread.
  20. Same numbers of heads

    Twenty coins lie on a table, with ten coins showing heads and the other ten showing tails. You are seated at the table, blindfolded and wearing gloves. You are tasked with creating two groups of coins, with each group showing the same numbers of heads (and tails) as the other group. You are only permitted to move or flip coins, and you are unable to determine their initial state. What's your plan?
  21. Dividend please (on steroids)

    It's an alternative representation to put the quotient to the right side. Here is the more familiar placement. The green (overlined) digits repeat forever. _________________ x x x x x x x x x x x x -------------- x x x x x x / x x x x x x x x x x x x x ----------- x x x x x x x x x x x x x ----------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x ----------- x x x x x x
  22. Dividend, please?

    Here are the placeholders for a long division, along with a single digit in the quotient. Can you piece together the dividend? x 7 x x x ---------------- x x x / x x x x x x x x x x x x ------- x x x x x x ----- x x x x x x x ----------- x x x x x x x x ------- - - - - -------
  23. Dividend, please?

  24. Things are not going well at the Acme Company. Executive talent is hard to come by, and it is not cheap. Folks at the water cooler have no ideas, and the coffee-breakers can't imagine how to improve things either. But those who party around the teapot, they came up with something. They suggested to the Board that Acme promote the newest hire in the mail room and make him the CEO! We need to shake things up, but good. Qualifications, job experience, brains, judgment, integrity, these are all things of the past. Some were not so sure. Doesn't make sense at all, the old timers said. Almost like appointing some guy with orange complexion to be President. That's exactly the idea, said the tea-people. Turn things on end, let the bull loose in the china shop, and see what happens. Hey -- how could it be worse than what we have now? Not surprisingly, the debate was long and heated. Such a risk merited proof of possible gain, so the old guard posed a challenge: produce a concrete example of where the idea had been tried with incontrovertible benefit. In fact, make it mathematical. You know, something that might make a good BrainDen puzzle. We'll promote the mail room guy, they said, if you can show us an integer that doubles in value when its least significant digit is promoted to its most-significant position. That is, give us a number { some digits } q that has half the value of q { same digits }. That all happened last week, and now we're looking for the mail room guy. Was he promoted? Did the tea people find such a number? Is there one? We need a number or a proof that one does not exist. T.L.D.R. What number doubles in value by by moving its last digit to the first position (if there is one)?
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