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V. GAME TYPE 2: CHARACTERISTIC GAMES

This next game type that we will examine are called characteristic or attribute games. As the name implies, these games require you to assign attributes or characteristics to each element of the game. In a later section, we will describe grouping games, which involve grouping together two or more elements of a game based on a shared characteristic. Characteristic games differ, however, in that you are assigning characteristic to the separate elements, and not grouping the elements together in any way. This may sound a bit confusing at first, so why don't we instead look at an example of a characteristic game.

Sample Game

Priscilla has a small store where she sells upholstered furniture. She sells armchairs, couches, loveseats, futons, and recliners covered in several different fabrics: floral, plaid, vertically striped, solid, or mixed. She wants to display examples of her work in the storefront window. Because the window is quite small, only one piece of furniture can be displayed at a time. She plans to display a different item of furniture for each of the next five months. Also, each of the five items of furniture will be covered in a different fabric. Priscilla plans her display according to the following conditions:

The couch must be displayed before the futon.

The loveseat can only be displayed after the recliner.

The plaid item of furniture must be displayed before the one covered in mixed fabric.

The vertically striped piece of furniture must be displayed in the third month.

The futon must be covered in floral fabric.

Solving

The first step is to sort out what problem we are being asked to solve. In this case, we are asked to assign five different items of furniture to five different months with five different fabrics. This is actually a very typical example of characteristic games: we are asked to make a schedule of some sort taking into consideration at least two different characteristics. (In this problem the two characteristics are items of furniture and fabrics.)

Earlier, we had mentioned that it is important to create a roster of the elements of a game. For characteristic games, however, the elements may be more straightforward, such as the months of the week, Monday through Friday, or the numbers one through seven, etc. For this game the elements are five months, which we can simply number as one through five. The different characteristics, however, may benefit from being abbreviated as we usually do in creating our rosters. In this case, we have a list of possibilities for each of the two characteristics:

Furniture = armchair, couch, loveseat, futon, and recliner
Fabric = floral, plaid, vertically striped, solid, and mixed.

Let's just write these over in abbreviated form using only the first letter for each word:

Furniture = A, C, L, F, R
Fabric = f, p, v, s, m

Notice that we have used upper case letters for one list and lower case letters for the other. This will help to keep the two characteristics separate. This sounds simple at this point, but it is very easy to get caught up in trying to solve a game and forgetting what the individual letters represent. (You then might end up with trying to schedule things like a futon recliner or a floral plaid, instead of a floral futon and plaid recliner!

Now that we have converting the lists to a more abbreviated form, what next? While there is not a single diagramming approach that works well for all characteristic games, these game problems are usually best solved with a matrix. A matrix can be thought of as a kind of chart that we create with the game's elements in columns and the characteristics in rows (or vice versa, either direction will work). Again, this is easier to show, than to describe. So, what would the matrix look like for the above sample game?

We can start by listing the game elements in a row across the top, and then list the characteristics along the left side, and then draw lines horizontally and vertically to form a grid like this:

Now, what information can we place in our matrix? As with all of our game diagramming, let's start with the fixed conditions. We have one easy one, the first condition. In this case, we know that the vertically striped piece of furniture must be displayed in the third month so we can put our vertically striped symbol (v) into the second row under column 3 (for the third month).

Can we put anything else into our matrix right away? Well, the first two conditions tell us that the couch must be displayed before the futon, and the recliner before the loveseat. That means that neither the futon nor the loveseat can possibly be displayed in the first month, since the couch and recliner have not yet been displayed. So we can fill these into the matrix also, as ^F and ^L for not futon and not loveseat.
Now, you might think that these are the only two pieces of information in these two conditions, but is there anything else that we can conclude from these statements?

Well, if we just turn around the logic a bit, we will realize that if the couch needs to be shown before the futon, and the recliner before the loveseat, the couch and recliner cannot be the pieces of furniture displayed in the last month. (If the couch were displayed in month five, the futon would not have been able to be displayed.)
We can add these two pieces of information to the matrix also, in the top box of the last column as ^C and ^R. Further, we still have the original statement that the couch precedes the futon and the recliner precedes the loveseat. To be sure we do not forget this important information, we will write these symbolically to the side of our matrix as C < F and R < L.

Can we do anything with the condition stating that the plaid piece of furniture needs to be displayed before the mixed one?
Yes, using the same reasoning we applied in thinking about the last two conditions, we can conclude that the mixed piece of furniture cannot be displayed in the first month, and the plaid cannot be displayed in the last month. Again, we represent this as ^m for not mixed and ^p for not plaid. Additionally, we will write this to the side of the matrix as p < m. Now we are left with one condition, that the futon will be floral. If we knew in which month the futon was to be displayed, we could of course fill that into the matrix along with the fabric, but we don't know yet. If we look at our matrix, we see that the futon cannot be displayed in the first month, so we can also conclude that the material of fabric for that first month cannot be floral. So now we fill in that second row, first month box with ^f for not floral. Are there any other conclusions we can add to our matrix before we look at the questions? Well, if we look over our matrix, we can see that we already know that the piece of furniture displayed in the third month is vertically striped. If it is vertically striped, we know it can't be the futon (which must be floral), so we can add a ~F to the top box of month 3. Great

Now, don't be discouraged if it seems like we still don't know much about which piece of furniture (and in what fabric!) will be displayed in which month. We have a lot of information about which pieces cannot be displayed in certain months, and that information will be very helpful to us. One other thing to note about the way in which information has been put into the matrix is that it is a good idea to put "not conditions" (like no futon or ^F for example in the top left box) to the top of the individual boxes, and to put "fixed conditions" (like vertically striped or v for the third month) in the center part of the box. Visually, this will help you to further differentiate between the conclusive assignment of a characteristic and a statement that just limits the assignment of a characteristic. (For example, a conclusive assignment would be: The dog is brown. A statement that is non-conclusive and only a limit of an assignment would be: The dog is not white. In this case, the dog could be gray, black, brown, spotted, etc.)

For this game, all of the conditions have now been used in some form and filled into the matrix, and all additional information has been written to the side. It will not always be the case that at least part of every condition will be able to be added in some form to the matrix. For example, if we were told that the armchair was not the piece of furniture covered in solid fabric, where would we put that in the matrix? Because it is not related to any of the other conditions or able to be related to any of the boxes of the matrix that we have already filled in, we have no place to put this information in the context of the matrix and would instead write it to the side for later use. So, if you complete your matrix and there are conditions that you are not able to translate into an appropriate box on the matrix, translate this condition symbolically and write it immediately to the right side of the matrix. In that way there will be less chance that you will forget about this additional information later, when you might actually need it to answer a question. Once you have convinced yourself that you have extracted all possible information from the conditions and either added it to the matrix or represented it to the side, it is time to look at the questions.

Let's now look at some sample questions and then work through how to best solve them with the help of our matrix.

Question One:

Which of the following items of furniture could be displayed in the first month?

(A) floral futon
(B) vertically striped recliner
(C) mixed armchair
(D) floral loveseat
(E) plaid couch

To solve this, let's look at what we have already written under the first month column. We have a ^F, ^L, ^f, and ^m. Now let's look through the answer choices. Choice A is not correct because the futon cannot be displayed. Just looking at the first column, we cannot rule out Choice B, so let's come back to that one. Choice C is not correct because we know that we cannot have the mixed fabric. Choice D is not correct because we cannot have the loveseat in the first month. Choice E, the plaid couch sounds reasonable, as there is nothing in our first column to indicate that this is not a possibility. So now we have narrowed it down to two choices, B and E. Let's now consider the rest of our matrix. Is there any other box filled in that has to do with recliners or futons? Just that the futon cannot be in the third month and the recliner cannot be in the last month, so those conditions do not help us figure out what can happen in the first month. What about the conditions of vertically striped or plaid? Are there any other conditions filled in to distinguish between those possibilities? Yes, we know that the vertically striped piece of furniture will be displayed in the third month, therefore we can eliminate Choice B as a possibility. Choice E is the correct answer.

Now, from this example you should have learned one more important thing about the effective use of the matrix. Once you have put all the conditions into the matrix, and summarized any additional ones to the side, you should always look over the matrix and see if there are any additional conclusions that you can make if you just consider each box of the matrix in relation to the other boxes. For example, we had already filled in the fabric box for the third month as vertically striped (v). What does this tell us about the fabrics in all the months? They can't be vertically striped! Thus, at that point (still before reading the question!) we should have gone through the fabric row and filled in ^v (for not vertically striped) for each other month. Our new, improved matrix looks like this:

Using this new matrix, let's try some more questions.

Question Two:

Which of the following could be the items of furniture displayed in the five months, from one to five, respectively:

(A) couch, recliner, armchair, futon, loveseat
(B) recliner, futon, couch, loveseat, armchair
(C) recliner, couch, futon, loveseat, armchair
(D) couch, futon, loveseat, armchair, recliner
(E) couch, loveseat, recliner, futon, armchair

This is relatively straightforward to answer, using the matrix we have created. Choice A seems to work, but let's check the other choices. Choice B is incorrect because the futon can only be displayed after the couch has been displayed. Choice C is incorrect because the futon cannot be displayed in the third month because the futon must be floral and the piece of furniture being displayed in the third month is vertically striped. Choices D and E are incorrect because the loveseat cannot precede the recliner. Therefore choice A is the correct answer.

Question Three:

If the loveseat is displayed in an earlier month than the vertically striped piece of furniture, which of the following must be true:

(A) The loveseat must be displayed in the first month.
(B) The futon must be displayed in the fourth month.
(C) The recliner must be displayed in the first month.
(D) The piece of furniture covered in solid fabric must be displayed in the fifth month.
(E) The armchair must be displayed in the third month.

First, let's see if we can add the new piece of information from the question to our matrix. What do we know about when the vertically striped piece of furniture must be displayed? It must be shown in the third month. Therefore the loveseat must be displayed before that, which would be either the first or second months. Now let's look at those boxes. In the box for the first month, we have the ^L symbol for not loveseat. Since this possibility is eliminated, the loveseat must be displayed in month 2, so we can now add this piece of information to our matrix. (Remember! Any additional information you add that is derived from a question and not from the initial premise or conditions, must not be carried over to the next question.)

It is critical to remember that whenever you add any piece of information to a box in the matrix, evaluate whether or not it has any impact on any of the other boxes. Now that we have put L in the column for month two, what implication does this have on month one? If we look at the conditions we have written to the side of the matrix, we remember that the couch must precede the loveseat. If the loveseat will be shown in the second month, we can conclude that the couch must precede the loveseat.

Now we can go to the question and work through the answer choices one at a time. Choice A is obviously not correct, since we have just deduced that the loveseat must be displayed in the second month. The futon in the fourth month? Certainly possible, but must it be in the fourth month? Let's wait on that one. Choice C, the recliner in the first month- well, we just figured out that this must be true! Choice C is the correct answer. What is the best strategy if you don't immediately see the right answer? When you are asked whether or not statements must be true, the easiest way to check this is to try to find another arrangement. Is it necessary to have the futon in fourth month? Well, see if you can come up with a viable arrangement with the futon being displayed in the fifth month. Does it work? Yes. You will find that the other three possibilities B, D, and E, (not A, since we ruled it out as not being possible) are only that, possibilities, and not mandatory "must" statements.

Summary of Characteristic Game Strategy

Let's summarize the strategy we have used to solve this game:

First, figure out what problem we are being asked to solve. What are the elements of the game? What are the characteristics you are being asked to assign to these elements?

If you need a roster of the elements, create that next. (Remember, often you will not need a roster of elements for this kind of game.) Next, convert the lists of options for the characteristics to a more abbreviated form. You may use upper case letters for one list and lower case letters for another. What happens if there are three (or more) characteristics? It is still a good idea to have the abbreviations for each list look different. Say we are asked to assign houses (colored red, yellow, and white) and professions (lawyer, teacher, carpenter) and favorite activities (gardening, sewing, and baking) to three people. One possibility is the use of subscripts, such as is shown for these lists:

Houses = Rh, Yh, Wh

Professions = Lp, Tp, Cp

Activities = Ga, Sa, Ba

Next, create a matrix by listing the game elements in a row across the top, and then listing the characteristics along the left side, and drawing lines in the form of a grid.

Start to add information to your matrix, starting with the fixed conditions. When you write information in the boxes, place "not conditions" (like ^F) to the top of the individual boxes, and place "fixed conditions" (like v) in the center part of the box.

Summarize any conditions that cannot be put onto the matrix to the right.

Look over the matrix and see if there are any additional conclusions that you can make if you consider each box of the matrix in relation to the other boxes, or if you consider the conditions summarized at the side with respect to any of the boxes in the matrix.

Read the question. If there are additional conditions within the question, add them to the matrix or to the side. (Of course, you will be sure to remember that you will not carry these new conditions over to the next question!)

Answer the question by using the matrix to evaluate each possibility.

1: Ordering Games
2: Characteristic Games
3: Grouping Games
4: Network Games
5: Non Linear Spatial Games
6: Map Games

Continue to:

III. GAME TYPE 3: GROUPING GAMES
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