Math Page
Math was my undergraduate major. Now it's just a recreation.
Links
Here are some of my favorite math links:
Puzzles
And here are some of my favorite puzzles:
A 100-lb sack of potatoes is lying in the sun. The potatoes weigh a total of 100 pounds, but 99% of that weight is made up of water. After hours in the sun, some of the water evaporates, with the result that the potatoes are now made up of only 98% water. How much water evaporated?
Two magicians perform the following card trick: Magician B leaves the room. Spectators shuffle a deck of cards and select five cards at random. Magician A looks at the five cards and places them face down on a table. Magician B returns. Magician A reveals four of the five cards one by one. Magician B now names the fifth card.
Of course the two magicians have consulted with each other beforehand and agreed on their method, but the method does not involve any covert signals, which hand Magician A uses to turn over the cards, how long Magician A takes to turn the cards, or anything like that. It depends solely on the cards revealed.
What is the method by which Magician A names the fifth card?
You, Sandy, and Pat have committed such terrible violations of each other’s honor that the three of you agree to fight a three-person duel, or “truel.” Standing at the vertices of a large equilateral triangle, each of you, in turn, will fire one shot (except, of course, that anyone killed before his or her turn will not fire). If there is more than one survivor after the first round, the survivors will, in the same order, each fire another shot. This process must continue until there is only one survivor.
You are the worst marksman in the group; at the distance used you hit your target only 30% of the time. Sandy is somewhat better; at the distance he will hit the target 50% of the time. Pat is so expert that at the distance she will hit the target 100% of the time. To even things up a little, the order of firing will be the reverse of the skill level, so that you go first, then Sandy, then Pat.
Nonfatal hits are to be neglected for purposes of the problem. Collusion among truelists is forbidden.
What is your best strategy, and what is your chance of survival?
This excellent puzzle was created by mathematician Paul Halmos.
Halmos and his wife attended a dinner party also attended by four other couples. During the cocktail hour, some of those present shook hands, but in an unsystematic way, with no attempt to shake everyone's hand. Of course, no one shook his or her own hand, no one shook hands with his or her spouse, and no one shook hands with the same person more than once.
During dinner, Halmos asked each of the nine other people present (including his own wife), how many hands that person had shaken. Under the given conditions, the possible answers ranged from 0 to 8. Halmos was interested to notice that each person gave a different answer: one person hadn't shaken anyone's hand, one person had shaken exactly one other's hand, one had shake exactly two hands, and so on, up to one person who had shaken hands with all the others present, except his or her spouse, that is, 8 handshakes.
Now, the question is this: how many hands did Halmos's wife shake?
I heard this great puzzle from Eugene Volokh.
Prove that for any positive integer n, there exists a positive integer consisting entirely of ones and zeros that is divisible by n.