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David Justice and Derek Jeter were professional baseball players. In 1997 they had the following conversation:

David: Did you know that in both 1995 and 1996 I had better batting averages than you?

Derek: No way, my batting average over the last two years was definitely higher than yours!

It turned out that both of them were right. How is it possible?

This is the so called Simpson’s paradox. The reason it occurred is that during 1996 both players had high averages and Derek Jeter had many more hits than David Justice. In 1996 both players had low averages and David Justice had many more hits than Derek Jeter. You can see their official statistics below.

Player199519961995-1996
Derek Jeter12/48 (.250)183/582 (.314)
195/630 (.310)
David Justice104/411 (.253)45/140 (.321)149/551 (.270)
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Can you figure out which word is depicted by this rebus?

Each of the images in the first row depicts HAIR. Each of the images in the second row (except the second cell) depicts a BUG. Each of the images in the third row depicts ER.

The I from HAIR on the first row gets moved right after the B from BUG on the second row. Also, the U from BUG gets flipped upside down. Therefore, we get HARBINGER.

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Alice secretly picks two different integers by an unknown process and puts them in two envelopes. Bob chooses one of the two envelopes randomly (with a fair coin toss) and shows you the number in that envelope. Now you must guess whether the number in the other, closed envelope is larger or smaller than the one you have seen.

Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?

Choose any strictly decreasing function F on the set of all integers which takes values between 0 and 1. Now, if you see the number X in Bob’s envelope, guess with probability F(X) that this number is smaller. If the two numbers in the envelopes are A and B, then your probability of guessing correctly is equal to:

F(A) * 0.5 + (1 – F(B)) * 0.5 = 0.5 + 0.5 * (F(A) – F(B)) > 50%.

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White to move and mate in 1.

Remark: There is only one correct answer.

The answer is Qa3#.