CP. The Prisoners Dilemma
After discussing last week’s Mismatched Pennies Game, Kyle and Chaim are hauled off to jail!
Archive for answers
CO. Mismatched Pennies
A few months ago, we offered our listeners up to One Million Dollars! Unfortunately, we didn’t have to give away a cent, as Game Theory predicted all along.
This week, to celebrate the recent “rock scissor paper” World Championship, Kyle and Chaim play a game of “mismatched pennies”. Each puts down a penny on the table, choosing to lay the penny down showing heads or tails.
If the pennies both show heads, Chaim wins 3¢; if both show tails, Chaim wins 1¢, and if one is heads and the other tails, Kyle wins 2¢.
Thinking this through, second and triple guessing one’s opponent, is ultimately fruitless, as wonderfully illustrated in this scene from The Princess Bride
(Incidentally, game theory shows up in many movies, television shows and works of fiction!)
John von Neumann tells us, in his celebrated minimax theorem, that there is an optimal strategy for both players; each assigns a percentage to each of his options; the choice of which option to use is made randomly, by these percentages. Von Neumann tells us that there is no way to take advantage of knowing what the opponent’s optimal strategy is– that’s what makes it optimal!
But the game still might favor one player or the other, even if both are using their optimal strategy. This week’s puzzle then, is to answer: does this game of mismatched pennies favor Chaim or Kyle?
CM. Crossing the Bridge
Save Indiana, his girlfriend, his father and his father’s sidekick from certain doom! They must cross a bridge across a gorge in no more than one hour!
Read the rest of this entry »
CL. Some Number of Numbers Sum
Niclas Hedell gives his solution to the third tree puzzle he posed last week, and we ask a puzzle about sums of numbers.
CK. The Third Tree
Niclas Hedell, a listener, poses a problem from his days in the Swedish military: given two trees in the forest, and a rope twice as long as the distance between the trees, how do you find a third tree so that all three make a right triangle.
And we explain how the Stork can catch the Frog.
Follow-up: The Stork and The Frog
Amusingly, this problem has exactly the same solution as the proof that there are as many rational numbers as there are counting numbers. And the proof generalizes: one stork can catch three frogs, or ten or fifty.
Here are some bonus problems:
- The stork can catch the frog even if it can start at any rational number and hop any fixed rational distance each step.
- However, if the frog can start at any real number or hop any real distance, the stork has no strategy that guarantees a catch. This is, in effect, the same as proving that the real numbers are not countable.
CJ. The Stork and the Frog
Lord Butler, Knight of the Garter, has never heard of the rule on last week’s show. But he notes he does move down the table each year…
When we posed this problem, we thought things would turn out a little differently, as we discuss here.
CH. Rayo’s Number!
A contestant for our Million-Dollar-Give-Away sent in Rayo’s Number, hitherto the largest number ever used for any real purpose: to wit, winning the
Check out the article by Scot Aaronson that inspired them to duke it out! And this thread on the math forum is quite interesting as well.
Follow-up: Graham’s Number
Graham’s number, as huge as it is, can be “described” or “named” in a very few symbols. Several people sent us programs that (in principle!) calculate Graham’s number— you can think of any of these programs as notation for Graham’s number.
Q&A: The Race
We never did resolve the question of which grows faster:
In this corner we have
Sequence 1 n^^n
1, 2^2, 3^3^3, 4^4^4^4, and so on.
And over here we have Sequence 2, defined recursively by
In short, we can define the second sequence as s(1) = 1; s(n) = n, followed by s(n-1) factorial signs.
Which sequence grows faster than the other??
We have many conflicting answers, and no decisive resolution; here was one idea .
The Math Factor in iTunes
RSS feed
Comments RSS
mathbun.com