jack

A friend on facebook linked to a variant of the 100 hats puzzle I talked about at: http://cartesiandaemon.livejournal.com/232415.html?nc=21

Suppose there are 100 scientists (or logicians, or philosophers...) and 101 hats. As in the other puzzle, the scientists are told the rules, allowed to confer, and then not allowed to speak. They're lined up all facing the same way, so the one at the back can see everyone else's hats, and the one at the front can see no-one's hats. And no-one can see their own hat.

The hats are numbered "1" to "101". The hats are placed randomly on the heads of the scientists, with one unknown hat left over. In any order the scientists can guess what hat they're wearing. But can't guess a number that was previously guessed.

What's the maximum number of scientists you can get to guess right?

If you have that, can you do it if there are 102 hats and two are left over? I genuinely don't know the answer to that one.

Hints in the comments.

A friend asked on twitter, if you could place two connected portals anywhere on earth (but not in space), what would you do with them. Personally, linking Cambridge and Liv's campus flat would be nicest! Geopolitically, probably linking two disparate regions might be most useful.

But of course, the question turned to free energy. Suppose the portal is 1mx2m, laid horizontally, one at ground level and one 100km up at the edge of space, and you diverted sufficiently much water into the lower one to get an endless waterfall. How much energy would you get out?

I'm not sure I have these equations right any more, but under those assumptions (and that the density of water and the gravitational constant for the whole height are rounded to the nearest power of ten and that the square root of 2 is 1.5), and assuming that extracting the energy from the falling water slows it to rest again, I tried to calculate the speed and the energy. I think freefall from that height (assuming you can somehow construct an airtight tube) lands at 1500m/s. And that translates to 3x10^12 Watts = 3TW. Can anyone confirm if I have that right?

It's hard to tell from wikipedia, but it looks like that world energy consumption was 15 TW the last time someone worked it out and updated that page. So this device would make a worthwhile dent in it, but not obsolete everything else.

Of course, that assumes there are sensible engineering solutions to "build an airtight tube to the edge of space" and "slow water from mach 5 safely without wasting any energy" which there probably aren't.

You could dig the tunnel _down_ instead, but you'd have to actually dig it, although you could put one portal down there and dig up through the other one, until the rock fell freely. And you'd have to be careful not to go down too far because if your experiment starts spewing pressurised magma you've invented your own personal volcano, and you have to hope that it solidifies and buries it well enough to withstand ten atmospheres of pressure.

Of course, if you did the above experiment with rock instead of water, the energy involved would be 10 times greater, although presumably the engineering challenge would still be the limiting factor.

However, leaving the practical impossibilities aside, something bothered me about the physics. It seems like, if you don't slow the falling substance completely but let it go through the portal already at high speed, you get correspondingly more energy out. I guess because the limiting factor for energy is that each atom going through a portal to 100km up creates a certain amount of potential energy, so you get the more energy the more you cause that to happen. It seems dodgy that the energy production could just keep on growing in that case, but I guess the assumptions violated physics, so there's no reason not to expect that to violate a wide number of other principles. Have I actually got that right?

Puzzle the first, I assume you have heard this one, but I'll repeat it in case you haven't.

Suppose a hundred people are buried in the sand one in front of another so each can see all the people in front but none of the people behind. An Evil Dictator (TM) places a red or blue hat on each of them. He goes along the line from the back asking each person what colour their hat is, and at the end kills everyone who gets it wrong.

They're allowed to discuss strategy beforehand, but the dictator listens and doesn't allow them to come with any tone-of-voice signals or the like, information can only be conveyed by saying "red" or "blue" when asked.

What strategy is sure to save the most people? If the hats are colored randomly What strategy has the best Expectaion of number saved? For instance the first person is always at risk because NO-ONE knows anything about his hat at all, so no solution can do better than 99 guaranteed saved or 99.5 expected saved.

( Solution )

Puzzle the second. What if the line was (semi) infinite?

( Solution )