I thought about it for about 5 minutes before deciding to script it, and got "fobs" and, annoyingly, dismissed "fres" as not a word.

I imagine if I had been more rigorous it wouldn't have taken long to get all the 4 letter ones, since they all have an internal vowel, which was the obvious place to start looking.

I wrote a check for this property for all the words in my system's inbuilt vim dictionary and got the following list:

Rubbish Words:

er, Livy, Lyly, na, ob, re, uh

Interesting Words:

an, fans, fobs, gnat, ravine, robe, serf, tang, thug

I'm in the same boat as you with regards to whether EPA/DHA has a bigger effect than ALA, but I was convinced enough to try to find some when I became vegetarian last year.

If you google "algal dha together" you'll find what I'm taking - meeting your criteria of vegetarian (vegan), eco-friendly and health-friendly (with aforementioned uncertainty)

ALA can also be found in flaxseed, soy/tofu, walnut and pumpkin, so you needn't stick to seaweed if you only want ALA.

I gave this a shot as well as since your value for E(T) → ∞ as T → ∞, while I would think the system should cap out at εN.

I get a different value for S(E), reasoning:

If E/ε is 1, there are N microstates, since 1 of N positions is at energy ε. If E/ε is 2, there are N(N-1) microstates. etc. etc, giving for E/ε = x that there are N!/(N-x)!

so S = ln [N!/(N-x)!] = ln(N!) - ln((N-x)!) = NlnN - (N-x)ln(N-x)

S(E) = N ln N - (N - E/ε) ln (N - E/ε)

Can you explain how you got your equation for the entropy?

Going on I get E(T) = ε(N - e^(ε/T - 1) )

This also looks wrong, as although E → ∞ as T → ∞, it also doesn't cap at exactly εN, and E → -∞ for T→ 0...

I'm expecting the answer to look something like: E(T) = εN(1 - e^(-ε/T))/2 which ranges from 0 to εN/2, which seems sensible.

EDIT: Nevermind, the answer was posted while I was writing this. I'd still like to know how you got your S(E) though.

We could really use a new Aral sea, but intuitively I'd expected that this would be a tiny dent in the depth of the oceans. So, to the maths:

Wikipedia claims that from 1960 to 1998 the volume of the Aral sea dropped from its 1960 amount of 1,100 km^3 by 80%.

I'm going to give that another 5% for more loss since then, as the South Aral Sea has now lost its eastern half enitrely.

This gives ~1100 * .85 = 935km^3 of water that we're looking to replace.

The Earth is ~500m km^2 in surface area, approx. 70% of which is water = 350m km^2 in water.

935 km^3 over an area of 350m km^2 comes to a depth of 2.6 mm.

This is massively larger that I would have predicted, and it gets better. The current salinity of the Aral Sea is 100g/l which is way higher than that of seawater at 35g/l, so we could pretty much pump the water straight in still with net environmental gain. Infact this is a solution to the crisis that has been previously proposed, although it looks like most people would rather dilute the seawater first.

To acheive the desired result of 1 inch drop in sea level, we only need to find 9 equivalent projects around the world. Sadly, the only other one I know of is Lake Chad, which is significantly smaller than the Aral Sea. However, since the loss of the Aral Sea is due to over-intensive use of the water for farming, the gives us an idea of how much water can be contained onland in plants: I would expect that we might be able to get this amount again if we undertook a desalination/irrigation program in the Sahara.

Who are we expecting to have buried things there? I can come up with 6 possibilites, is there another you were thinking of?

Modern humans. In this most likely case it's probably not interesting, maybe some Propaganda Preservation Program from the Cold War.

Recent aliens. I would expect if any aliens were about to jaunt over, notice our space-faring potential and bury a cache for us to discover to mark our readiness to join the Galatic Federation, we would have probably noticed them in other ways by now.

Ancient aliens. Why would visitors before intelligent terrestrial life think it worthwhile to bury stuff just in case we evolved? You've got to have a whole lot of faith in your civilization's stability to think that leaving tags everywhere is a better strategy for continuity than just colonising.

Ancient, non-human but earthbound civilization - Silurians. I could believe that another society might do this, and I think this is who the grandparent is suggesting we aim for - but since we're speculating over geological times the location of the poles is quite variable. Unless we have a fair idea of when the sender lived we don't know where to look, and to find out when they lived we'd need to find the cache... Or you could be saying "hmm, those big extinction events kind of look like the one we're causing now, I wonder where the poles were at those times?"

Some recent but forgotten technological human civilization - Atlantis. Maybe, but like the recent aliens I would expect there would be other signs.

The whole of human history is a lie! - Hiigarans. Fun times.

I don't think it's worth specifically scouting around for something, but maybe if we're buying anyway and it's cheap it'd be worth checking.

Goldbach's conjecture is "Every even integer above four is the sum of two primes," surely?

Also, Gödel's incompleteness theorem states that there are theorems which are true but non-provable, so you get something like:

P(X) = (P(will be proven(X)) + P(is true but unprovable(X))) / (P(will be proven(X)) + P(will be disproven(X)) + P(is true but unprovable(X)))

Is there a reason to suspect that a counterexample wouldn't be a very large number that hasn't been considered yet? Consider sublime numbers: the first (12) is a number which will be checked by any search process, but there is another which has 76 digits and, I would suspect, could be missed by some searches.