- Thu Jun 06, 2024 10:27 am
#106845
Hi g_lawyered,
This is a defined, balanced, moving grouping game with a 2 value system. In other words, all of the variables are used once and spread across two groups (semesters), so if a variable isn't in semester 1, then it must be in semester 2, and vice versa.
The moving designation means that we know all 6 variables are used, but we don't know exactly how many go in each group, and this can vary from question to question (similar to undefined and partially defined games, except here every variable definitely gets used).
I set this game up vertically, with a stack for semester 1 and a stack for semester 2.
The six variables are G,H,L,M,P,R. (There are no randoms.)
The first rule is that M and P are in the same semester, which I diagram as a vertical block (since that matches my game setup).
The second rule is that P and R do not both go in semester 2, which I diagram as a vertical not block (but also write 2 under the not block to specify that this only applies to semester 2. In other words, it is completely fine for P and R to go together in semester 1.
One implication from rule 2 is that at least one of P or R must go in semester 1. (Since we know from rule 1 that M and P are always together in this game, anytime another rule mentions P, I treat it as the MP block.)
In the main diagram, for semester one I add a MP/R dual option in one of the spaces. This shows that we will always have either MP or R (or both) in semester 1. (Again, the MP would be diagrammed as a vertical block.)
The third rule is conditional and can be diagrammed:
H2 -> G1 + L1
When taking the contrapositive, use the 2 value system to more clearly show the meaning. In other words, rather than writing "not G1," it is more helpful to write "G2" because if G is not in 1, then it must be in 2.
So the contrapositive would be:
(G2 or L2 -> H1)
One implication of this rule is that either H is in semester 1 or (G and L) are in semester 1, or both. I'd diagram this as a H/(G+L) "dual" option in semester one on the main diagram.
At this point, we know from our main diagram that semester 1 has a minimum of 2 variables since we have 2 separate dual options with different variables.
The fourth rule is also conditional and can be diagrammed:
M1 + G1 -> L2
And the contrapositive would be:
(L1 -> M2 or G2)
Again, because M and P are always together from rule one, I'd diagram the M in this rule as the MP block, although technically that is an inference rather than literally what the rule says. (Understanding that distinction could come into play in a rule suspension or substitution question.)
One implication of this rule is that either L is in semester 2 or MP are in semester 2 or G is in semester 2 (or some combination of these). I'd diagram this as a L/MP/G "triple" option in semester 2 on the main diagram.
This shows that there is at least one variable that must go in semester 2.
The only other inference I made was linking the 3rd rule to the contrapositive of the 4th rule.
H2 -> G1 + L1 -> MP2
The reason that we know that MP must be in 2 is that the contrapositive of rule 4 is (L1 -> M2 or G2); however, since we know that G is not in 2 in this situation (G is in 1 from rule 3 as shown above), then that means that M (and P with it of course) must be in 2 since that is the only other option.
As is typical of moving grouping games, there are several fixed numerical distributions.
They are 2-4, 3-3, 4-2, 5-1 fixed (with the first number referring to semester 1 and the second number referring to semester 2.)
