Earlier, we 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, so we can keep the two characteristics separate. (Otherwise you 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 converted 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?

Let’s 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.

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.

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?

If we think about the logic behind it, 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.

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?

We can also conclude that the mixed piece of furniture cannot be displayed in the first month, and the plaid cannot be displayed in the last month. We represent this as ^m for not mixed and ^p for not plaid. We are left with one condition, that the futon will be floral. 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 (^f). 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.

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” to the top of the individual boxes, and to put “fixed conditions” in the center part of the box.

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.

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 other months? They can’t be vertically striped!

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

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