Problem number nine. Nyssa has three different types of cards in a game. Apple, cherry, and grapes. They can be arranged in sets of five. In each set, Nyssa can choose just two cards and swap their places. She wants to arrange the cards so that all the cards with the same fruit are next to each other. For which set is this not possible? A, B, C, D, or E? We're going to have to look at each set and figure out if we can move just two cards and have all the fruits of the same kind next to each other. In A, we have multiples of two fruits, grapes and apples. We need to have the grapes next to each other and the apples next to each other. We can try moving one of the apples. If we move it here and switch its place with these grapes, the grapes would not be next to each other. If we moved it here, switching it with these grapes, the grapes still would not be next to each other. If we move this apple over here instead, we'd get the apples next to each other but not the grapes. If we tried moving the grapes, other than switching it with the first apple, we could switch with this apple, getting the two grapes to each other, but the apples would not be next to each other. The same thing would happen if we moved this set of grapes here. Also, if we moved the grapes here, that would not put the apples next to each other. And if we put this set of grapes here, the grapes would not be next to each other and the apples would not be next to each other. So we cannot switch just two cards in A and get the fruits of the same kind to be next to each other. Let's look at the other sets and make sure that we can do it there. If all the other ones are okay, then we've shown that answer is A. In B, we only have apples repeated, so we just need to worry about putting the apples next to each other. We can do it in several ways. We can take this first apple and switch it with the grapes so that the apples are on these three cards. Or we could take this apple, put it here, and the three apples will be on these three cards. So we can do it here. In C, we have multiples of apples which are already next to each other, and we have two sets of grapes. We can switch one of the grapes so that it's next to the other grapes, and then the apple still ends up being next to the other apple. The same happens if we take these grapes, switch them with this apple, the two apples are then next to each other on this end, and then the grapes are over here. So it works for C. In D, we have three cherries. We can move the one from the end and switch its place with the apple, and we have three cherries in a row. So that's doable. In E, we have two cherries and two bunches of grapes. We can switch these grapes with the cherry here, so we have two grapes here and two cherries here. Or we could put the these grapes, put them here, and then the cherries end up being right here. And we would have cherries and then grapes. So that works. So the only set for which we cannot switch two cards and have all the fruits of the same kind next to each other is A. So the answer is A. That's the one that we cannot do this for.

card arrangement

fruit types

set rearrangement

two-card swap

set A