DATCrusher PAT Breakdown:
Cube Counting

Ahh, cubes. Yet another favourite of mine. Besides being one of the four sections on the PAT that you can easily ace, it’s pretty fun. Cube counting can also raise your confidence levels going into paper folding (assuming you follow natural order of the PAT). Since this is one of the easier sections, I won’t beat around the bush any longer. Let’s get into it with a quick overview of what this section is all about.

Overview

You are given a figure that consists of multiple cubes that have been glued together. Imagine someone taking a bucket and pouring paint on to the figure from all directions like the figure above. The task then becomes to correctly identify how many cubes have a specific number of their faces painted. At this point it is good to outline some rules regarding the figure and how it is painted:

  • The bottom of the figure rests on the ground and is therefore not painted
  • Individual cubes cannot be solely connected to the figure via line angles
  • If a cube is not visible, it is assumed present only if it is supporting another cube or if it needs to be present for the figure to be fully connected
  • Ambiguity of individual cubes will not be present

Pretty straight forward right? Alright let’s get into what you really came here for, the strategies!

Approach

1. T-Table Technique

In this method, the student draws a T-table on the provided sheet of paper and systematically analyzes the figure while categorizing cubes by the number of faces that are painted. For organization, it is best to pick a row or column to begin with and move on only after marking each cube in the row/column. However, the order in which each cube is counted does not matter so as long as no duplicate marking occurs. Let’s see this method in practice.

1. Identify any row/column to begin with

In this example, the light shaded red cube in level A, located at the left-most side of the figure was chosen.

2. Begin analyzing cubes and work your way back and up

The shading of the colors represents the progress with light shades occurring before darker ones. The two red cubes on level A were recorded first with levels B, C, and D following respectively.

3. Continue until all cubes have been accounted for

Upon completion of the red section, the two green cubes on level A were accounted for next, followed by the two cubes above, on level B. Finally, the last section comprised of the single blue cube was analyzed and the tally completed.

4. Answer the question(s)!

It is important to note that the approach to where to start and how to progress along the figure is highly customizable. Start with the side that feels most natural to you and with practice, this will become second nature! It is also important to mention that although the tally chart in the example above included “0” for completion, this will likely never be tested on the DAT so feel free to omit it for efficiency purposes. Similarly, do not waste time labeling the tally chart with the header text. At the end of the day, your goal on test day is to be as efficient as you possibly can! For example, as a final check, one might want to count all tally marks to see if it matches up with the total cube count in the figure. However, with practice and confidence, this checking step can be ignored as it takes up valuable time.

2. Direct Marking Technique

In this method, instead of drawing a t-table, the student writes the number of faces painted for a specific cube on one of the faces of the same cube. Organization in this method is exactly the same as the t-table method. Simply pick a row/column to begin with and move on only after marking each cube in that row/column. Now this is where it gets a little tricky. Because of the nature of this strategy, it’s impossible to mark hidden cubes as no faces will be visible for writing. As a solution, whenever a hidden cube is encountered, the student can either write the number outside of the cube adjacent to the hidden cube, or simply write the number to the side. The below image portrays these two variations and the strategy itself.

It is important for you to try both strategies and see which one suits you best. You are also encouraged to modify strategies so they fit your test taking style. Now let’s look at another consideration that can push you from being just good at cube counting, to being great!

 

Considerations

Symmetry

Another important factor in becoming a master at cube counting concerns figure symmetry. In the examples above, I scanned each cube in the figure separately but if a particular row/column has a sister row/column, then tally marks can be doubled. This drastically reduces the amount of time one has to spend on each figure. In writing this may be a bit confusing so let’s get some pictures involved.

To begin this question, we proceed as normal and pick a row/column to begin with. However, this time it is in our best interest to begin at one of the portions of the figure that displays symmetry as shown in the figure. In the example above, simply doubling the tally marks for the cubes in purple will have essentially allowed us to analyze the cubes in blue by way of symmetry.  This subtle tactic has the ability to streamline the section even more! With that said, it is not wise to scout the figure for any particular symmetry. Only utilize this tactic if the symmetry is obvious and make sure that you do not end up double marking the symmetric cubes.

 

With that said, we have reached the conclusion of cube counting guide. You should now have an idea of how you can approach and excel at this section but nothing beats good ol’ practice so click below to try some questions out for yourself using our exclusive Cube Counting Generator.

Cube Counting Generator