Math Chat: Halloween Cards, Sprinklers, & Mirrors
Old Halloween challenge
Ralph Thomas proposes a new card game for you and an opponent for Halloween. Each hand, a goblin deals both players a certain number of cards. The highest card wins (e.g., ace beats king, two aces tie). If you win you get two points; if your opponent wins he gets just one point. (You have an advantage.)
Do you prefer to use a regular deck, a double deck, or a pinochle deck (which has two of each card from 9 on up to ace)? How many cards would you like the goblin to deal out each hand? Do you prefer four or eight hands in the tournament?
In the winning answer, Stephen Jabloner writes: "If I were playing that Halloween card game, I'd opt for one regular deck, one card per hand, and eight hands in the tournament. Each of these choices is based on the notion that I want as few ties as possible since I have an advantage when someone wins.
"A single regular deck is less likely to end in a tie game than either a double deck or a pinochle deck. For example, if I hold one card from a regular deck, there's only a 3/51 (5.9 percent) chance that my opponent will tie me, whereas there is a 7/103 (6.8 percent) chance with a double deck and 7/47 (14.9 percent) chance with a pinochle deck."
Similarly, the more cards dealt, the more likely a tie. If the goblin deals out all the cards, there is a good chance we will both have an ace and tie.
Finally, the longer the tournament, the less likely my opponent will win by a fluke.
More on sprinklers
Although most readers agreed with the published answer that an underwater sprinkler sucking in water will not turn, Erik Randolph for example argues that conservation of momentum requires that if the water in the sprinkler is going one way, the sprinkler must turn the other way.
Leaf Turner points out that this backwards effect, much smaller than that due to all the water spurting out in normal sprinkler operation, and almost undetectable, has been measured in experiments with a special, nearly frictionless sprinkler. (See Amer. Journal of Physics, Vol. 57, p. 654, and references to previous articles with different conclusions.)
New mirror challenge
Why do mirrors reverse left and right and not up and down?
Chris Snyder writes, "I'm taking two philosophy courses and in both the issue of simplicity has come up over and over again. I was wondering if there is a math-type definition for simplicity."
Math Chat replies: Mathematicians usually measure simplicity by the length of the description of a solution or alternatively by the time required to compute a solution. For example, the simplicity of a method to alphabetize a list could be the length of the instructions or alternatively how long the process takes (and how that time grows with the length of the list).
Gregory Sahagen notes that the sun's gravitational force on the earth is greater than the moon's and asks why the tides are primarily due to the moon.
Math Chat replies that the tides are due not to the magnitude of gravitational force but on the variability of gravitational force, pulling harder on nearer waters (and causing them to swell), average on middle waters (which remain relatively low), and weaker on distant waters (and letting them swell in the opposite direction). Since the moon is closer than the sun, the percentage change in distance from one side of the earth to the other is greater, and as it happens this effect is enough to make the moon's tide's greater. Note that since the tides bulge in both directions, there are two high tides a day, not just one.
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