Question number 30. In the forests of a magical island, three kinds of animals roam. We have lions, wolves, and goats. Wolves can eat goats, and lions can eat both wolves and goats. However, because this is a magical island, if a wolf eats a goat, the wolf will turn into a lion. If a lion eats a goat, the lion will turn into a wolf. And if a lion eats a wolf, the lion will turn into a goat. So I have written down that operation where I represent the number of animals here at any point as a triple, lion, wolves, and goats. If a lion eats a wolf, so that's LW, the number of lions and wolves is decreased by one. We get one more goat. If a lion eats a goat, that's LG. Lions and goats are decreased by one. We get one more wolf. If a wolf eats a goat, however, the number of wolves and goats is decreased by one, and we get one more lion in that case. So originally, we have six lions, 55 wolves, and 17 goats on the island. And I have represented that as a point in space where we have the lion axis, the wolf axis, and the goat axis. And this point in space moves in various directions, indicated here by the black, green, and purple vectors corresponding to the three operations that we have. So our job here is to determine what is the highest possible number of animals remaining on the island after it is no longer possible for any animal to eat another. And that would happen if only one species remains. If we only have goats, then they can't eat themselves. If we only have lions, they can't eat themselves. Likewise, wolves cannot eat themselves. But if we have any of those two, then they can keep eating each other. So we are looking for the x-intercepts with the coordinate axes. So just goats would be this point here. Just lions would correspond to a value here on the L axis. And only wolves remaining would correspond to the W value on the W axis. And we have to find the largest coordinate here out of these three. So heuristically, what we should notice is that the direction we should be moving here is towards only lions, because that point here is farthest from the point 6, 55, 17. So if it's farthest, when we move in that direction, we're going to get the highest coordinate in magnitude. So we should be looking for only lions remaining on the island. And the upper bound here would be if we eliminate all the wolves. So eliminating all the wolves leaves 23 animals. And that would be the upper bound here for this problem. So we can choose answer D here. And then we should just check if it's actually possible to obtain 23 lions. So here's my procedure for doing that. What we are starting with is 6 lions, 55 wolves, 17 goats. Then I will apply the operation wolf eats goat 17 times. So all the goats are eaten. That becomes 0. We have 17 less wolves, because they became lions, and we have 23 lions now. So now lions can eat wolves. But each time that happens, we have a new goat. So we string the operations. Let a lion eat a wolf, then let that wolf eat a goat. So in that case, the numbers of lions and wolves are first decreased by 1. We get a goat. Then the wolf eats a goat. Wolves are decreased by 1 again. It becomes a lion. The lions are back to what they were before, and the goat is gone. So what happens is the number of lions remains the same after these two operations. Goats remains the same, but wolves decreases by 2. So we need to do that 13 more times, and then we see that we end up at the coordinate here on the L axis, 23,00, corresponding to 23 wolves. So that checks that our heuristic guess is indeed correct, and the answer here is 23D.

magical island

animal transformations

lions wolves goats

species optimization

transformation rules