You die and the devil says he'll let you go to heaven if you beat him in a game. The devil sits you down at a perfectly round table. He gives himself and you an infinite pile of quarters. He says, "OK, we'll take turns putting one quarter down, no overlapping allowed, and the quarters must rest flat on the table surface. The first guy who can't put a quarter down loses." You guys are about to start playing, and the devil says that he'll go first. However, at this point you immediately interject, and ask if you can go first instead. You make this interjection because you are very smart and can place quarters perfectly, and you know that if you go first, you can guarantee victory. Explain how you can guarantee victory.
You place a quarter right in the center of the table. After that, whenever the devil places a quarter on the table, mimic his placement on the opposite side of the table.. If he has a place to place a quarter, so will you. The devil will run out of places to put a quarter before you do.
There are 1 million closed school lockers in a row, labeled 1 through 1,000,000.
You first go through and flip every locker open.
Then you go through and flip every other locker (locker 2, 4, 6, etc...). When you're done, all the even-numbered lockers are closed.
You then go through and flip every third locker (3, 6, 9, etc...). "Flipping" mean you open it if it's closed, and close it if it's open. For example, as you go through this time, you close locker 3 (because it was still open after the previous run through), but you open locker 6, since you had closed it in the previous run through.
Then you go through and flip every fourth locker (4, 8, 12, etc...), then every fifth locker (5, 10, 15, etc...), then every sixth locker (6, 12, 18, etc...) and so on. At the end, you're going through and flipping every 999,998th locker (which is just locker 999,998), then every 999,999th locker (which is just locker 999,999), and finally, every 1,000,000th locker (which is just locker 1,000,000).
At the end of this, is locker 1,000,000 open or closed?
Locker 1,000,000 will be open.
If you think about it, the number of times that each locker is flipped is equal to the number of factors it has. For example, locker 12 has factors 1, 2, 3, 4, 6, and 12, and will thus be flipped 6 times (it will end be flipped when you flip every one, every 2nd, every 3rd, every 4th, every 6th, and every 12th locker). It will end up closed, since flipping an even number of times will return it to its starting position. You can see that if a locker number has an even number of factors, it will end up closed. If it has an odd number of factors, it will end up open.
As it turns out, the only types of numbers that have an odd number of factors are squares. This is because factors come in pairs, and for squares, one of those pairs is the square root, which is duplicated and thus doesn't count twice as a factor. For example, 12's factors are 1 x 12, 2 x 6, and 3 x 4 (6 total factors). On the other hand, 16's factors are 1 x 16, 2 x 8, and 4 x 4 (5 total factors).
So lockers 1, 4, 9, 16, 25, etc... will all be open. Since 1,000,000 is a square number (1000 x 1000), it will be open as well.
Three playing cards in a row. Can you name them with these clues? There is a two to the right of a king. A diamond will be found to the left of a spade. An ace is to the left of a heart. A heart is to the left of a spade. Now, identify all three cards.
