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# harey

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1. ## 100 white marbles

@jasen: a computer simulation for such a problem is IMHO an overkill. Bonanova's approach is correct, he just did not finish it.
2. ## 100 white marbles

@bmad: yes, "non-mathematical" equivalence of "the jar is quite large" and "the marbles are quite small" @jasen: In overwhelming majority of cases, I base my decision on probability, very exceptionally on luck. @bonanova: The answer I expected. I would reason something like that: if there is one black marble, p(black)=1/100. Repeating 100 times, I would estimate p(all white) about 95% and would be very surprised if I were told it is less than 90%. Knowing the solution, I recall the problem of n people having birthday the same day. However, your formula is missing something. If you complete it, you will get another surprise. (Please hurry, I do not want to be faced with the problem of two best answers again.)
3. ## 100 white marbles

There are 100 (quite small) marbles in a (quite large) jar. Pull out one marble, look at it, put it back. You have done it 100 times. All marbles were white. Would you bet 5:1 that they are all white? Bonus (and most interesting) question: Suppose you have 15-20 seconds to decide.
4. ## Random bullets

The speed is a real number. As there is an infinity or real numbers, p(two bullets have the same speed)=zero, at least for a "small" n. The same way we do not suppose that 3 (or more) bullets can collide. However, there is something else that gives me headaches. It works when there are n bullets in line AFTER all n bullets were shot. What if the first two bullets collide and then n-2 bullets are shot? More generally, what if there are collisions before all bullets are shot?
5. ## Interesting question

Yes, the "official" one. But as I know it, it would be unfair to publish it. So I will leave it to someone else. We gave some hints now, did not we?
6. ## Interesting question

The list of given examples is not exhaustive nor limiting. You can tailor a property to any number and we can argue ad eternum whether this property is interesting or not. Just my personal opinion
7. ## Interesting question

Well, I know the "official" answer to this problem, but I do not agree with it. Human language is very imprecise, "interesting" is quite subject to interpretation. I.e. my ex does not find 4 interesting at all. 2+2=2*2=2^2? So what?
8. ## Random bullets

I hope I got it:
9. ## Need an Alibi

I calculated very roughly. Interpolating an exponential is quite dangerous. If the temperature was rounded by 0.2 degrees and measured 2-3 minutes before/after the whole hour, you can get quite a different result. Proof is left to the reader.

11. ## Random bullets

Not necessarily. v3>v2>v1 and diff_1<>diff_2 assumed diff_1=v2-v1 diff_2=v3-v2 if(diff_2>diff_1) then [3 reaches 2] else [2 reaches 1] I just cannot figure out p(diff_2>diff_1).

14. ## Half-balls

@hhh3: Can you tell how?
15. ## numbers by the book

What about a supplementary rule excluding Fred has 354?
16. ## numbers by the book

I have got 3 solutions, but I do not really understand the point (9)
17. ## Can't find the receipt

I calculated with 660.60 and the price is 600.60. Just a small typo as I often do...
18. ## Can't find the receipt

Set aside the sets calculated by Bonanova: 13,519.90-(4*660.60)-(1*1501.50)=9,376 As 600.60*5=1501.50*2=3003=3*1001, remain TVs for 231, 273, 429, 1001 (multiple solutions reserved). 231=3*7*11 273=3*7*13 429=3*11*13 -> all divisible by 3 9+3+7+6=25; so 9,376 is not divisible by 3; -> there are at most 8 TVs priced 1,001 and only 7,374 and 1,368 fits. The same way substract multiples of 231 and consider multiples of 39 (3*13, common divisor of 273 and 429). Alternatively, I think the problem could be solved by Diophantine equations, but I never studied them.

20. ## Highest of the lows

I do not follow here. Worst case: the cards are sorted. Bonanova wrote: What is the expected number of cards in the left pile after all N cards have been drawn? OK, that answer is not difficult to determine, since the definitions of the piles are symmetrical.

100% loss
22. ## Airplane with or without wind

Even here, it increases the travel time. A non-zero component of the plane's speed is used to compensate the wind.
23. ## Illuminating a Convex Solid At Higher Dimensions

This assumes that a point is (at least partially) illuminated if a straight line can be drawn uninterrupted between it and a light source.

25. ## Illuminating a Convex Solid At Higher Dimensions

These thoughts suggest what I think is an interesting question: for each N, find an example of a convex solid that can be illuminated with the fewest lights. That's the start to the original question What bothers me with points and lines: We do not illuminate an object as such, we illuminate it's surfaces. (Already a problem with a sphere and alike: in this case, we may not consider the sphere has one surface, but infinity.) If a surface is illuminated, does it imply that it's edges and vertexes are illuminated, too? One answer would be that this question does not have a sense, but it seems a little week as argument.
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