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Somewhere in a forest, there are seven turtles on a log, between a pond and a trail. The seven turtles weigh 1, 2, 3, 4, 5, 6 and 7 pounds respectively. They are arranged in a random order. When someone walks by, it scares the closest turtle, who jumps into the water. But the next turtle will only get scared and jump if the first turtle was larger than it. (Otherwise, it figures that whatever scared the first turtle isn't big enough to hurt it). If the second turtle does jump, then the third turtle will jump if the weight of the first two is greater than or equal to it. The fourth turtle will jump using the same criterion. So, for example, if the first turtle is three pounds, the second one pound, and the third four pounds, the third will jump. The question is, if someone walks by, on average, how many turtles jump in?

This is an interesting problem, because while the arithmetic is simple, the answer is less so. There is a pretty simple solution to find an answer, but it depends on what answer you are looking for. We know right away that if the one pound turtle jumps first, that no other turtles will jump. We also know that if the seven pound turtle jumps, all of the other turtles will jump. Since 1+7=8, and that divided by 2 is 4, we can see that on average, 4 turtles will jump between these two groups. If we look at the 2 pound turtle and the six pounds turtle, we come to a similar average: in the case of the 2 pound turtle, 5/6th of the time, only 1 turtle jumps, and in the case of the 6 pound turtle, 5/6th of the time they all jump. 2+6/2 equals 4. So in general, we can say that for each of these 5040 possible combinations, there is a counterpart with the complementary answer. Therefore, the answer is 4.

Notice, however, you can never have four turtles jump in. The minimum weight of four turtles is the triangular number of four, which is 10. That is, four turtles could never weigh less than 10 pounds, and the fifth, sixth and seventh turtle would therefore jump in as well. In fact, even a three turtle combination is highly unlikely: it would have to begin with 2-1-3-7, 3-1-2-7, or 3-2-1-7, three combinations out of our 5040 possibilities. There is a very skewed distribution: the answer is either 7, or 1, a few cases of 2, and a tiny number of cases of 3. So the mean answer is around 4, the mode answer is 1.

Thus, the puzzle is more ambiguous than it seems, because the "average" answer depicts a case that can not actually occur.

While the set-up of this puzzle might seem silly or irrelevant, the principle can be applied to things other than startled turtles. It can be used to model systems where change is dependent on the decision of a single large company. When a large retailer decides to promote a product, when a state gives its vote in a primary to a certain candidate, when the coolest kid at school starts listening to a new band...all of them are the heavy turtle jumping in. So this puzzle represents why dynamic systems either change shape, or stay the same.

The answer to the puzzle I gave here explains the concept, but is not a good mathematical demonstration. If you have different reasoning about how many turtles jump in, please message me in a comment or submit a writeup with your answer!

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