Blimix

"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?

I was listening to one of my 700 or so favorite songs recently, when it struck me that, while this song was musically a force to be reckoned with, the lyrics made it something of a douchebag anthem. See how quickly you can figure out the song from my paraphrasing.

Verse 1:
I have plans tonight for my significant other, whom I refer to by an insultingly diminutive term of endearment. I must keep these plans secret from her parents, presumably to avoid being thwarted by their inevitable disapproval. The questionable ethicality of this is ameliorated by our respective, uncommonly good, appearances.
I interrupt my initial plan of getting us both drunk at a small but charming tavern because I disparage the entire genre of music that suits the proprietor's musical tastes, and I insist that my significant other drop everything to leave with me immediately.

Refrain:
Our behavior toward, and within, this municipality shall be so egregious that I must describe it with a metaphor that would be fatal if it were anatomically possible. It shall be so egregious that the residents will object vociferously, but we will persist until we explode and lose consciousness.

Verse 2:
We delightedly engage in a decades old version of East Coast Swing dancing. I look with derision upon another patron's outdated fashion sense. He notices me, so I threaten him with physical violence. As if I had to emphasize this, our enjoyment of this evening will be characterized by massive destruction. (It appears that my increasingly inebriated state softens my understanding of this as being purely metaphorical.)

[Repeat refrain.]

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. )

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. )

A good president...

me aygyj nmeqjkzqyn pc "tkzqe".
fjdtmqe tjdb hkj zjmbye.
me kpdgy qsy xkh.
ayyne ad jykeda.
jmle qsy pkxxdq.
yezkfye akqrjkx eyxyzqmda.
ykqe pkpmye.