December 2003 - mathfun@mailman.xmission.com

Re: [math-fun] Hot and cold & The separate taps
by asimovd＠aol.com 05 Dec '03

05 Dec '03

I should clarify that I'm thinking of two handles to adjust the flow of hot and cold water separately, but only one faucet through which the combined water flows. --Dan << I find having separate hot-flux and cold-flux adjusters easier to use than the hardware I've seen based on overall temperature and overall flux. >>

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RE: [math-fun] 7.11 puzzle
by Schroeppel, Richard 05 Dec '03

05 Dec '03

Probably still legal tender for small debts, but I doubt 7-11 would take them. But it raises an interesting generalization: Prices are frequently "2 for 1.29". Suppose we ignore the custom of rounding up (.645 -> .65), and allow (positive) rationals in the solution. It's not clear to me that the set of solutions is still finite, and my enumeration methods would only work if the denominators are bounded. Rich -----Original Message----- From: R. William Gosper To: math-fun(a)mailman.xmission.com Sent: 12/4/2003 11:19 PM Subject: Re: [math-fun] 7.11 puzzle Are half-cents still legal tender?-) 1.125, 1.28, 3.125, 1.58 _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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RE: [math-fun] re: The Cooler & math fun at the casino
by Andy Latto 05 Dec '03

05 Dec '03

> -----Original Message----- > From: Joshua Singer > Sent: Thursday, December 04, 2003 4:58 PM > To: math-fun > Subject: Re: [math-fun] re: The Cooler & math fun at the casino > > This is the only point I was making in my initial response. > When S draws a card, > it certainly doesn't affect > A's expected return on the round in question. True. But it can affect the expected return in the following round. > > The argument that S by his play makes a hot deck end sooner > seems valid, but I'd > need to see a careful argument, All you really need to prove that the other player can hurt A's chances is an example situation where this is the case. Let's suppose that there are 2 cards left in the deck, both red. Without the other player, A gets 2 guaranteed wins against the house. If the shill plays the next hand, A gets only one guaranteed win. So it's pretty clearly in the house's favor to have the shill play this hand. While this is theoretically interesting, I don't believe this is actually used as a countermeasure by casinos in the real world. In a hand-held game, you can achieve the same effect by having the dealer count, and shuffle early if the deck becomes favorable to the player. This technique, known as "preferential shuffling" is used by casinos. In a shoe game, where the time to reshuffle is fixed in advance (by inserting a plastic "cut card" into the shoe, and shuffling when this point in the shoe is reached), the house could profit by having a shill who played when the deck was favorable. But this uses up a seat at the table, and is an extra salary to pay. Doing this at every high-stakes table would be overkill. And if you're going to find the advantage players, and use this countermeasure only at their tables, you have to detect the skilled players. And once you've done that, why not just bar them? > > Here's an interesting related problem: Consider the > following version of > red/black: you post one dollar up front, > and then the dealer will start rolling the deck one card at a > time until you tell > him to stop. At that point, the house pays 1-to-1 if the > *next* card is red; > otherwise, you lose. Surprisingly, this is a fair game. > That is, you cannot make money at it. The easy way to see this is true: When you say to stop, the order of the remaining cards in the deck is random. So it wold be the same expectation if, when you placed the bet, the dealer dealt the next card from the bottom instead of the top. Now the game is "you win if the bottom card is red, and lose if it's black, but before you make the bet, you can look at cards from the top of the deck, as many as you like". As long as you are ultimately required to bet on the bottom card every round, it makes no difference whether or how many cards you look at from the top of the deck. Andy Latto andy.latto(a)pobox.com >

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RE: [math-fun] Hot and cold & The separate taps
by Schroeppel, Richard 05 Dec '03

05 Dec '03

This might make a good example of a coordinate conversion problem. Rich -----Original Message----- From: Thane Plambeck To: math-fun(a)mailman.xmission.com Sent: 12/5/2003 1:19 PM Subject: [math-fun] Hot and cold & The separate taps When I lived in England I considered the separate taps for hot and cold water a considerable hardship, right up there with warm beer and brussels sprouts. Content from "Old Fashioned Faucets: Unique British Standard" http://homes.wsj.com/housegarden/indoorliving/20021031-hagerty.html Boris Johnson, a Conservative Party member of Parliament representing Henley, congratulates "the higher civilizations" that have adopted advanced plumbing technology. But he argues that having the choice of either hot or cold for washing hands "is an incentive to get it over and done with and not waste water." Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Hot and cold & The separate taps
by Thane Plambeck 05 Dec '03

05 Dec '03

When I lived in England I considered the separate taps for hot and cold water a considerable hardship, right up there with warm beer and brussels sprouts. Content from "Old Fashioned Faucets: Unique British Standard" http://homes.wsj.com/housegarden/indoorliving/20021031-hagerty.html Boris Johnson, a Conservative Party member of Parliament representing Henley, congratulates "the higher civilizations" that have adopted advanced plumbing technology. But he argues that having the choice of either hot or cold for washing hands "is an incentive to get it over and done with and not waste water." Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm

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[math-fun] re: The Cooler & math fun at the casino
by Dave Dyer 05 Dec '03

05 Dec '03

Inspired by the current film "The Cooler" (where the title character is employed to cool off hot streaks of gamblers) In casino games where each player plays against the house, are there any mathematically sound strategies that a player working for the house could employ to disadvantage other players? Assume that the winnings or losses of such players are unimportant because it all goes back to the house.

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RE: [math-fun] Hot & Cold
by asimovd＠aol.com 05 Dec '03

05 Dec '03

Marc writes: << we could score higher on the Martian intelligence test by universally adopting saner basis vectors for the control space in the first place--namely temperature and total water volume, rather than hot-volume and cold-volume. >> I find having separate hot-flux and cold-flux adjusters easier to use than the hardware I've seen based on overall temperature and overall flux. This is especially true for adusting temperature. Perhaps it's a little like when you want to quietly close a door (which normally makes a loud noise when being closed), so you pull it against your foot while the foot pushes it closed gently. --Dan

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Re: [math-fun] Re: Hot & Cold
by asimovd＠aol.com 05 Dec '03

05 Dec '03

Can someone please provide the most up-to-date definition of a^b where a and b are surreal numbers? (I hope this coincides with a^b = exp(b * log_e(a)) for a > 0, where exp is given by the usual Taylor series and log_e is its inverse function.) --Dan A.

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[math-fun] Correction Re Exponentiation of surreal numbers
by asimovd＠aol.com 05 Dec '03

05 Dec '03

Can someone please provide the most up-to-date definition of a^b where a and b are surreal numbers? (I hope this coincides with a^b = exp(b * log_e(a)) for 0 < a < omega, where exp is given by the usual Taylor series and log_e is its inverse function.) --Dan A.

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[math-fun] 7.11 puzzle
by Richard Schroeppel 04 Dec '03

04 Dec '03

My daughter Mercury passed along a good puzzle over Thanksgiving. I'll delete the story and just put the math question: Find prices for 4 items that total to $7.11, and that also multiply to 7.11. Implicit side conditions: no sales tax, all prices a positive whole number of cents, product exact. If you solve it, please delay posting the answer till Thursday morning, to give others a chance to work on it. Rich rcs(a)cs.arizona.edu

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