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bonanova

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Posts posted by bonanova


  1. Al and Bert are among 100 passengers assigned to one hundred seats on an airplane. Al was first to board, and Bert was last. Strangely, the first 99 passengers ignored their boarding passes and took random unoccupied seats. Bert liked the seat he was assigned and is not happy with the situation. If he's lucky, his seat is unoccupied and there's no problem. Otherwise, he insists the passenger erroneously occupying it move to his own assigned seat. The displaced passenger must then move, possibly displacing another person. This process continues until all passengers are seated.

    What is the probability that Al must move?


  2. Consider a random walk in the plane where each step is taken, beginning at the origin, in either in the positive x or positive y direction, i.e. either east or north, each choice being made by the flip of a fair coin. The length of each step is 1/2 the length of the previous step, and the first step has length √2. After infinitely many steps have been taken, what is your expected distance from the origin?

    Edit: Ignore the original text in pink. Instead,

    What is the distance to the origin of the centroid of the possible termination points? You find the centroid of a set of points by averaging respectively their x- and y- coordinates.

    First correct answer wins, but style points will be awarded as well. B))


  3. @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. On 3/26/2018 at 10:33 PM, plasmid said:

    If you do that and the (uv) term is x(F3-1), then at x=infinity wouldn't you end up with infinity times zero?

    If there's a rule about the limit of multiplication between x and functions with e-x as x goes to infinity, then maybe I could use that in the route I was taking.

    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:

    Spoiler

    Here's a plot of  uv  =  x (F3 -1)   =   x { - 3e-x + 3e-2x - e-3x }  vs x for x in [ 0, 10 (scaled x10) ]

    uv.jpg

     

    It's intuitive that the exponential dominates for large x if you think of say x / e3x, instead of  x  e-3x.


  5. I guess we can compute expectation value as well:

    Spoiler

    E = Sumk { outcomek } x { probability of outcomek }

    One million simulations:

    # holes       %  prob      product
    ----------------------------------

    1 hole    1.01%  0.0101    0.0101
    2 holes  50.40%  0.5040    1.0079
    3 holes  32.85%  0.3285    0.9856
    4 holes  11.98%  0.1198    0.4790
    5 holes   3.04%  0.030427  0.1521
    6 holes   0.60%  0.006024  0.0361
    7 holes   0.10%  0.000993  0.0070
    8 holes   0.01%  0.000125  0.0010
    9+ holes  0.00%  0.000021  0.0002
                               ======
                              
    2.6792  

     

     


  6. 9 hours ago, harey said:

    Nice try, but they do not share and they survive all.

    Hint:

      Hide contents

    Very few will find 100 nuts and do not dig a second hole.
    Some will find find 100 nuts in two holes and do not dig a third hole.
    How many will dig a third hole? And a fourth?

     

    Spoiler

    One million simulations:

    # holes  % of squirrels that find 100+ nuts in their holes

    1 hole    1.01%
    2 holes  50.40%
    3 holes  32.85%
    4 holes  11.98%
    5 holes   3.04%
    6 holes   0.60%
    7 holes   0.10%
    8 holes   0.01%
    9+ holes  0.00%

    Spoiler

    Every hole has a 1% chance of containing a single nut.
    If you want to guarantee all squirrels survive, they must each dig up 100 holes

     


  7. The King has decreed that his daughter the Princess shall marry the most wonderful Prince in all the land. One hundred suitors have been selected from their written applications, and on a certain day the King arranges for them, in turn, to interview the Princess. Each suitor must either be chosen or eliminated on the spot. If the Princess does not choose any of them, she will marry the last Prince to speak with her.

    You have been chosen as the Royal Advisor to the Princess and tasked with implementing her best strategy to choose the Most Wonderful Prince of the realm. You devise an evaluation scheme by which the princess can assign a unique "wonder number" to each prince as she meets him. The strategy then is to have the Princess reject, but record the highest score of, the first N princes that she meets. The Princess will then choose the first Prince that she subsequently interviews whose score exceeds that recorded score.

    That's it. The puzzle is basically solved. Except, of course, to decide on the optimal value of N. It requires some thought. If N to too high, the most wonderful prince is likely to be eliminated at the outset, and she ends up with the last guy. If N is too small, the Princess will likely settle for a fairly undistinguished prince.

    What value of N optimally balances these two risks? What is the probability that the Most Wonderful Prince will be chosen?

    Disclaimer: I recall this puzzle being posted before, with different flavor text. And it's somewhat of a classic. To give it a fair play here, I'll ask not to post any links and not to just give the answer if you know it, at least not without "showing your work."


  8. There is survival in numbers.

    Spoiler

    Say the colony numbered N squirrels, and they each dug up 2 holes.

    They would create N piles of nuts whose average size is 101 and whose standard deviation is about 40.
    So if they share among themselves, they will all get through the winter.
    If each eats only what s/he finds, maybe only 2/3 of them will survive.

     


  9. 12 hours ago, harey said:

    Though they are similar, I see at least one huge difference. In the traffic jam puzzle, you cut ANY part. In the marble problem, you cut the REMAINING part. Enough for different formulae.

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


  10. 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.

    .


  11. That agrees with simulations I ran, which gave, approximately,

    Spoiler

    4, 10, 30, 84 and 244 (geometric series in e) as the number of marbles that require integer numbers 2, 3, 4, 5 and 6 grabs.
    So if you multiply the number of marbles by e, it takes one more grab.

    exp { 1.4  2.4  3.4  4.4  5.4 } =~ { 4  10  30  84  224 }.

    That's interesting.

    Nice solve.


  12. On 3/22/2018 at 10:27 AM, Molly Mae said:
      Hide contents

    This is a thought exercise I've considered in the past.  The answer that I've always come up with is that it doesn't matter when disregarding thickness.  If the raindrops are constant and fill any cubic meter equally, he'll encounter the same number of raindrops on his "face" regardless of speed. 

    The .20m thickness, though, needs to be considered.  We know that when he moves at 0m/s, he will encounter some amount of rain along that thickness and he will have made no progress.  So to minimize the number of raindrops that lands on top of him, we just increase his speed infinitely.  We also know that this can cause severe issues, however.

    If you'd like some math to support this argument:

      Reveal hidden contents

    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. :blush:


  13. On 3/21/2018 at 1:18 AM, aiemdao said:

    ??? there are 3 digits before green digits , but there are 4 subtraction different from all after

     

    More clues...

    Spoiler

                                  _________________
                            x x.x x x x x x x x x x  < -- Showing the decimal point in the quotient
                 --------------        
    x x x x x x / x x x x x x x
                  x x x x x x        <-------------------+
                  -----------                            |

                  x x x x x x 1 <-- clue                 |
                    x x x x x x                          |
                    -----------                     (These create the first three digits)
                    x x x x x x 0 <-- clue               |
                    x x x x x x x                        |
                    -------------                        |
                     
    x x x x x x 0  <-------------------+
                        x x x x x x                      |
                      -------------                      |
                        x x x x x x 0 <-- etc.           |
                        x x x x x x x                    |
                        -------------                    |
                          x x x x x x 0                  |
                          x x x x x x x                  |
                          -------------             (These create the repeating digits)
                            x x x x x x 0                |
                            x x x x x x x                |
                            -------------                |
                              x x x x x x 0              |            
                              x x x x x x x              |
                              -------------              |
                                        x x 0 0 0 0      |
                                        x x x x x x      |
                                        -----------
          |
                                        x x x x x x <----+

     


  14. 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?


  15. 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?


  16. 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?


  17. 12 hours ago, aiemdao said:

    I dont really understand the question, why there are seven red x, what is symbol ( in the middle of dividend.

    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


  18. 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


  19. 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
                     -------
                     - - - -
                     -------

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