Saturday, June 21, 2014

Puzzle: Gambler’s ratio

There is a rather dark bar called “Der Brocken”, near the “Mystery Spot” in Santa Cruz. Last Friday night as I went by I noticed a strange shadow cast by the bar on the surrounding fog. I went in and ordered some fish, which the barman assured me was second grade fresh. As I was nursing a masterly margarita, a strange person materialized next to me, with one dead grey eye, one piercing yellow eye and very arch eyebrows. He introduced himself as Wolande. Wolande was rubbing his knee with one hand, and with the other, he was playing with a rather odd-shaped coin, which looked like an asymmetric double convex lens.

From what I could see, on one side it had the image of a head on a platter, and on the other, words that began with, “There is a rather dark bar called …”. At times the coin itself appeared to be the platter, with the head presented on it. As Wolande tossed the coin in the air, it not only tumbled but also seemed to shrink as it rose and expand as it fell, vanishing entirely sometimes as it moved upwards and then after a short time reappearing moving downwards at the same height. As I looked at it I had an odd sensation of imbalance, as if I was tilted and should fall out of this world, but there wasn’t any particular direction I was leaning towards.
Wolande asked me if I was a gambler, and when I said I was, he offered me a wager. When I asked if it was fair, Wolande replied cryptically that I could make it so. So without thinking, I accepted – yes, yes,  jumping in headfirst where angels fear to tread and a fool and his money … . Wolande explained: “So it is on. I’ll wager a thousand prutot, no, thalers -- wait, nowadays you call them … dollars, don’t you? --  on the toss of this coin, if it lands tail up, you win the thousand dollars, if it lands head up you lose the amount you wager.”
“Is it a fair coin?”, I asked, “It doesn’t look it.”
“Oh what an enchanting question!” Wolande replied, “Of course not! But, this much I can tell you, the odds against your winning are numerically the same as your probability of winning. Now, you can wager what you want, and if it is fair to me I’ll accept it. I’ll toss in the respect and credibility of your colleagues on the 3rd floor, and then we’ll toss the coin.”
A sense of premonition hit me as Wolande continued, “But if your wager is not fair to me … I’ll keep your soul.”
1. How much would you have wagered, in dollars, correct to a cent?                                   
2. What book inspired this puzzle?


Anonymous said...

$1618.04 ??
G. E. B.

Ranjeet said...

Yup, Golden Ratio in thousands of dollars, rounded UP to be fair to Wolande!
"GEB"? Good guess, based on the self-reference I suppose. I thought that you would be a Hofstadter fan from what our common friend has told me about you. However the character Wolande with the mismatched eyes, the eyebrows and the gimpy knee, as well as the reference to "second grade fresh" fish is from "Master and Margarita" by Bulgakov.
The behavior of the coin can be explained as that of a 4D object that Wolande is tossing in 4D under the action of an oblique gravitational field.