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- Mon Sep 11, 2017 11:00 pm
#41098
Setup and Rule Diagram Explanation
This is a Partially Defined, Balanced, Identify the Templates, Grouping game.
The game requires us to assign six variables (rangers) to three different groups (areas). Since each variable is used exactly once, the game is Balanced. However, the game is only Partially Defined, because the number of variables in each group can vary from one to three.
With all the rules represented, your initial setup should look like this:
That last rule requires some additional analysis. You can see above that it gives a pair of double-arrow relationships, where the first half of the rule creates the O2 arrow to the JK block, and the second half of the rule creates an arrow in the reverse direction (J and K together means O must be in 2). It effectively functions as an if but only if statement: “O is in area 2 if but only if J and K are assigned to the same area.” From that we can take a double-arrow contrapositive, where the absence of either condition (O not in 2, or J and K not together) tells us the other cannot happen.
At first glance, it may be tempting to delve into a deep analysis of the Numerical Distributions governing the assignment of rangers to areas. After all, there are only two distributions in play: 3-2-1 and 2-2-2. However, these distributions are largely unfixed because none of the rules limit the precise number of rangers assigned to each area, and this fact tends to limit their utility somewhat. (There is no rule telling us, for instance, that more rangers are assigned to area 2 than to area 1, or that exactly twice as many rangers are assigned to area 3 as to area 2).
More specifically, unfixed distributions are always worth noting, and depending on the game can prove quite significant, but as we will see they do little for us here. Why? Primarily because the three areas are not functionally identical: almost all the rules seek to limit the rangers’ assignment to particular groups, rather than distribute the rangers into functionally identical groups. Thus, a distribution of 3-2-1 would be much more useful if we knew precisely which groups have 3, 2, or 1 rangers assigned to them, i.e. if the distributions were fixed. So the numbers matter, but our ability to assign sizes to individual groups is extremely limited in this game and that restricts their overall impact.
A better way to proceed is to combine the rules in the hopes of making a few inferences. For instance, since L must be assigned to the same area as either K or M, but not to the same area as both, it is clear that K and M can never be assigned to same area as each other. Thus, K cannot be assigned to area 3.
And as noted previously the last rule deserves our attention: not only is it a complicated rule and thus risky to treat dismissively, but it is also one that “splits” the solutions to this game into two distinctly different directions: either O is assigned to area 2, or it is not. Let’s examine them separately:
Template 1: O is assigned to area 2.
In this template, K and J must be assigned to the same area as each other. Since K is never assigned to area 3 (see above), J cannot be assigned to area 3 either. So, it seems that the JK block must be assigned to either area 1 or area 2, however this is only partially correct. If J and K are both assigned to area 2 (along with O), then area 2 would be maxed out. In accordance with the second rule limiting the placement of L to the same group as either K or M, we need to assign L to area 3. Unfortunately, this leaves P to be assigned to area 1, in violation of the second rule. Since at least one ranger must be assigned to each area, this solution is impossible. Therefore, in Template 1 the JK block must be assigned to area 1:
Template 1: O is assigned to area 2.
The alternative scenario created by the last rule is to not assign O to area 2. In that case, we need to ensure that J and K are not assigned to the same area as each other. Furthermore, since O is never assigned to area 1 (second rule), this scenario requires that O be assigned to area 3 (along with M).
Template 2: O is not assigned to area 2.
We could continue building upon this template by using the LK block or the LM block, which would require splitting the solution off in three different directions (the LK block can be assigned to either area 1 or area 2, whereas the LM block can only be assigned to area 3). This approach could yield some benefit, but the risks (the time required as well as the potential for leaving a possibility out) outweigh the benefits for all but the most advanced test takers. It is better to develop a comfortable relationship with degrees of uncertainty, and move on to the questions!
This is a Partially Defined, Balanced, Identify the Templates, Grouping game.
The game requires us to assign six variables (rangers) to three different groups (areas). Since each variable is used exactly once, the game is Balanced. However, the game is only Partially Defined, because the number of variables in each group can vary from one to three.
With all the rules represented, your initial setup should look like this:
That last rule requires some additional analysis. You can see above that it gives a pair of double-arrow relationships, where the first half of the rule creates the O2 arrow to the JK block, and the second half of the rule creates an arrow in the reverse direction (J and K together means O must be in 2). It effectively functions as an if but only if statement: “O is in area 2 if but only if J and K are assigned to the same area.” From that we can take a double-arrow contrapositive, where the absence of either condition (O not in 2, or J and K not together) tells us the other cannot happen.
At first glance, it may be tempting to delve into a deep analysis of the Numerical Distributions governing the assignment of rangers to areas. After all, there are only two distributions in play: 3-2-1 and 2-2-2. However, these distributions are largely unfixed because none of the rules limit the precise number of rangers assigned to each area, and this fact tends to limit their utility somewhat. (There is no rule telling us, for instance, that more rangers are assigned to area 2 than to area 1, or that exactly twice as many rangers are assigned to area 3 as to area 2).
More specifically, unfixed distributions are always worth noting, and depending on the game can prove quite significant, but as we will see they do little for us here. Why? Primarily because the three areas are not functionally identical: almost all the rules seek to limit the rangers’ assignment to particular groups, rather than distribute the rangers into functionally identical groups. Thus, a distribution of 3-2-1 would be much more useful if we knew precisely which groups have 3, 2, or 1 rangers assigned to them, i.e. if the distributions were fixed. So the numbers matter, but our ability to assign sizes to individual groups is extremely limited in this game and that restricts their overall impact.
A better way to proceed is to combine the rules in the hopes of making a few inferences. For instance, since L must be assigned to the same area as either K or M, but not to the same area as both, it is clear that K and M can never be assigned to same area as each other. Thus, K cannot be assigned to area 3.
And as noted previously the last rule deserves our attention: not only is it a complicated rule and thus risky to treat dismissively, but it is also one that “splits” the solutions to this game into two distinctly different directions: either O is assigned to area 2, or it is not. Let’s examine them separately:
Template 1: O is assigned to area 2.
In this template, K and J must be assigned to the same area as each other. Since K is never assigned to area 3 (see above), J cannot be assigned to area 3 either. So, it seems that the JK block must be assigned to either area 1 or area 2, however this is only partially correct. If J and K are both assigned to area 2 (along with O), then area 2 would be maxed out. In accordance with the second rule limiting the placement of L to the same group as either K or M, we need to assign L to area 3. Unfortunately, this leaves P to be assigned to area 1, in violation of the second rule. Since at least one ranger must be assigned to each area, this solution is impossible. Therefore, in Template 1 the JK block must be assigned to area 1:
Template 1: O is assigned to area 2.
The alternative scenario created by the last rule is to not assign O to area 2. In that case, we need to ensure that J and K are not assigned to the same area as each other. Furthermore, since O is never assigned to area 1 (second rule), this scenario requires that O be assigned to area 3 (along with M).
Template 2: O is not assigned to area 2.
We could continue building upon this template by using the LK block or the LM block, which would require splitting the solution off in three different directions (the LK block can be assigned to either area 1 or area 2, whereas the LM block can only be assigned to area 3). This approach could yield some benefit, but the risks (the time required as well as the potential for leaving a possibility out) outweigh the benefits for all but the most advanced test takers. It is better to develop a comfortable relationship with degrees of uncertainty, and move on to the questions!
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