Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping/Linear Combination, Identify the Templates game.


The scenario establishes that a tour will visit exactly four cities that are selected from a pool of six cities. The selection of the four cities from six is a Grouping exercise, where as the ordering of the four chosen cities then is a Linear exercise. Thus, this is a Grouping/Linear Combination game.

• H J M O S T 6
  
  ___ ___ ___ ___ ___ ___ ___ ___
       1 2    3  4
  Group of 4 Cities Order of 4 Cities

Rule #1: This rule has two parts: first, it establishes that H and T are both in the group of four cities selected, and second it then separates them in the ordering of cities. As an opener, this is an incredibly powerful rule, and you should immediately recognize the major effect it has on this game.

First, from a Grouping standpoint, we now know 50% of the cities to be selected:

• _H_ _T_ ___ ___
  Group of 4 Cities

This means that of the four remaining cities—J, M, O, and S—only two can selected. That is extremely limiting, and is the first sign that perhaps this game will only have a small number of solutions as far as the group of four cities.

Additionally, since H and T must both be selected but cannot be consecutive, they are then limited in where they can be placed on the tour. Specifically, they cannot be in spots 1-2, 2-3, or 3-4, so they can only be 1-3, 1-4, or 2-4.



Rule #2: This is a straightforward grouping rule that impacts the selection pool:

• O S

Because O and S can never be included on the same tour, the remaining selection pool for the final two spaces is further limited, and is now restricted to: J, M, O/S. Since those cities are the only remaining options for the final two spaces, we can infer that either J or M must always be selected (because if you don't select at least one, then you would only have a maximum of three cities):

• _H_ _T_ _J/M_ ___
  Group of 4 Cities

The remaining space can be either the remainder of J/M, S, or O. This is extremely limiting, and it means that there are only five possible groups of cities that fulfill the rules:

• 1. H T J M
  
  2. H T J O
  
  3. H T J S
  
  4. H T M O
  
  5. H T M S

The ordering options for these five are further restricted, a point we will revisit after examining the final two rules.


Rule #3: This rule is also fairly straightforward, and simply indicates that when J is included, J must be visited third:

• J 3

This will clearly limit the options under each of the five grouping templates above. More on this shortly!


Rule #4: Similar to Rule #3, this is another rules that affects ordering when certain cities are included. In this case, if J and M are selected, they must be a rotating block:

• J3
  + JM or MJ
  M

Because the third rule stipulates that J is 3rd when selected, this means the block initially appears to have only two possible placements, in 2-3 or 3-4 (this is then affected by the HT rule, which prohibits J and M from being in 3-4, since that would force H and T into 1-2).

At this point, you could easily jump into the questions and be successful. The options for selecting cities is limited to just five choices, and then there are several rules that further limit what can occur. All in all, it's a very limited game despite having both a Grouping and Linear component! However, just to see some of the effects of the various rules, let's look at a few additional implications that occur under various scenarios.

• 1. Cities selected: H T J M. This scenario meets the conditions in the last rule, and so J and M must be a block. The other two cities—H and T—cannot be consecutive per the first rule. The combination of these two rules means that J and M must occupy the middle two spaces, which then allows H and T to be separated. And, since J must be third from the third rule, we can infer that M is second, resulting in the following template for this particular group of cities:
  
  • _H/T_ _M_ _J_ _T/H_
     1 2    3    4
  
  
  2. Cities selected: H T J O. With J in the group, J must be third from the third rule. From the first rule, H and T cannot be consecutive, and so they cannot jointly occupy the first two spaces. Thus, one of H and T must be fourth:
  
  
  • ___ ___ _J_ _T/H_
     1    2    3    4
  The first two visits are occupied by H/T and O, in some order.
  
  
  
  3. Cities selected: H T J S. This group of cities has the same basic setup as the prior group:
  
  • ___ ___ _J_ _T/H_
     1    2    3    4
  The first two visits are occupied by H/T and S, in some order.
  
  
  
  4. Cities selected: H T M O. The only active rule affecting this group of four cities is that H and T cannot be consecutive, so this template contains a fair number of solutions.
  
  
  
  5. Cities selected: H T M S. The only active rule affecting this group of four cities is that H and T cannot be consecutive, so this template contains a fair number of solutions.

While the last section of analysis is not necessary to do prior to attacking the questions, you will have to realize those inferences once you are working with the questions.

[midisson]

What is the set up here? I get so lost in sequencing with the OR pieces

[Dave Killoran]

Hi Midisson, there is now a complete explanation of the setup above. Thanks!