And so, once again, the average distance the grasshopper had to travel was simply 1/2. This time around, it could never return to positive territory, even if all the remaining jumps were to the right. Now suppose the grasshopper’s first jump was to the left, meaning it was now at -1/2. Because you never had to worry about crossing the origin again, the average distance the grasshopper had to travel was simply 1/2. Even if all the remaining jumps were to the left, the grasshopper would have been at 1/4, then 1/8, then 1/16 and so on, returning to the origin after infinitely many jumps. What’s more, there was no way for the grasshopper to ever cross over to the left side of the origin again. Since subsequent jumps were equally likely to be left or right, its average position from this point forward remained at 1/2. Suppose the grasshopper’s first jump was to the right, meaning it was now at 1/2. So while the average position was always zero, the average distance to the origin was positive as soon as the grasshopper took its first jump. However, this puzzle asked for the average distance to the origin, which was defined as positive whether the grasshopper was to the left or to the right. And so some readers thought zero was the answer. That is, averaging the various positive landing positions to the right of the origin and the negative positions to the left of the origin always gave you exactly zero. Since the grasshopper was equally likely to jump left or right at any given point, by symmetry its average position always remained zero. On average, what was the expected distance it traveled to return home? (Note that no matter which side of the origin the grasshopper was on, “distance” was defined as being zero or positive, but couldn’t be negative.) For each jump, it hopped left or right along the number line with equal probability.Īfter infinitely many jumps, the grasshopper’s head was once again clear and it wanted to return home to the origin. However, before the jumping began, it drank a little too much grasshopper juice and lost all sense of direction. Its Nth jump had length 1/2 N, so its first jump had length 1/2, its second jump had length 1/4, its third jump had length 1/8 and so on. Last week, a grasshopper was jumping on a number line and started at its home at zero (i.e., the “origin”). The game is a lot of fun, and this week’s Express is inspired by it.Ĭongratulations to □ Shantanu Gangal □ of Mumbai, India, winner of last week’s Riddler Express. If you instead place those 1s in a two-by-two square, you get 6 points - there are now six adjacencies (including the two diagonals). Whenever two of the same digits are placed in adjacent squares (whether horizontally, vertically or diagonally adjacent), you get a number of points equal to the value of those two digits.įor example, if you place four 1s in a row, you get 3 points - there are three adjacencies. In the game Digit Party (of which Vince was one of the creators!), you place 25 digits one at a time on a five-by-five board. Riddler Expressįrom Vince Vatter comes a puzzle about a “party” game that’s all the rage these days: Please wait until Monday to publicly share your answers! If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter or send me an email. Submit a correct answer for either, 1 and you may get a shoutout in the next column. Two puzzles are presented each week: the Riddler Express for those of you who want something bite-size and the Riddler Classic for those of you in the slow-puzzle movement. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability.
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