Blimix

Dragon math

2022-10-11T19:38:59Z

"Oh, well enough, I suppose." The dragon sipped their cinnamon tea.

"Uh huh." The witch raised an incredulous eyebrow. "There's something bothering you."

The dragon sighed. "I've been finding things around my lair, since I got back from vacation. Bits of tin, and sticky notes with rude pictures and drawings."

The witch nodded. "Fairies."

"That's what I thought. The problem is that they've hidden them so I can't ever find them all. The first day, I couldn't get comfortable on my pile of gold. Eventually, I discovered a tin coin in it."

The witch scowled with indignation. "They didn't!"

"They did. I found half a coin in my squash soup the second day. The third day, a quarter of a tin coin turned out to be jamming my LaserDisc player. I swept up an eighth of a coin from a lava tube the next day. And so on."

"This can't keep going indefinitely, can it?" The witch popped half a hot Brussels sprout in her mouth.

The dragon shook their head. "It can, and I suspect it will. Dragons live forever. We have a close relationship with infinities."

The witch swallowed. "Huh. How will you deal with infinite age?"

"I'll figure that out when I get there." The dragon smirked slightly.

The witch chuckled. "Fair." She leaned back. "If you keep finding half the coinage you did before, that'll make two whole coins. At least, the total will approach two as time approaches infinity."

The dragon nodded. "Yes, my gut told me that even over infinite time, this would still only be a finite amount of tin. I have a feel for infinities, and I could tell that this wasn't one. Only..."

The witch leaned forward in interest.

The dragon frowned. "The notes. I found one stuck on the wall the first day: A drawing of a dragon with chicken wings and reindeer horns. Two days later, I found a note in the cutlery drawer with a toothless, googly eyed face captioned, 'Fierce dwagon. Oh no!' Four days after that, there was a note under the soap by the hot spring. It said-" The dragon's eyes closed and their voice caught.

The witch reached over and put a hand between the dragon's shoulders. "It's okay, you can tell me."

The dragon choked. "It said, 'Your hoard demonstrates an inexpert and undiscerning interest.'"

"Oh, fairies can be cruel."

"It really got to me. I know it shouldn't — they were just trying to hurt me — but it still did. I feel ridiculous for taking it personally!"

"Your feelings are valid. Those were hurtful words, and you're allowed to feel hurt by them." The witch rubbed the dragon's back.

The dragon broke the silence after a few minutes. "The whole thing is confusing. I found the next note eight days later. Then sixteen days after that. You can see where this is going."

"Indeed." The witch picked up her bubble tea and sipped.

"So the notes are getting more scarce, a lot like the remaining amount of tin coin is. Or, they should be, but they aren't quite."

The witch pondered. "Oh dear."

"Exactly. So here's the thing. My gut tells me that the notes are infinite. So they must be. But I don't get why." The dragon arranged the Brussels sprouts on their plate in decreasing size. "Each new day, I find half as much additional coin. Whereas each new note takes twice as many additional days." The dragon started using finger quote gestures. "If 'additional days per additional note' is getting doubled, that's the same as if 'additional note per additional days' is getting halved. Which is exactly how we'd describe the coins! 'Additional coins per additional day' keeps getting halved. So how is one of them finite, and the other infinite?

The witch closed her eyes. "I can tell you the formula for the notes, and your gut is right: It doesn't converge to a finite amount. But that won't answer your question: If the notes and the coins follow the same pattern of halving over time, why don't they add up the same way? Hmm." She walked to the hanging basket. "This calls for half a recently picked lime."

The dragon tilted their head. "Are you using witchcraft to solve this?"

"Yes, I am." The witch cut the lime and squeezed it onto her plate. She smelled the rind, then dipped half a Brussels sprout in the puddle and ate it. "Mmmm." She chewed, eyes closed, with a growing smile. One second after she swallowed, her eyes flew open. "Oh! That's it! Here's what makes the decreasing notes materially different from the decreasing coins..."

What did she tell the dragon?

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The hidden solution

2020-04-24T00:31:42Z

I've had an interesting encounter with a logic puzzle.

Raymond Smullyan's book What is the Name of This Book? includes many puzzles with the premise of an island populated by knights, knaves, and normals. Knights always speak the truth, knaves always speak falsely, and normals sometimes do either.

A typical (if easy) example is problem 103: At a trial, you (an inhabitant of this island) must make a statement proving that you are a normal.

There are many possible solutions, but a clear and elegant one is, "I am a knave." Neither a knight nor a knave could say this, so you must be a normal.

Unstated but critical in the solution is the idea of "proof by contradiction" employed by the jury. They consider the possibility that you are a knight, but this quickly leads to the conclusions "You always speak the truth" and "Your statement is a lie." This is a contradiction; therefore, its premise (you are a knight) must be discarded as false. Similarly, the premise that you are a knave leads to a contradiction: You always speak falsely, but you just made a true statement. So they know you're not a knave. The premise that you are a normal is the only one that does not lead to a contradiction. Therefore, they know that you must be a normal. (I apologize if this is obvious. It'll be important shortly.)

Problem 106 inverts this: How many statements would it take to convince the king that you are not a normal? (Answer for both a knight and a knave.)

In case you want to think about that for a while, I'll leave some spoiler space.

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Smullyan helpfully provides answers to his riddles, and for this one, he simply says: It can't be done. Even if you're truly not a normal, any statements you make to try to prove this could just as easily be uttered by a normal (who after all can say whatever they please). You can't prove you're not a normal.

That was not my solution.

My solution for the knight was a single statement, "I am a knight or this sentence is false."
A knave could instead state, "I am not a knave and this sentence is false."

In both cases, the premise that the speaker of this statement is a normal leads to a contradiction, regardless of whether the statement is assumed to be true or false.

Could a normal utter it anyway? Maybe. It is unclear in the book whether normals are free to tell only truths and falsehoods, or whether they are empowered to string together any words they wish, including self-contradictory statements. (Normals "sometimes lie and sometimes tell the truth". Can they also do other things? (Everyone, please no nitpicking about "lies" versus "falsehoods". It's beside the point.)) However, even if we generously grant that normals can say anything at all, I will note that the challenge was not to make a statement that a normal could not make, but rather to convince the king that you are not a normal.

The king, like the jury, employs logic with ruthless efficiency. Nowhere is it implied that we might count on the king to make a logical blunder, such as entertaining a premise that leads to a contradiction. That would undermine the entire challenge. Since you have just made a statement which results in a contradiction if you are anything but a knight, he must conclude that you are a knight, or he must abandon logic. (The knave's answer is just as powerful.)

Mr. Smullyan, I am prepared to claim from your ghost the five and a third Internet Points which I have just won in fair combat. (Also, I loved your book.)

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p.s. On the same piece of paper where I had written my solution in a dentist's waiting room last year, I had also noted, with amusement, the following:

"This sentence is false" is contradictory. But "I am a knave" is not!

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Decaffeinating tea

2019-07-04T11:49:10Z

In response to my post about getting to sleep, someone has kindly pointed out that the common wisdom regarding water decaffeination of tea is bunk. (My wife had heard, and told me, that the first minute of steeping removes 90% of the caffeine, which is clearly quite wrong.)

This page has tables showing the caffeine found in first, second, and third brewings of various kinds of tea, where each steep is three minutes long.

Normally, Assam (our tea of choice) wants to steep for four minutes. However, my wife decaffeinates her Assam for one minute, then steeps it for six minutes (to make up for the flavor loss) before we make second pot tea (another six minute steep). The tl;dr of the following is that my second pot tea appears to contain 5.3% of the caffeine of fully caffeinated first pot tea.

Dr. Branan's test has the first pair of three-minute steeps showing a 70% reduction in caffeine per serving (86.3 mg reduced to 25.75 mg), and the second pair showing a 67% reduction (25.57 mg down to 8.5 mg). Using the latter ratio (as it involves the time frame we're concerned with), we can estimate that each minute takes 31% of the caffeine remaining in the tea leaves out into the tea. Summing the series gives us 38.44 mg of caffeine remaining in the leaves at the three minute mark. So the caffeine in the leaves at the seven-minute mark should be 5.87 mg, and at the thirteen-minute mark should be 0.64 mg. The difference (what's in my second pot tea) is 5.23 mg of caffeine.

By a similar calculation, if the initial three-minute test had been the standard four minutes, it would have contained 98.17 mg. My second pot tea thus appears to contain 5.3% of the caffeine of fully caffeinated first pot tea.

The one minute mark, interpreting geometrically from the ratio of the caffeine in the first and second pots, should leave 86.09 mg of caffeine in the leaves, of which 80.22 mg comes out in the first six-minute steep. This is 81.7% of the caffeine found in four-minute, first pot tea.

However, both of these figures are really lower than shown by an unknown amount, because the initial minute of decaffeination involves vigorous stirring to increase the dissolution of caffeine. We have no data on this effect.

(Note that I used more significant figures in my calculations than I show here, so final digits might appear to be off by one. This doesn't matter at all, because the experimental margin of error has got to be way too high to justify even the significant figures that I show.)

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Adventures and accomplishments at work

2018-12-15T00:34:58Z

( Lots of cool stuff I've done at work. Cut for length. )

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Hedging and Bitcoin: Not a great combination?

2017-12-14T18:55:27Z

I've been thinking lately about possible hedges against the collapse of the dollar, and in particular about Bitcoin. Please note that nothing I say here constitutes expert opinion. These are my own inferences and concerns, based on very occasional investigations.

I used to think that gold was a good hedge, until six years ago. The price of gold was rising until the Federal Reserve and the European central banks brought it back down. Apparently, it's been suspected for an awfully long time — and confirmed since 2004 — that they work hard (through covert selling and adjusting interest rates) to keep the price of gold down, in order to protect fiat currencies, and also to protect bullion banks that have been "borrowing" and selling gold (effectively short-selling it).

All right, so screw that. If the dollar sinks and the Fed makes gold sink with it, that's not a hedge.

I looked into Bitcoin last year, and it was looking like a great investment. [A personal aside starts here. If you don't want to see me vent, you can safely skip this paragraph and the following asterisk.] Technical issues kept piling up and preventing me from buying, at least in a manner that felt safe. (There were "give us your banking password" options, but people can and do get their accounts cleaned out by sharing their banking passwords.) I got past most but not all of the issues, which took months.* Then I got sick, and ran out of cope. I figured I'd dive back into it when I felt better. I feel better now. The price was below $4,000 when I was trying to buy it, and is above $16,000 now. Given the amount I was planning to invest, the opportunity I lost represented (what is for me) a very large amount of money.

* This involved changing my (obsolete but very fast) operating system, but there were also server-side problems. CoinBase neglected their increasingly buggy web interface in favor of their app, and I don't trust the security of smart phones. Other bitcoin exchanges allowed me to go through the entire sign-up process before revealing that they don't do business in New York. The shady one that takes credit cards with no proof of identity had higher fees, which I was willing to pay, but then the options for disposable credit cards I looked into (to protect myself from fraud) wound up either having exorbitant fees or being unavailable. I finally got signed up at Coinbase, but they wouldn't let me engage in transactions until I sent a picture of my driver's license — which their system wouldn't accept, for no stated reason.

You'd think I could still get into Bitcoin now, and expect to at least make some money and/or use it as a hedge, right? Not so fast. One of the touted reasons that Bitcoin could conceivably reach $1,000,000 involved estimating its demand as currency, and knowing the hard-coded limits on its supply.

The Bitcoin system as a whole consumes a tremendous amount of energy. (Thanks, Colin!) Much of that is used by Bitcoin mining operations that I expect will taper off as the remaining Bitcoins become too scarce/hard to mine profitably. However: "Number of U.S. households powered for 1 day by the electricity consumed for a single transaction: 7.93" This seems to mean 234 kWh of electricity. The biggest countries seem to have electric rates of 8 to 12 cents per kWh, so we'll call it $23.40 per transaction, a cost that is presumably getting eaten collectively by the bitcoin nodes and exchanges.

I shit you not, I wrote the above before looking up the following. The Recommended Bitcoin Transaction Fee for getting a transaction to happen within the next hour is, as I write this, an amount of Bitcoin worth $23.928. (It'll be more or less depending on what priority/urgency you seek, of course.)

So, what we're looking at is a system that, even aside from environmental costs, is only sustainable as long as people are willing to pay $23.40 transaction fees. The system currently offers $9 to $24 fees. (Well, user-chosen fees, really, but that's the practical range. If you choose too low a fee, your transaction just might not get processed at all: Its priority will be too low, and it might even look like a spam transaction.) With the lower fees, the system as a whole is eating the electric cost, but individual node operators probably don't care, perhaps because most transactions use the higher fees, or because they're making more than enough money by mining Bitcoins. I don't know. (Some exchanges might even just be happy to be taking in money as people continue to buy more than they sell, regardless of sustainability. This would be wonderful for them until there's a run on the bank.)

Still, let's compare these fees to other transaction fees, which (whether absorbed by buyer or seller) often work out to somwhere around 3%. Even if the transaction fees become (or stay) high enough (on average) to cover the electric costs, a 3%, $23.40 average means average transactions have to be (at least) $780 for the fee to stay competitive. For the last few years, average Bitcoin transaction values have mostly been hovering at about 10 bitcoins, which is way more than enough.

However, if Bitcoin becomes a viable currency that you can use for all your everyday transactions, the median transaction is going to be roughly the price of a fast food meal. The only way this is sustainable is if there are enough large transactions to make up for this in the average, and if transaction fees become percentages rather than flat rates so that large transactions can support small ones. (I'm not paying $23 for the privilege of spending $5 at a hardware store.)

The first of those requirements cannot be met, I think. Statistics on wealth in the U.S. show families having a mean net worth of $342,300, and a median of $85,060. That's a 4:1 ratio, and even rich people still have to make small purchases, so the ratio of mean to median transactions in the full economy must be less than that.

The second requirement also seems unlikely: Electricity costs for blockchain nodes are per transaction, not per value. They have no financial incentive (that I know of) to subsidize small transactions at the cost of large ones. (It might also invite users to start gaming the system, for all I know. Arbitrage between subsidized transactions and a credit card with a lucrative reward system might even be possible, but I'm not going to spend the time and energy to figure that out now.)

Between these two concerns, I think the best scenario for Bitcoin's long-term value (though still not great for the environment) is one in which a Bitcoin economy develops for transactions above $800, but people still use cash and credit cards for their day-to-day business. Even this scenario utterly cripples the forecasts that Bitcoin will be used enough to reach $1 million on the strength of its utility as a currency.

This doesn't impugn the other reasons that Bitcoin value might get that high, but it implies (to my mind) that it won't stay that high, if it gets there at all. What we're looking at is people buying into Bitcoin (and thus raising its price) as a means to profit, and as a hedge against the dollar collapsing. Both of these rely on continued general faith in the value of Bitcoin, rather than its actual utility.

So unless I've missed something big (which I might well have! Not an expert, remember?), I'd expect Bitcoin to keep rising (especially as people continue to lose faith in the dollar), get huge, and suddenly crash the moment that a random drop in price scares people enough.

What's "huge"? $20,000? $5 million? I don't know. But it feels like a heck of a gamble now. Like deliberately investing in the early part of a Ponzi scheme (people do that), reaping the reward, and getting out before the whole thing collapses. But the potential "reward" in this case is gigantic, as the scheme is taking a long time to collapse. It still likely has a while to go (especially in light of the looting of the U.S. economy in progress), so there's plenty of money to be made, but please be aware of the risk and don't bet what you cannot afford to lose.

While Bitcoin might still be a decent gamble, I no longer consider it a good long-term hedge. I'd love to find one, so that I don't risk having to spend my entire savings on a loaf of bread in a few years. Any suggestions?

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Difference between natural language and formal logic rules.

2016-01-26T00:10:38Z

I believe that the confusion that some students experience with conditional statements in math and formal logic courses, over the idea that one cannot assume the inverse/converse of a conditional, stems from their intuitive understanding of Grice's maxim of Quantity. I think it would be most helpful, upon the introduction of this subject, for the teacher (or the course material) to explain roughly as follows:

Conditional statements take the form of "If something, then something else." Like, "If it's raining, then I wear a hat." Let's say I told you, "If anyone's around, I don't pick my nose." So what do you think I do when I'm alone? [Pause.] I pick my nose, right! But do you really know that? If I never picked my nose, my statement would still be true, right? [Longer pause, possibly repeating the original statement for consideration.] You're right: If I never picked my nose, I wouldn't have said it like that. I would have just said, "I don't pick my nose." What we just demonstrated is a linguistic rule called Grice's maxim of Quantity, which states that I don't tack on phrases like "If anyone's around," unless I actually have a reason to do so. I said it, so you figured I had a reason for saying it, and that's how you knew that I pick my nose when nobody's around. Incidentally, the statement "If nobody's around, I pick my nose" is the inverse of my original statement, "If anyone's around, I don't pick my nose." You get the inverse just by negating both parts of the statement.

The maxim of Quantity, the rule that you used to figure out the inverse, is not a rule in formal logic! It's just a rule of natural language. In formal logic, I could say, "If anyone's around, I don't pick my nose," and you would have no way of knowing what I do if nobody's around. Unlike with natural language, when you're given a conditional statement in formal logic, that's not enough information to know whether the inverse of that statement is true. That should be obvious with the statement, "If it's raining, then I wear a hat." Can you conclude the inverse, that if it's not raining, then I don't wear a hat? Of course not! Maybe I'll also wear a hat to keep the sun off my face, or just because it's stylish. So we can't assume the inverse.

Now let's look at the contrapositive, which is just a fancy name for a concept that you already understand, as illustrated by the famous quote by Dan Quayle, "If Al Gore invented the Internet, then I invented spell check..."

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Puzzle Answers (sane and less sane)

2013-02-25T22:48:08Z

I posted puzzles about Rock-Paper-Scissors on Thursday. One person got both answers, in the comments on DreamWidth, and you can check that out if you want the short version. I'll throw in extra thoughts here, including an idea of how even the "right" answer might become problematic.

( Solutions and rambling. Simultaneous equations, and an abortive look at the Prisoner's Dilemma. )

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Puzzles with Rock-Paper-Scissors

2013-02-21T22:44:51Z

Today, I thought of some variants on an otherwise trivial puzzle.

This will be Rock-Paper-Scissors, as played by a mathematician and/or game theorist, rather than (as it is played in real life) a psychological contest of reading your opponent's intentions.

In these puzzles, we will posit that your opponent knows your strategy, and plays optimally for their own benefit. (Assume each player cares only about their own income. No generosity and no spite for the other player.) So if your strategy is to play Rock, your opponent will play Paper. Your saving grace is that you are allowed to use a random ("mixed") strategy.

I hope that even those among us who are not mathematically inclined can intuit that the ideal strategy is to pick randomly from Rock, Paper, and Scissors, with equal chances of each.

But what happens if we throw off the symmetry of the game, not by changing the system, but by introducing different rewards for different wins?

Problem 1: Good old Rock. The payout from the loser to the winner is as follows: $9 if the winner throws Rock; $3 if the winner throws Scissors; $1 if the winner throws Paper.

Now you will lose by keeping the probabilities equal: Your opponent would throw Rock and count on winning much more to Scissors than they would lose to Paper. And you couldn't just throw Paper in anticipation of them throwing Rock, because they get to know your strategy, and they would just throw Scissors instead.

What strategy can you employ, that your opponent can't beat even if they know what it is?

( Basic hint from game theory. )

Problem 2: On the House. The payouts are the same as in problem 1, but instead of coming from the loser, they come from the bank. Does your strategy change, and if so, how?

I won't screen comments, so beware of spoilers if you read them. I'll post my answers in a day or two.

( Cut for white space. )

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Logic puzzles for you!

2012-09-28T07:09:17Z

I wrote a puzzle today, and in correcting it, I discovered that it lends itself to a series of similar puzzles with intriguing variations. Have a crack at them yourself, and expect to see some spoilers if you view comments. I know that some of you won't be able to resist this.

Puzzle 1:

The sphinx crossed a secret chamber deep in the castle's basement, trying to keep a level head and ignore the sounds of battle coming from outside. Her friends would survive only if she could quickly find and disable their enemy's source of power. There was, she had been informed, only one path to that source. As she approached the three doors, each one in turn formed a mouth and spoke, then went silent and became again a plain, wooden door with a number on it.

The first door said, "Door two speaks truly and leads to doom."
The second door said, "Door three speaks truly and leads to power."
The third door said, "Door one lies and leads to power."

An inscription above them read, "At least one door speaks the truth."

( Puzzles two through four are behind the cut. )

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