CaptainEd

Oops! Thanks Bonanova, not only I was backward, but also biased! Durn!

First part

Is your first if-statement necessary and sufficient?

Too bad I can't think things out before I post.

@Bonanova, yes, it's true, I'm working on the wrong problem still.

I did a few simulations, with n = number of points (2, 3, 4). Generated n uniform points in (0,1), sorted, calculated section lengths (including round-the-corner length from max to min), and picked max length. Averaged over 2000 tries. I expect that our varying numbers derive ultimately from using subtly different experiments. .754 (n=2), .6166 (n=3), .5466 (n=4)

maybe...

I feel like such a buffoon! (I could sum this series in high school, but not now...Please nobody needle me :-( )

Aiming at gold star...

Are you sure multiplication is only concatenation? 4 * 3 and 3 * 4 are different strings, and only 4*3 looks like S(12).

Here's my guess about the rules. I have no answers for your questions, Bonanova. S(i) is the string for number i. As I read the examples, I infer that * S(1) is the empty string * if j is non-prime, replace j with the string of its factors (e.g. replace 12 with 2 2 3) * replace the i-th prime with angle brackets around S(i) (counting 2 as the 1-th prime)

Short form of Bonanovas solution

Sorry, with a chute of length 13: 6 spaces on the left, one space in the center for the 1-ball niche, and 6 spaces on the right. As Bonanova said earlier, it's trivial with 15-length chute. If the niche can slide around, we can do it with fewer spaces. You said that the niche is "in the middle". I have assumed that meant "equidistant from both ends". BMAD, are we allowed to assume an asymmetrical chute? (ie, with the niche in a position specified by us?)

Durn! I still can't do it, even with 6 on a side plus a one-ball niche. I can easily get to BWWW /_\ BBBW but I don't see how to get that rightmost W out of the B's way, without putting 7 on the left side and one in the niche. I'm assuming that inserting/removing a ball into/out of the niche requires that there be no ball at the niche's position in the main chute.

Another good insight! It's funny to introspect--I thought the guests in positions 2 and 8 were particularly disadvantaged, but they start the game with condition ( 1 ) already true, so in some sense it's EASIER for them to win. Nice analysis, Bubbled! Nice puzzle, Bonanova

(sigh--hard to post from phone...)

My guess

If I understand the picture properly, the chute looks like this, with the hole on the Right end: BBBB /niche\ WWWW and I think balls can the niche freely even when the niche is occupied

Thank you, dgreening. That's the answer (as you know). When that was explained to me, I was tickled by the apparent bizareness--for example, a 30% bazfaz would actually be rather precise around one or the other tail of the distribution, because, as you point out, the ones in the center of the distribution were moved to a higher-priced product. As a result, a 30% 100 wZ item will turn out to be an 8% 125 wZ item OR a 12% 85 WZ item. What a scream! All the purchaser has to do is the same thing the supplier did--test each item, and put them into other bins. Buy the cheap ones, and discover their actual characteristics. I don't see how to mark your answer as The Answer...

As the Design Engineer at Precision Dornicks, Ltd., you have designed, and PDL has built, numerous dornicks over the years, with good quality, though slightly high price. This next dornick, however, requires a design that hinges on the availability of a high-precision bazfaz. Good news is: Amalgamated Bazfaz Corp sells them in bulk. Bad news is, NOBODY can manufacture a bazfaz with the needed precision--you need 0.001% tolerance on a 10-whozatz bazfaz. ABC offers a dozen precision grades of bazfaz: 30%, 10%, 3%, 1%, 0.3%, 0.1%, 0.03%, and 0.01%. The whozatz values include 1-wZ, 3-wZ, 6-wZ, 10-wZ, 30-wZ, 60-wZ, 100-wZ, etc. You have spot-checked hundreds of examples of ABC bazfaz, and they are always within the advertised tolerance: for example, every single 1% 60wZ bazfaz that you've ever tested fell within +/- 1%, that is, between 59.4 and 60.6 wZ, as measured on your industry-standard whozatometer. The funny thing is, we all know that technology does not permit manufacturing control of the whozatz parameter of a bazfaz more closely than 30%. Two questions: ( 1 ) How can ABC reliably sell product at a higher precision than manufacturing process technology can achieve? ( 2 ) How will your design allow PDL to manufacture the next dornick to achieve .001% tolerance using the ABC .01% 10-whozatz bazfaz? ("higher precision" means "lower tolerance"--a 10% bazfaz is said to have a higher precision than a 30% bazfaz, because the actual measured value is closer to the nominal value)

oooh, that makes more sense than my gibberish! Thanks, plainglazed and dgreening.

Thanks, BMAD, I've never heard of Stern-Brocot before, a fascinating topic!