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Everything posted by bonanova

  1. The conclusion that they must be parallel?
  2. No embarrassment needed. It's not simple math. you can Google "pursuit curves" to get an idea. Simulations are the best answer in a lot of cases. Here's something to think about. If the tamer runs straight away along their joining line, he (obviously) keeps his full distance from the lion. Now we know that if he runs at right angles to his vector from the lion, the separation shrinks to 1/2. This suggests that if he runs at some acute angle to his lion vector (partially towards the lion) the distance would shrink to exactly zero. I don't know, and now I wonder, what angle that would be. Possibly 45 degrees?
  3. Right, CaptainEd, middle does not imply equidistant, and your answer is correct..
  4. In a previous puzzle we asked whether a lion can catch his tamer if both are contained by a circular cage and ran at the same top speed. Considering the radius of the cage to be r, the lion begins at the center and the tamer at some point on the edge of the cage. We now remove the cage. Assume the tamer begins running in a direction perpendicular to the direction of the lion and continues a straight-line course. Further, assume the lion follows a "curve of pursuit." That is, the lion's bearing is always in the direction of the tamer. After a long time, their two paths will become collinear, and the lion will trail the tamer by a certain distance. What is that distance. Again, consider the lion and tamer as points.
  5. Three asteroids are traveling in a gravity-less region of space. At time t = t1 they occupy collinear positions. At a later time t = t2 their positions are also collinear. The asteroid in the middle at t1 is also in the middle at t2. Will the asteroids always be collinear? Assume the asteroids are light enough, and distant enough, that their mutual gravitational attraction is negligible.
  6. I don't know how to do without calc or trig ... interesting puzzle
  7. bonanova


    If mating is random, a male (female) could mate with their mother (father). How is that % calculated? Or (a) it doesn't matter because it's >10%, or (b) must citizens mate only with those of their own generation, thus prohibiting aunts, uncles, cousins of differing generations, etc. Actually the more I think of it, probably ANY relation closer than 2nd cousins (common great grandparents) will not bear offspring. Can we just make that simplifying assumption?
  8. bonanova


    This is an intriguing problem that i find myself mulling over in my mind, I have the same questions that DejMar raises, plus some others. Does every male mate mate exactly once every day with a random female? Do the original ten citizens live for exactly 4 days? How is percentage relationship reckoned? For example: siblings=100%; cousins=50%; 2nd cousins=25%; father-daughter=50%? Just guessing here. Does "sustain" require only that after 7 days at least one citizen is living?
  9. Good catch. It couldn't have been that easy. The additional paths number because
  10. Hi wiseabel, and welcome to the Den. So clearly there are several ways in which these numbers are similar and, equally clearly, we're to look for something beyond the fact they are integers. Three are odd, and a different group of three are prime. Three of the four contiguous pairs are descending. Only one of the digits (1) repeats among the numbers. At first glance, I don't see a common similarity, but I'll give it some more thought perhaps in a later post. Thanks for submitting a puzzle!
  11. I think only at the North Pole this could happen. Or on a non-spinning globe. See Coriolis Force. In real life, a ball hit up in the air by a bat has great spin, and follows a curved path. In an idealized sense, with no wind resistance, and in an inertial frame of reference, a ball could go straight up and straight down. But the thing that makes this an interesting question (earth's rotation) also prevents that in general from happening. Rotation is acceleration, and it introduces forces (centrifugal and Coriolis) in addition to gravity. Over short distances and small times, their effect is small.
  12. 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.
  13. 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.
  14. 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
  15. 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.
  16. 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.
  17. 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.
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