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- Wed Jun 29, 2016 10:03 am
#26794
The third game appears at first glance to be a standard Basic Linear game, with six auction items--H through V--placed sequentially into six spots, 1-6. Simple enough. The numbers match for a 1:1 distribution, there's no excess or empty spaces, no recycled variables...if anything the scenario here is the easiest on the test thus far.
The rules however proved to be anything but simple.
Well, I should clarify that statement. Two of the rules, #1 and #3, are quite simple: S cannot go in 1, and must go earlier than both M and V. Those should be familiar to even a novice test taker. They also lead to a handful of clear Not Laws: S is not in 1, 5, or 6; M is not in 1 or 2; V is not in 1 or 2. (M and V can't go in 2 because that would put S in 1 and break the first rule)
It's rules 2 and 4 that caused issues. Let's look at them individually:
Rule 2: this is a conditional sequencing rule, where H earlier than L triggers the sequence M before L. That sounds clear enough, but it caused people a lot of problems. Essentially this rule allows for three unique sequences, while only disallowing one:
Allowed: H/M --- L (this is what happens when H is before L: H and M are both before L)
Allowed: L --- H/M (this is what could happen when H is NOT before L: H and M are both after L)
Allowed: M --- L --- H (this is also possible when H is not before L)
Not Allowed: H --- L --- M (this is the only sequence that violates the rule; with H ahead of L, M must also be ahead of L)
So really what that rule does is eliminate a single possibility, H --- L --- M. Beyond that anything goes. L ahead of, or after, both H and M? Fine. L between H and M? Fine, as long as M is first. These possibilities are a killer if you don't catch them all. Be careful with conditional sequencing.
Rule 4: this is also a conditional sequencing rule, albeit less obviously conditional than rule 2. We talk a lot about this exact construct in our books and courses so I'll spare you too much detail here, but suffice it to say that only two possible sequences can occur with H, T, and V:
H --- T --- V
V --- T --- H
Essentially T must always be placed between H and V (it's always earlier than one, but not the other, so it gets sandwiched), whatever H and V's order. That also produces two Not Laws for T: T can't be first or last.
That's great, and leaves us with the following diagram thus far:
Now, with the rules comfortably in hand, we need to quickly consider additional inferences that might be made, either from ideas we've noted but not explored, or from rule linkage (shared variables in multiple rules that begin to affect one another).
The only thing we've really found so far but not probed for consequences is the outlawed sequence from rule 2, where we can never have H --- L --- M. Is there anywhere we could put L that would force H --- L --- M? Yes. Two places, in fact. If we put L in 2, as you can see from the diagram above, that places H in 1. Meaning H 1, L 2, and M somewhere after L in 3-6. That's the forbidden sequence, so L can never be in 2. What about L in 3? Once again, that puts H in 1, but can we get M ahead of L and avoid the violation? Nope. We know that M is after S from rule 3, and there's no room for S ahead of M if H is 1 and M is 2 (we also have a Not Law saying no M in 2). So M between H and L is impossible with L in 3, meaning we'd end up with the impossible H --- L --- M again, and thus L cannot go in 3 either.
So your final diagram (or at least the one I took to the questions with me), is this:
As for the questions, they become much, much easier with the setup above, but there are still a few twists and turns to navigate.
The rules however proved to be anything but simple.
Well, I should clarify that statement. Two of the rules, #1 and #3, are quite simple: S cannot go in 1, and must go earlier than both M and V. Those should be familiar to even a novice test taker. They also lead to a handful of clear Not Laws: S is not in 1, 5, or 6; M is not in 1 or 2; V is not in 1 or 2. (M and V can't go in 2 because that would put S in 1 and break the first rule)
It's rules 2 and 4 that caused issues. Let's look at them individually:
Rule 2: this is a conditional sequencing rule, where H earlier than L triggers the sequence M before L. That sounds clear enough, but it caused people a lot of problems. Essentially this rule allows for three unique sequences, while only disallowing one:
Allowed: H/M --- L (this is what happens when H is before L: H and M are both before L)
Allowed: L --- H/M (this is what could happen when H is NOT before L: H and M are both after L)
Allowed: M --- L --- H (this is also possible when H is not before L)
Not Allowed: H --- L --- M (this is the only sequence that violates the rule; with H ahead of L, M must also be ahead of L)
So really what that rule does is eliminate a single possibility, H --- L --- M. Beyond that anything goes. L ahead of, or after, both H and M? Fine. L between H and M? Fine, as long as M is first. These possibilities are a killer if you don't catch them all. Be careful with conditional sequencing.
Rule 4: this is also a conditional sequencing rule, albeit less obviously conditional than rule 2. We talk a lot about this exact construct in our books and courses so I'll spare you too much detail here, but suffice it to say that only two possible sequences can occur with H, T, and V:
H --- T --- V
V --- T --- H
Essentially T must always be placed between H and V (it's always earlier than one, but not the other, so it gets sandwiched), whatever H and V's order. That also produces two Not Laws for T: T can't be first or last.
That's great, and leaves us with the following diagram thus far:
Now, with the rules comfortably in hand, we need to quickly consider additional inferences that might be made, either from ideas we've noted but not explored, or from rule linkage (shared variables in multiple rules that begin to affect one another).
The only thing we've really found so far but not probed for consequences is the outlawed sequence from rule 2, where we can never have H --- L --- M. Is there anywhere we could put L that would force H --- L --- M? Yes. Two places, in fact. If we put L in 2, as you can see from the diagram above, that places H in 1. Meaning H 1, L 2, and M somewhere after L in 3-6. That's the forbidden sequence, so L can never be in 2. What about L in 3? Once again, that puts H in 1, but can we get M ahead of L and avoid the violation? Nope. We know that M is after S from rule 3, and there's no room for S ahead of M if H is 1 and M is 2 (we also have a Not Law saying no M in 2). So M between H and L is impossible with L in 3, meaning we'd end up with the impossible H --- L --- M again, and thus L cannot go in 3 either.
So your final diagram (or at least the one I took to the questions with me), is this:
As for the questions, they become much, much easier with the setup above, but there are still a few twists and turns to navigate.
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L and L 
I found the questions to be much more difficult and much more time consuming. 