plasmid

Biologists don't tackle the question quite like it was posed in the OP. If you read a DNA sequence like ATATCAGG, the read in the opposite direction wouldn't be GGACTATA. Nucleotides have a directionality; the "preceding" nucelotide would be attached to the 5' carbon of the nucleotide you're looking at, and the "following" nucleotide would be attached to the 3' carbon, and all (or nearly all) of our techniques start from the 5' end and work toward the 3' end. So this would look unambiguously like 5'-ATATCAGG-3'. The tricky part is that DNA is double stranded, where each A is paired to a T on the opposite strand and each C to a G. And the strands run in opposite directions. So for the above sequence, you would have the two strands 5'-ATATCAGG-3' ...||||||||... 3'-TATAGTCC-5' If you turn the complementary strand to show it in the conventional 5' to 3' orientation it would be 5'-CCTGATAT-3'. And there's generally no way to tell if you're looking at the strand of interest or its complementary strand.

This is a problem where it's actually advantageous NOT to apply "without loss of generality".

Getting the probability that a set of N people all have a birthday that's not in January-March (and those specific months) would be straightforward. Phil's simulation might be the only reasonable way of answering the case where the "gap" can be anywhere. Analytically solving for the probability that if you pick N birthdays you will get a largest gap between birthdays of at least X seems tricky.

Unfortunately the propelling device is handheld. If you must hold it in one hand, then conservation of momentum forbids that solution (I think).

Very neat problem. The answer is certainly unintuitive. Edit: renamed states for clarity

If no one else is going to go after this one... I can give an answer based on, well, working in the spirit of the best solution the program could find. But I can't prove that there aren't any larger squares that can fit in the unit cube.

Sorry if you find this disappointing, but I don't think the question can be answered in any meaningful way that you would be hoping for. At least not as far as I know. Statistics can answer certain questions, and there are well-developed tools for answering questions along the lines of "I have two groups of patients: those that were treated with a drug and those that were treated with placebo. I have a null hypothesis that the the two groups had the same change in blood pressure after treatment. I'll test the probability that the observed changes in blood pressure for the two groups could have been drawn from the same distribution, assuming that the distributions are Gaussian. If that probability is very low (usually <5% by convention), then I'll feel comfortable rejecting the null hypothesis and therefore concluding that the drug had an effect." Those types of questions can be answered. In this case, if your null hypothesis is that the birthdays were drawn from a uniform distribution, then most of the tools that are based on statistics for Gaussian distributions wouldn't apply. I think you would be stuck having to say that any possible distribution of birthdays would be equally likely to have been drawn from a uniform distribution. Edit: if you had a very large sample of people, you could perhaps ask the question: "What is the probability that the number of people with a birthday on each day of the year follows a distribution that you would expect to see if birthdays fell on random days?" But this example obviously has too few people for that sort of analysis. If you have a different null hypothesis, like you noticed that there are no birthdays in January-March and want to compute the probability that that would occur, then it would be possible. But then you run into problems associated with post-hoc analysis. For example, if you have a group of 11 people and you notice that none of them have a birthday in January, then you can compute the probability that for any group of 11 people you would have no one with a January birthday (which I think would be ~38%) and therefore claim that your distribution where there are no January birthdays was improbable. But it would miss the point that with a group of 11 people, there MUST be at least one month without a birthday, so you would be hard pressed to really claim that your distribution was improbable.

This is what my brute force approach came up with. If you look at the coordinates of the square that it fit into the unit cube, it should become clear how to imagine that it's oriented, and allow you to come up with a more analytical approach to solving the problem. Which I'm not going to attempt myself for now.

Ah, I understand Anza's answer better now. If you actually do "warp the map" by cutting out the river completely and dragging the two halves of land together and stitching them up, then drawing a straight line over the remaining land would give you the shortest path (excluding the bridge), and the point at which that line crosses the scar left by the removed river would tell you where the bridge should be. That as well as the spoiler in my post above should, I think, qualify as two ways to go about it without the hard math. Anza's approach is better because it would tell you exactly where to place the bridge, not just tell whether or not you've found the best spot and which way to nudge the bridge if it's not optimally placed.

So far the best that my horribly inefficient and possibly buggy java code came up with is an area of 1.0984. But it's still running. I probably ought to have searched for a reasonably efficient Windows C compiler instead. Unfortunately I don't see any easy way of modifying the code to work in 4 dimensions. Especially since it uses cross products which I don't think are defined in 4D.

Are you sure about that? Consider the scenario where the river does not lie halfway between city A and city B. In fact, suppose it lies right next to city A and is at a 45 degree angle. A drawing would help. If the center of the bridge lies on the line from A to B, then you follow the solid line. If the start of the bridge lies at the city gates of A, then you follow the dot-dash line. If you were to make city A very small (unlike in the drawing), and you omit the potions of these paths that cross over the bridge (because they will be the same length on the two paths), it should be clear that you could move that bit of the solid line from city A to the bridge up so it would connect the end of the solid line from city B to the bridge and the end of the dot-dash line from city B to the bridge so they form a triangle. And it will then be instantly obvious that the dot-dash path is shorter than the solid line path. I started trying to do the math to solve for the distances explicitly, and it looks too hard for me. Edit: Saw that Bonanova edited. And:

In that case, if it's a friend who you know recently read a book or watched a movie or got taught something in class etc., that would probably be the best place to check for any characters that he might be talking about. Or ask him if he's willing to reveal where he found it. Otherwise, I think my first guess would be... but it still seems more like it's talking about a particular character.

I get the feeling it might be riddling about either a concept, along the lines of honor or justice or righteousness (it's difficult to distinguish between them with these clues), or a fictional character that might or might not be very well known. Can you give us any clues about how/where you came across the riddle that would point us into the right ballpark to look for answers?

Apologies for the late reply as I was out of town, but Wilson has this one.

I'm neither a planet nor a liquid metal. They shouldn't be scared of little ol' me, I'm just tossing one out there to keep folks on their toes in case Shakeepuddn decides to drop by and really whallop them with a stumper. If it's any help, consider the "I" in this riddle to be the one acting whenever you see a verb.

I would say that I only need four mirrors, as long as they're large enough.

Bound by peers to hinder my crimes For fear I'll defile your eyes Source of strength in previous times I once laid a path to the skies Fire, gold, or night of new moon Bring these or the earth or the snow Warm the thinkers blessed by my boon And gashed, shall I ever regrow

No other answers for this one?