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[ed boy]

Spoiler: That is wrong, because: (click to show/hide)

You are given that initially it is at one station and after one hour it is at another station. In your solution attempt, you assumed that the two stations are adjacent to each other, that is not necessarily the case.

Spoiler (click to show/hide)

The same logic applies. The train's velocity has to be some integer number of miles per hour, and its starting position has to be an integer number of miles from station 0 (wherever it's chosen to be). The set of pairs of integers (x, v) can be well-ordered: sort them first by |x|+|v|, and then within that by x, and then by v. Then you can check each (x, v) in order by calling station (x + v*t).
So at t=0, check (0, 0) = 0.
Then check (-1, 0), then (0, -1), then (0, +1), then (+1, 0).
Then (-2, 0), (-1, -1), (-1, +1), (0, -2), (0, +2), (+1, -1), (+1, +1), (+2, 0).
And so on. The terms of the problem rule out the v=0 cases, so skip those. But all of the possible (x, v) pairs are on this list.
Once you find the train's position, do something similar to determine its velocity: one hour later, call station x+1, then station x-2, then station x+4, then x-6, then x+12, and so on until you find the train again.

Got it. For added fun, try to adapt that to the situation where the velocity and initial position are rational numbers.

[ed boy]

Here's a new one:

You are in charge of distributing pills to sick people are part of a medical trial. You are given lots of bottles of pills (an arbitrarily large amount). Some of them contain placebos, and some of them contain experimental drugs (and are thus heavier). They are otherwise identical. Unfortunately, during delivery of the pills, all the labels fell off the bottles and you can no longer tell the difference between them. You must therefore identify which of the bottles are which. To aid you in this task, you have an electric scale. However, due to battery constrainst, it only has enough energy to weigh two loads.

How do you identify the pills?

[Strife26]

I'm at a loss here. I don't think that there's any way to figure out which is which, given an infinite number of bottles and an unsure ratio. I suppose that the easiest thing to do would be to make your own balance scale.

[Ampersand]

Let's say you have two bottles, an arbitrarily large amount, one placebo, one medicine. In this case, you can do it in one measurement, obviously.

Suppose you have three, 2 placebo, one medicine. In this case, you can still do it in one measurement. If two bottles weigh the same, they are the placebos, and if one is heavier than the other, then you've found the one medicine. Assuming you know how many you have of each, that is. Otherwise, you'd need to measure both.

With 4 bottles, you can do it in two measures. Put two bottles on each side of the scale, If one side is heavier than the other, you're done. If equal, remove one bottle from each. One side will drop down. Put the bottle you removed from the other side with that one, and vice versa.

But as the number of bottles exceeds 4, the ability to make these comparisons swiftly deteriorates.

[ed boy]

Apologies, I believe you will need three weighings in order to work it out.

You can take pills from the bottles, you don't have to weigh the bottles themselves.

If it helps, if you know the weights of both, then you can figure out which is which in one measurement.

[Ampersand]

I'm still not sure what the idea is. Using only three weighings, are we supposed to be able to identify all of the bottles?

Because, I've seen this type of puzzle before, but only with a single out of place item that needs to be separated, not many.

[irmo]

Yeah, this seems to be incompletely specified. There's an arbitrarily large number of bottles, how do you identify them with a finite number of weighings?

[ed boy]

Consider the following situation, in which you know the weight of the placebo (A) and the weight of the real pill (B). Let N be the number of bottles.
Number the bottles from 1 to N.
find some integer K such that A<K and B<K.
put on the scales one pill from bottle 1, K pills from bottle 2, etc, so you have K^N-1 from bottle N.
weigh the pills, and put the weight in base K.
You should now have an N digit number, where each of the digits is A or B. The rightmost digit corresponds to bottle 1, the next to bottle 2, etc. If a digit is A, then it is a placebo. If a digit is B, then it is a real pill.
So if you know the weight of both, then you can do it in one measurement.

[Sowelu]

Ohhhh.  That kind of scale.

See, there's a very similar and very common problem involving a mechanical scale, where you have two trays and can only measure which one is heavier or if they're equal.  I skipped right over reading 'electric scale'.

[ed boy]

Here's another one:

There is a man who wants to escape from prison. This particular prison only has one way in/out: a rickety bridge over a large chasm. It takes fifteen minutes to walk across it, and it is too rickety and unstable to be ran across. There are no locked doors or anything between the man and the bridge; if we wants to, he can try to walk across at any time.

However, there is a guard stationed near the prison end of the bridge. If someone comes to him from across the bridge, he will tell them that the area is off-limits and send them away. If he sees anyone trying to leave the prison across the bridge, he will shoot them (unless they are one of the people who approached from the other side and have been sent away).

However, the guard is a very lazy man, and can't be bothered to watch the bridge all the time. Instead, he only checks it every ten minutes.

How does the prisoner escape?

[Darvi]

He waits until the guardsman looks, then walks across for about 9:30 minutes, walks back for half a minute, and gets sent away. Profit!

[ed boy]

He waits until the guardsman looks, then walks across for about 9:30 minutes, walks back for half a minute, and gets sent away. Profit!

Yep, that was a fairly simple one.

[Darvi]

Simon the sorcerer 3D had a similar riddle.

[ed boy]

On the pill bottle question front:

I've been examining my solution, and I can confirm that it is doable in only two measurements, if you are given that there is at least one bottle of real pills and at least one bottle of placebos.

EDIT: It also requires the ratio between the two pill weights to be a rational number

[Strife26]

That's a really elegant way to do it, however, doesn't it fail if you've got a lot of bottles with only a few pills per bottle?