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About bonanova

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    Retired Expert
  • Birthday November 03

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  • Gender Male
  • Location New York
  • Interests Music [performing and directing], photography.

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  1. Drop two sticks

    Simulation shows the probability to be 0.75. This corresponds to 100% intersection if the needles align with opposite diagonals and 50% if the align with the same diagonal. Here is a figure that suggests how this comes about: We chose here a needle length of about 1.26. A simple construction shows (light red) the area where a needle of that length could land and (dark red) the region that its center point can occupy. The center point of a green needle landed on the green dot. The green bow-tie is the region the needle can occupy with its center fixed at the dot. The blue dot is a reflection in the diagonal, and the blue bow tie is the region the mirror image of the green needle can occupy. Note the coincidence of the lower left boundaries of the bow-ties. As the green needle moves through its allowable positions in a CW motion, the blue needle does the same in a CCW motion. During exactly 1/2 of that motion the two needles intersect. During the other 1/2 they do not. This is only a suggestive proof. The bow-ties can take different locations for different landing points. In some cases the blue and green needles never intersect, in others they always intersect and in still others they intersect for fractions of their motion that differ from 1/2. All the cases average out to 1/2.
  2. The lion and the tamer

    Greetings, Prof. T. Great to hear from you. The lion does start from the center, sadly for the tamer. The prevailing thought is for the lion to maintain the tamer's azimuth (his radius is shorter) and inch his way outward. By maintaining his azimuth, never leading nor lagging, the lion does not permit the tamer to gain advantage by reversing his direction. Although, since he can do so instantaneously, neither would the tamer lose advantage.
  3. Playing baseball

    First thoughts:
  4. Drop two sticks

    It turns out to rational. When the two sticks lie close to the opposite diagonals they always intersect, when they're near the same diagonal they intersect with a very simple probability that I think can be proved geometrically
  5. The lion and the tamer

    Good thinking. Let's add the condition that the lion and tamer are point objects. Can they coincide? Also, could the lion reduce the radius of his circle to that of the lion and maintain any angular separation he might at some point have obtained? (Your solution prohibits this, but suppose the lion made one misstep and just for a moment he lagged the angle of the tamer.) This question has an amusing answer.
  6. The lion and the tamer

    Hi and welcome Chandra. Let's add the condition that the tamer (and the lion) can change/reverse directions instantaneously. Tamer can switch from CW to CCW at his pleasure.
  7. The lion and the tamer

    If a lion and his tamer can run at precisely the same speed without tiring, would the tamer be safe inside the lion's circular cage? Assume the lion sits on a stool at the center of the cage as the tamer enters though a door on the perimeter of the cage. You can also neglect the size of their bodies. i.e. consider the lion and tamer as points.
  8. Dangerous safari?

  9. 3 remarkable numbers

    The puzzle is interesting, especially for those who are into number theory. Last week I looked at this puzzle and decided I would not be able to find the connection among the numbers. So I searched OEIS out of curiosity. For some reason I didn't find it. (Strange.) Next I Googled the numbers themselves, and that turned up the football reference. <moderator hat on> I agree with Mike's sentiments. The Forum is for solving puzzles; it's not an Internet scavenger hunt. If you can't solve it, say so and maybe start a (spoilered) group effort. Or ask the OP for a clue. The spoiler function permits multiple users to have a go at solving a puzzle. So solutions belong there. (Even funny ones about football.) <moderator hat off>
  10. 3 remarkable numbers

    I'll bet this is not the answer you're looking for, but it does qualify as a remarkable matchup:
  11. What are these?

    Good one. But I can't identify all of the movies, either.
  12. Principal's age

    Assuming that they bought
  13. Drop two sticks

    In this puzzle, limits can lead us astray.
  14. In the case of 80 mph the fly, Ferrari and truck all meet on the fly's first and only collision. The OP asks the fly to move to the truck, back to the Ferrari, back to the truck and finally back to the Ferrari. Then, "upon returning from the second trip," the three collide. The 80 mph case has them colliding upon reaching the truck for the first and only time.
  15. Drop two sticks

    Yes. Sorry I didn't make that clear. In reply to your answer,