PrepTest 73, Game 4, Setup
Game four. The scenario here gives us another floating grouping game. We have three different groups, which are the three bouquets, and five things to put in those groups, the five flowers. This game is a lot more open-ended than a lot of games. They do tell us that we have to have at least one flower in each bouquet, but they don't tell us that we have to use each flower.
So for all we know, we could just use P, R, and S, and never use L and T. It also doesn't tell us how many flowers go in each bouquet. So for all I know, we could put three flowers in bouquet 1, and only one flower and bouquet 2. And finally, it doesn't tell us that we have to use any of the flowers only once. So we could possibly put roses in all three bouquets.
So when the test maker doesn't specify some stuff that we normally expect to be specified, we need to make sure that we recognize that, that we don't assume that it's just like every other game. Still, what we have is enough to let us set up some things. We know we have three bouquets, so we can make three categories. We know that each bouquet needs at least one flower, so we can put one space under each.
And we know that we have five flowers to work with, so we can list them out to the side in a roster. And that's about all we can do before we go into the rules, so let's go onto those rules. There's a lot of rules here, let's take them one at a time. The first rule says that bouquets 1 and 3 can't have anything in common.
We can represent this as a formal logic rule or a conditional rule. If something is in bouquet 1, it's not allowed in to be in bouquet 3, and vice versa. So 1 means no 3, and 3 means no 1. A given flower can't be in both bouquets, but it doesn't have to be in either. So we know we can't see an L in both 1 and 3, but we don't know that we have to see an L in either.
Now rule 2 says that bouquets 2 and 3 need to have exactly two kinds of flowers in common, which is a lot to unpack. The first thing it's gonna mean is that we're gonna need an extra row of spaces for bouquets 2 and 3. You can't have two things in common if you only have one thing. We also need a symbol to indicate that those spaces are shared, so we'll put some equal signs in between them.
And finally, we need a symbol to say that those are the only shared spaces that the two bouquets get. Any additional spaces won't be allowed to be the same. So we can handle that by drawing a line to say no more. And then, putting a note that anything that comes after that line isn't allowed to be shared between 2 and 3.
Rule 3 is a lot simpler. It just says that bouquet 3 needs to have a S, which we can symbolize by putting right over the top of the column. And the reason we're gonna do that is, we don't know if that S is going to be shared with bouquet 2 or not. So we don't know whether to put it above the line or below the line.
Of course, if bouquet 3 has something, according to rule 1, bouquet 1 is not allowed to have it. So we'll also need to indicate that bouquet 1 doesn't get an S. Rule 4 is a bit of formal logic, and a little bit of complicated formal logic. If a bouquet has L, then it must also have R, and it can't have S. So when we write that into formal logic, we're gonna see, L means R and not S.
Of course, we also need the contrapositive. Remember, when you make the contrapositive, flip, negate, and then any ands need to swap over to ors, and any ors become and. So the contrapositive here means, if you don't have R, or you do have S, you're not allowed to have L. When you have multiple formulogic rules in the same game, keep their arrows lined up so that you can use it as sort of checklist.
Now, this rule is going to have an immediate effect, because since bouquet 3 has an S, it won't be allowed to have an L, and we can note that over the top. The final rule, another piece of formal logic, if something has a T, then it also has to have a P. So in formal logic, T means P, and then, the contrapositive, no P means no T. Everything all organized as our little list.
Ironically, when a game has a whole lot of conditional rules, it's assigned to not go trying to figure out a whole lot of inferences and deductions. We have a lot of rules and they can interact in a lot of different ways. One rule might trigger another rule, and so on. But generally, it's best to let the local question's conditions trigger the rules, follow the chain of events that gets triggered, and then see where you are, rather than trying to figure out every possible iteration at the beginning.
So we're gonna go to the questions. And of course, the optimal question order here is gonna be, actually, the way that they're written. We like to do solution questions first, and 19 is a solution question. We like to do local questions next, and 20 and 21 are locals. And then, after that, we would do cleanup with 22 and 23, which are global.
So on to the questions.
Read full transcriptSo for all we know, we could just use P, R, and S, and never use L and T. It also doesn't tell us how many flowers go in each bouquet. So for all I know, we could put three flowers in bouquet 1, and only one flower and bouquet 2. And finally, it doesn't tell us that we have to use any of the flowers only once. So we could possibly put roses in all three bouquets.
So when the test maker doesn't specify some stuff that we normally expect to be specified, we need to make sure that we recognize that, that we don't assume that it's just like every other game. Still, what we have is enough to let us set up some things. We know we have three bouquets, so we can make three categories. We know that each bouquet needs at least one flower, so we can put one space under each.
And we know that we have five flowers to work with, so we can list them out to the side in a roster. And that's about all we can do before we go into the rules, so let's go onto those rules. There's a lot of rules here, let's take them one at a time. The first rule says that bouquets 1 and 3 can't have anything in common.
We can represent this as a formal logic rule or a conditional rule. If something is in bouquet 1, it's not allowed in to be in bouquet 3, and vice versa. So 1 means no 3, and 3 means no 1. A given flower can't be in both bouquets, but it doesn't have to be in either. So we know we can't see an L in both 1 and 3, but we don't know that we have to see an L in either.
Now rule 2 says that bouquets 2 and 3 need to have exactly two kinds of flowers in common, which is a lot to unpack. The first thing it's gonna mean is that we're gonna need an extra row of spaces for bouquets 2 and 3. You can't have two things in common if you only have one thing. We also need a symbol to indicate that those spaces are shared, so we'll put some equal signs in between them.
And finally, we need a symbol to say that those are the only shared spaces that the two bouquets get. Any additional spaces won't be allowed to be the same. So we can handle that by drawing a line to say no more. And then, putting a note that anything that comes after that line isn't allowed to be shared between 2 and 3.
Rule 3 is a lot simpler. It just says that bouquet 3 needs to have a S, which we can symbolize by putting right over the top of the column. And the reason we're gonna do that is, we don't know if that S is going to be shared with bouquet 2 or not. So we don't know whether to put it above the line or below the line.
Of course, if bouquet 3 has something, according to rule 1, bouquet 1 is not allowed to have it. So we'll also need to indicate that bouquet 1 doesn't get an S. Rule 4 is a bit of formal logic, and a little bit of complicated formal logic. If a bouquet has L, then it must also have R, and it can't have S. So when we write that into formal logic, we're gonna see, L means R and not S.
Of course, we also need the contrapositive. Remember, when you make the contrapositive, flip, negate, and then any ands need to swap over to ors, and any ors become and. So the contrapositive here means, if you don't have R, or you do have S, you're not allowed to have L. When you have multiple formulogic rules in the same game, keep their arrows lined up so that you can use it as sort of checklist.
Now, this rule is going to have an immediate effect, because since bouquet 3 has an S, it won't be allowed to have an L, and we can note that over the top. The final rule, another piece of formal logic, if something has a T, then it also has to have a P. So in formal logic, T means P, and then, the contrapositive, no P means no T. Everything all organized as our little list.
Ironically, when a game has a whole lot of conditional rules, it's assigned to not go trying to figure out a whole lot of inferences and deductions. We have a lot of rules and they can interact in a lot of different ways. One rule might trigger another rule, and so on. But generally, it's best to let the local question's conditions trigger the rules, follow the chain of events that gets triggered, and then see where you are, rather than trying to figure out every possible iteration at the beginning.
So we're gonna go to the questions. And of course, the optimal question order here is gonna be, actually, the way that they're written. We like to do solution questions first, and 19 is a solution question. We like to do local questions next, and 20 and 21 are locals. And then, after that, we would do cleanup with 22 and 23, which are global.
So on to the questions.
