Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is a Grouping: Defined-Fixed, Unbalanced: Overloaded game.

The game scenario establishes that five candidates are available for three ambassadorships:

J K L N O5

powerscore_M12_T4_O2011_LG_explanations_game_2_diagram_1.png

Because there is only one available ambassadorship for each country (for a total of 3 ambassadorships) and five candidates total, this game is Unbalanced: Overloaded (5 into 3). Let us examine each rule:

First Rule:

The first rule establishes that at exactly one of K and N is assigned an ambassadorship:

• K/N 1

Because both K and N cannot be assigned, the following diagram can be used to capture that aspect of this rule:

• K N

With only three available ambassadorships and five candidates, the removal of K or N has a significant impact on the game because removing K or N mean that the pool for the three ambassadorships is reduced as follows:

• K/N, J, L, O

From a Grouping standpoint, this can also be represented visually as:

powerscore_M12_T4_O2011_LG_explanations_game_2_diagram_2.png

Consequently, if any one of J, L, or O is removed from consideration, the other two must be assigned ambassadorships. Understanding this relationship makes the game considerably easier.

Second Rule:

The second rule is a simple conditional rule, and can be diagrammed as:

• J K

Note that although this rule is simple in nature, the contrapositive of this rule has a major effect on the game because of the numerical limitations present. If K is not assigned an ambassadorship, then via the contrapositive J cannot be assigned an ambassadorship, and consequently the other three candidates—L, N, and O—must be assigned ambassadorships:

• [ K J ] L, N, O

Note: From the fourth rule, L would have to be assigned to Zambia.

The first and second rules can be combined easily, but we will address that connection after examining each of the four rules individually.

Third Rule:

The third rule is the first to specify a country, and it establishes that O cannot be assigned to Venezuela while K is assigned to Yemen:

• OV KY

Fourth Rule:

The fourth rule indicates that if L is assigned an ambassadorship, it must be to Zambia:

• L LZ

Consequently, this rule also creates two Not Laws for L under Venezuela and Yemen. And, while this rule appears innocuous at first, it can have a powerful effect on the game. More on this shortly.

With the four rules diagrammed, let’s take a moment to consider the inferences produced by combining the rules.

First and Second Rules Combined:

The first and second rules are naturally connected through K:

• J K N

This connection yields the inference that:

• J N

This inference is the answer to question #8.

First and Fourth Rules Combined:

The fourth rule, as mentioned previously, plays a significant role in this game. Because there are a limited number of “extra” candidates for the three ambassadorships, and because L is restricted to being assigned only to Zambia, if other variables are assigned to Zambia, further restrictions ensue. Specifically, consider what occurs when K or N is assigned to Zambia, because the assignment of either K or N eliminates the other variable from the candidate pool.

• When K is assigned to Zambia:
  
  When K is assigned to Zambia, from the first rule N cannot be assigned an ambassadorship, and from the fourth rule L cannot be assigned to Zambia, and thus L cannot be assigned an ambassadorship. Thus, when K is assigned to Zambia, N and L cannot be assigned ambassadorships, and J and O must be assigned ambassadorships:
  
  
  • KZ N + L = KZ, J, O
  When N is assigned to Zambia:
  
  When N is assigned to Zambia, from the first rule K cannot be assigned an ambassadorship, and from the fourth rule L cannot be assigned to Zambia, and thus L cannot be assigned an ambassadorship. But, with K removed from the candidate pool, from the second rule we can infer that J cannot be assigned an ambassadorship, leading to a situation where J, K, and L are no longer in the candidate pool. Because there must be three ambassadors, this situation produces an invalid result, leading to the inference that N cannot be assigned to Zambia:
  
  • NZ

Continuing on in this vein, because the fourth rule is so restrictive, let us examine what occurs when the remaining two variables­—J and O—are assigned to Zambia.

• When J is assigned to Zambia:

• When J is assigned to Zambia, from the second rule K must be assigned an ambassadorship, and from the fourth rule L cannot be assigned to Zambia, and thus L cannot be assigned an ambassadorship. Additionally, from the first inference, N cannot be assigned an ambassadorship, meaning that the three ambassadorships must be assigned to J, K, and O.
  
  With J is assigned to Zambia, K and O must be assigned to Venezuela and Yemen. But, K and O both figure in the third rule, and applying that rule we can infer that K must be assigned to Venezuela and O must be assigned to Yemen:

powerscore_M12_T4_O2011_LG_explanations_game_2_diagram_3.png

• Thus:
  
  • JZ KV, OY, JZ

• When O is assigned to Zambia:

• When O is assigned to Zambia, from the fourth rule L cannot be assigned to Zambia, and thus L cannot be assigned an ambassadorship. This leaves J, K, and N in the selection pool. But, from the contrapositive of the second rule, because the removal of K would also remove J from the pool (and thus leave an insufficient number of candidates), K must be in the pool. Applying the first rule, N therefore cannot be in the pool, and the candidate pool must be J, K, and O. There are no restrictions on the placement of J and K at that point:

powerscore_M12_T4_O2011_LG_explanations_game_2_diagram_4.png

• Thus:
  
  • OZ JV/Y, KY/V, OZ

Combining the previous information produces the final setup for this game:

J K L N O5

powerscore_M12_T4_O2011_LG_explanations_game_2_diagram_5.png

Inferences:

powerscore_M12_T4_O2011_LG_explanations_game_2_diagram_6.png

[anahi78]

How do you determine when to use a double error. The rules says if Ong is assigned as ambassador to V, Kayne is not assigned. The explanation says that O cannot be assigned to V while K is assigned to Y. Do we need to paraphrase the rule in this way? cannot/while. Thanks for the clarification!

Anahi

[KelseyWoods]

Hi Anahi!

You can use the Double Not Arrow whenever you have a rule that states: "If something, then NOT something else."

If we look at this rule, it says if O is assigned to V, then K is not assigned to Y.

You could diagram that like this:

Ov Ky

To make the contrapositive, we flip and negate:

Ky Ov

Therefore, if O is in V, then K cannot be in Y and if K is in Y, then O cannot be in V. The Double Not Arrow comes from combining those two contrapositives to give you a more efficient single diagram.

The specific wording "cannot/while" is not necessary to paraphrase this relationship but it is necessary for you to understand that the relationship is basically that you can't have both O in V and K in Y.

Hope that helps!

Best,
Kelsey

[anahi78]

Yes it...thank you!

[lorettan102]

Is the best way to handle this set up and problems by diagramming each ambassador assigned to Zambia? I did not and found it significantly slowed me down when it came to some of the questions.

Thanks.

[Claire Horan]

Because this game is quite restrictive, I decided to diagram all of the possibilities before tackling the questions. In all, there ended up being only eight solutions. One way of beginning the process is starting with what happens if L is included, since L must be placed in Zambia or not at all. The entry point I would recommend, though, was to first determine which combinations of three are possible, then figure out within those groupings, where each ambassador can be placed. I hope this was helpful!

[Sufficient_Necessary]

I split templates on “N” and then on “L”. My rationale was “N” constrains the board the most: if “N” is in, the out group is filled by “J” and “K” and everything else is in.

If “N” is out, then “L” is the most constrained variable with only two possible positions – in at Z, or out.

Once that’s built, I think it’s helpful to put a little arrow from J K and from V Y to remind me that those variables/slots have potential relationships.

Yields three templates, and then flew through the questions.

[Adam Tyson]

That looks like a great approach to me, as it appeals to my inclination to do templates whenever possible. In the template where N is out and L is in, you must have K, and then you have an option for either J or O. At that point the only rule you still need to worry about is the one about K and O at Y and V. Avoid that and you're home free!