Hi tfab! I'm sure that Dave will post an official explanation and diagram soon, but until then I'll do a quick run through of it.
We have 5 prospective volunteers: L, M, N, P, Q. They are grouped into 3 days. Each day has 3 spots. Like so:
T: _ _ _
F: _ _ _
S: _ _ _
As always with Grouping games, I want to take note of the number of spots available (9) and how those spots can be filled. Initially, we have basically no limitations on how the spots can be filled. Can any of the people be assigned to all 3 days? What about being assigned to 0 days? Luckily, the first rule is very helpful in this regard.
Rule 1: No volunteer works every day. (No need for short-hand, I'd write this rule out in its entirety)
This means that each of the five volunteers can only work a maximum of 2 days. How many of the volunteers will work the maximum 2 days? We can figure out the answer - since there are 9 spots available, four of the volunteers must work two days. The fifth volunteer will work one day. It is not possible for any of the volunteers to work 0 days. So we have a strong inference: four volunteers work two days, the fifth volunteer works only one day.
Rule 2: M
L (contrapositive:
L M)
Since there are only 3 spots per day, this rule is important to keep in mind since it means M always takes up 2 of those 3 spots (M and L) on any day it appears.
In addition, we can make one inference based on Rule 2 and our inference about the number of volunteers: L
cannot be the volunteer who only works one day (if L only works one day, then M could only work one day and we wouldn't be able to fill all 9 spots).
Rule 3: N = F
Rule 4: P =/ S
If P does not work on S, that means we're left with only the four other volunteers (L, M, N, Q) to fill those 3 slots. However, from our contrapositive of Rule 2, we know that if L is not present, then M cannot be either. That would leave only 2 variables (N and Q) to fill the 3 spots on S, so that doesn't work. That means L must work on S.
Since L works on S, we are limited in where the M + L block(s) can go. This means the placement of M can be very instructive. For example, if M (and, because of Rule 2, L as well) goes on Thursday, then we know that M can't go on Friday (because that would mean L would have to be used all 3 days, which can't happen). That leaves us with only 2 volunteers to fill in the open Friday slots: P and Q. Similarly, if M (and therefore L) goes on Friday, we know that M and L can't go Thursday, which means Thursday has to be filled by the three remaining volunteers of P, N, and Q. And since P can't go Saturday, and we've already used N twice, the last two Saturday spots have to be M and Q. I don't think we can write down an exact inference about this, but the takeaway is that the placement of M is something to watch very closely.
Anyways, our
final diagram will look like this:
T: _ _ _
F: N _ _
S: L _ _ (P)
Along with two rules not represented in the diagram:
- M L (and we know it's a rule to watch very closely!)
- Rule 1, which is more helpfully expressed as our inference that four volunteers get used exactly twice, and the fifth volunteer (which cannot be L) exactly once.
From there, the questions become easier to answer.
Hope that helps!
