Old dice challenge (S. H. Logue)
The six faces of four dice have these numbers on them:
A 1 1 5 5 5 5
B 4 4 4 4 4 4
C 3 3 3 3 7 7
D 2 2 2 6 6 6
You and your opponent each choose one die to use throughout the game. Each round you both roll, and the high number wins. Which is the best die to have?
Timothy Dawn observes that "This game has the intriguing property that while D beats A, A beats B, and B beats C, C beats both D and A."
Christopher Burns concludes that "If you get to choose a die already knowing what die your opponent has chosen, it becomes a sweet game - for you. If your opponent chooses A, you should choose D; if B, A; if C, B; and if D, C."
In each case you have a 2/3 chance of winning. B beats C whenever C rolls a 3, which occurs 4/6 or 2/3 of the time. C beats D the 1/3 of the time that C rolls a 7, plus half of the 2/3 of the time that C rolls a 3, for a total of 2/3 of the time.
At Mendocino Middle School in California, Joan Carlson's eighth-graders actually constructed and rolled C against D 400 times and found C won 66.5 percent of the time.
Burns continues: "The game gets much more interesting if you don't know what die your opponent chooses. If your opponent chooses at random, pick C and you win 56 times out of 108. If you think your opponent is going to pick C, pick B. If you think your opponent is trying to think ahead of you, the trick is to know how many steps!"
Mathematical game theory says that the player who goes first has some "mixed strategy" (selecting various dice with various probabilities) that the player who goes second cannot beat. For this game, the safe mixed strategy is to use B and D at random (each with probability 1/2). Now whether your opponent chooses A, B, C, or D, she wins just half the time.
For developing this theory, mathematician John Nash won the 1994 Nobel Memorial Prize in Economic Science.
New Valentine Challenge
The heart clover has eight symmetries in space. You can rotate it by 90 degrees (a quarter turn), 180 degrees, or 270 degrees. You can flip it about a horizontal or vertical line through its center. You can flip it about either diagonal line through its center. Finally, you can leave it alone (mathematicians like to count that). It is the same as the symmetries of a square.
What are the symmetries of a cube in space? (Or a 4-dimensional cube in 4-dimensional space?)
Geometry can deal not only with simple shapes but also with complex soap bubble clusters, black holes, and crystal growth, thanks to the work of modern mathematicians, such as Frederick J. Almgren Jr., who passed on last week. He established, for example, the existence and structure of mathematical soap bubble clusters in 3-dimensional space, 4-dimensional space, or even 100-dimensional space, consisting if necessary of millions of bubbles. He was my PhD thesis adviser and the mathematician I admired most.
A high school student found a mistake on the SAT. The testers claimed that for a series such as 1, a, a2, a3, a4, the a2 term is the "median" or middle value, which is true if a is positive. Colin Rizzio pointed out that it is false if a is negative. For example, if a is -2, the series is 1, -2, 4, -8, 16, and the median is 1.
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