Monthly Archive for May, 2013

I Too Have Riddled Boxcars Boxcars Boxcars

Okay, Tuesday’s boxcar problem has gotten pretty interesting. I thought I knew the answer, but the comments on Wednesday’s followup post have sowed major seeds of doubt. There are a lot of excellent comments there.

I am thankful that I acknowledged in advance (at the bottom of Wednesday’s post) that I’m less sure of this one than I am of many others. I’d cheerfully bet $1000 (subject to agreement on a suitable referee) that I’m right about this relativity puzzle. (My answer is here.) And as far this old chestnut goes, my answer is here and I hereby cheerfully renew my offer to bet up to $15,000 on the outcome of a computer simulation. (Or any other amount, as long as it’s over $1000, to make this worthwhile.) Email me if you’re interested.

(This is on my mind because I’ve just had a very unpleasant encounter with a troll in another venue, who, like other trolls, is happy to bluster but runs away when you offer to put money on the line.)

For the first time ever, I am turning off comments on this post, because I don’t want to dilute yesterday’s interesting discussion by allowing it to take place half over there and half over here. Go there to participate. Many thanks to the commenters who have forced me to think harder about this, and thanks to anyone else who can help resolve the controversy.

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About That Boxcar

Yesterday’s puzzle was this: A boxcar filled with water sits on a frictionless train track. A mouse gnaws a small hole in the bottom of the boxcar, near what we’ll call the right-hand end. What happens to the boxcar?

(Spoiler warning!)

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Boxcar Willie

I’ve just been pointed to this notice of a conference in honor of the topologist Tom Goodwillie‘s 60th birthday.

This reminded me of several things, not all of them related to the relentless march of time.

For example, once a very long time ago (though it sure doesn’t seem that way) Tom asked me a simple physics question that troubled me far more than I now think it ought to have:

A boxcar full of water sits on a frictionless train track. A mouse gnaws a hole through the bottom of the boxcar, in the location indicated here:

The water, of course, comes gushing out. What happens to the boxcar?

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News From The Math World

Big news from the math world:

One of the oldest problems in number theory is the twin primes problem: Are there or are there not infinitely many ways to write the number 2 as a difference of two primes? You can, for example, write 2 = 5 -3, or 2 = 7 – 5, or 2 = 13 – 11. Does or does not this list go on forever? There are very strong reasons to believe the answer is yes, but many a great mathematician has tried and failed to find a proof.

Here’s a related problem: Are there or are there not infinitely many ways to write the number 4 as a difference of two primes? What about the number 6? Or 8? Or any even number you care to think about? It seems likely that the answer is yes in every case, though no proof is known in any case. But….

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What’s a Bitcoin Worth? (Wonkish)

This might be one of those questions I’ll eventually be embarrassed for asking, but…..

Imagine a future in which Bitcoins (or some other non-governmental currency) are widely accepted and easily substitutable for dollars, at an exchange rate of (say) $X per Bitcoin.

Then if there are M dollars and B bitcoins in circulation, the money supply (measured in dollars) is effectively M + X B .

Money demand is presumably P D, where P is the general price level and D depends on things like the volume of transactions and the payment habits of the community. (If it helps, we can write D = T/V where T is the volume of transactions and V is the velocity of money.)

Equilibrium in the money market requires that supply equals demand, so

M + X B = P D

Now M is determined by the monetary authorities; B is determined by the Bitcoin algorithm, and D, as noted above, is determined outside the money market.

That leaves me with two variables (X and P) but only one equation. What pins down the values of these variables?

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Seven Trees in One

When you met the late Armen Alchian on the street, he used to greet you not with “Hello” or “How ya doin’?”, but with “What did you learn today?” Today I learned that there are contexts in which the most ludicrous reasoning is guaranteed to lead you to a correct conclusion. This is too cool not to share.

But first a little context. The first part is a little less cool, but it’s still fun and it will only take a minute.

First, I have to tell you what a tree is. A tree is something that has a root, and then either zero or two branches growing out of that root, and then either zero or two branches (a “left branch” and a “right branch”) growing out of each branch end, and then either zero or two branches growing out of each of those branch ends, and so on. Here are some trees. (The little red dots are the branch ends and the big black dot is the root; these trees grow upside down.)

A pair of trees is, as you might guess, two trees — a first and a second.

There are infinitely many trees, and infinitely many pairs of trees, and purely abstract considerations tell us that there’s got to be a one-one correspondence between these two infinities. But what if we ask for a simple, easily describable one-one correspondence? Well, here’s an attempt: Starting with a pair of trees, you can create a single tree by creating a root with two branches, and then sticking the two trees from the pair onto the ends of the two branches. Like so:

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To Hold You Over….

Sorry to have been so silent this week; various deadlines have kept me away from this corner of the Internet. I’ll be back in force next week for sure. Meanwhile, if you’re looking for some good reading, this is the best thing I’ve seen all morning.

Edited to add: “Best all morning” was not intended as damning-by-faint-praise. It’s actually the best of many mornings.

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