DATCrusher PAT Breakdown:
Paper Folding

Many people have difficulties when it comes to this section because they think that they need to exercise a great deal of concentration in order to “fold” the given pattern and deduce which answer choice represents the correct 3D model. Well fret not my friends, I am going to show you three approaches that if utilized correctly, can help you conquer the pattern folding section with barely having to mentally fold the pattern!

Different Techniques

1) Odd One Out Technique

This technique works best when the the question contains an array of unique faces.

 1. Scan the option choices for a unique face

Once you have identified a face with varying geometry you can focus on it and move on to the second step.

2. Analyze the unfolded pattern for the presence/absence of the unique face

If the identified face IS NOT present in the unfolded pattern, then that specific option choice has no possibility of being the answer. If the identified face IS present, then that specific option choice has potential to be the correct answer but will have to be further analyzed to confirm.

3. Repeat steps 1 & 2

Steps 1 & 2 should be repeated until one answer option remains. At this point, you will be ready for the fourth and final step.

4. Answer the question!

Lets look at an example to put the above steps into practice!

At first, this question seems challenging. However after careful examination, we can see that the following colored shapes within option choices B,C and D are NOT found in the unfolded pattern and thus these answer options cannot be valid answers. So, based on the process of elimination and simple observation, we are able to easily determine that option A, is the correct option choice.

2) Face Exclusion Technique

This method requires us to focus only on the relevant faces within the unfolded pattern and rotate them so that they can be positioned relative to the option choices provided to us. This method is optimal for unfolded images that contain identical sides with patterns, and is best described by using an unfolded die. People often have difficulties with these types of questions because they either try to mentally fold and manipulate the pattern OR mentally unfold and manipulate the answer choices. With this technique however, these type of patterned questions become easy. Lets look at an example and work our way though each answer option in order to determine which is the correct one.

Option A

In this option choice, we are only presented with the five, three, and two faces of the die. Thus we can ignore the other faces within the unfolded pattern and focus solely on the relevant faces

We now need to rotate the unfolded pattern 90 degrees to the right. In other words, we need to rotate the unfolded pattern so that the 5-face is positioned on top, 3-face is positioned on the bottom right, and the 2-face is positioned on the bottom left.  This ensures that the orientation of the unfolded pattern matches the orientation of option choice A. At this point we are able to compare our modified unfolded pattern with the option choice A to determine whether it is correct or not.

By comparing the two, we can quickly conclude that option choice A is incorrect as the orientation of the 3-face does not match the orientation seen in the unfolded pattern.

Option B

 For this option choice, we again isolate the relevant faces and rotate the unfolded pattern by 90 degrees but this time to the left. After comparing the modified unfolded pattern with the option choice, we see a situation similar to that of option choice A. The 3-face of the die does not match the orientation seen in the unfolded pattern.

Option C

This option choice requires us to rotate the modified unfolded pattern by 180 degrees. Once again, the culprit is the 3-face of the die. This option choice is also incorrect.

Option D

By process of elimination we have deduced option D to be the answer for this question. In order to confirm we must rotate the modified unfolded pattern 90 degrees to the right and focus on the five-face, three-face, and the two-face. By analyzing the picture below, we can see that the orientation of all faces is represented correctly in option choice D, therefore it is confirmed to be the answer to this question.

The Face Exclusion Technique is more complicated than the Odd One Out technique so we will strengthen the principles by tackling a question that deals with a less familiar pattern and shape. Again, solving this question will barely involve the need to fold the unfolded pattern. Instead, we will be required to know how specific faces are oriented relative to one another. Lets begin with the last option choice!

Option D

This option choice focuses on white triangles touching black triangles with their bases. We have added red stars to the relevant white triangles on both option choice D as well as the unfolded pattern in order to aid the visualization process.

Analyzing the picture above, we can see that the unfolded pattern does in fact contain two white triangles whose bases are in contact with black triangles (Outlined in red). However, these segments will never be in contact with each other as other segments exist between them (Outlined in green). Furthermore, the two red segments of interest will point in the same direction according to the unfolded pattern. This also contradicts option D which portrays the two red segments to be in opposite directions. If one point was not enough to mark option choice D as the incorrect answer, we now have two!

Option C & B

Since these option choices differ only by a 180 degree rotation, we will consider them together. Inspecting the two, we can see that there are two side-by-side white triangles that are contacting black triangles with their bases. This is in fact the same situation we faced in option D however this time the two segments are pointing in the same direction. We have again outlined the relevant segments in order to aid the visualization process.

Again, we can see that the two red segments of interest will never be in contact with one another as other segments exist between them (Outlined in green). Therefore, option choices C & B can be marked as incorrect.

Option A

By process of elimination, we have deduced option A to be the answer for this question. In order to confirm, we can focus on the relevant faces and compare the two. By doing so, we can see that we are dealing with a segment of two black triangles and a segment of a white triangle in contact with a black triangle through its base. These segments have been outlined in red to aid in the visualization process.

Analyzing the picture above, we can see that option A matches the unfolded pattern perfectly. The graphic below portrays the the exclusion of irrelevant faces and the final rotation that was done in order to match the orientation of option choice A.

Important Note: When you are first starting to tackle these questions, you might find that you have to rotate your head to the side in order to rotate the unfolded pattern. However, as you become more acquainted with the PAT, you will be able to mentally rotate either the answer choices or the unfolded pattern to your liking, which will ultimately this section much easier.

3) Cut and Paste Technique

This method is best used when trying to discern option choices that contain patterned shapes as seen in the example we will be working with below. Initially, this modified die question seems like a hassle to deal with, especially if one is to mentally fold the pattern into the correct 3D shape. However, by mastering the cut and paste technique, you can effectively come to visualize how different patterns on faces will touch one another all without ever folding the given pattern. Lets look at the example and work our way though each answer option in order to determine which is the correct one.

Option A

This option choice requires us to determine how the patterned square faces are oriented relative to one another. If we focus our attention  at the patterned faces in option A, we see two faces which show black rectangular patterns in contact with white rectangular patterns along their long side (outlined in red). Furthermore, we can say that the two black rectangular patterns contact each other through their corner points (circled in green).

The unfolded pattern that is given to us contains three faces which contain the black and white rectangular pattern and I have labeled them 1, 2, and 3 in order to help us keep our thoughts organized. Further analyzing the pattern, we can see that the scenario in option A, where two patterned faces are contacting each other can only occur with faces 1 and 2, or 2 and 3. Faces 1 and 3 will never be in contact with each other as face 1 is surrounded by other faces.

To progress in solving the question, let us first focus on faces 1 and 2 to determine if they give rise to the pattern combination found in option choice A. Looking at the junction of the two faces, we can see that the two black rectangular patterns do not contact each other through their corner points as was circled in green in the above image. Therefore it is easy to tell that this combination will not result in the orientation seen in option choice A.

Now let us focus on faces 2 and 3 to determine if they give rise to the pattern combination found in option choice A. Before determining how these two faces will contact each other, we first need to see how they are oriented relative to the surrounding faces (in this case, the white cube that is separating faces 1 and 3). And remember, one of the rules for paper folding is that the pattern must always be folded inwards. This is what allows us to be able to “cut”, “rotate”, and “paste” specific faces and determine their orientation relative to neighboring faces as shown below.

In the above example, we cut face 2, rotated it 90 degrees counter-clockwise, and pasted it to the right adjacent white face. However we want to know the orientation of face 2 in relation to face 3, therefore we must cut the face as we did above, but this time rotate it 90 degrees clockwise so it contacts the left adjacent white face, and repeat once more to have it in contact and pasted with face 3. The graphic below demonstrates the steps more clearly.

Now that face 2 is where we want it to be in relation to face 3, we can focus on it and compare with what it presented in the option choice to see if they two are compatible. By inspecting the image below, we can see that the new unfolded pattern we generated gives rise to a pattern that does not match option choice A. Therefore, we can now eliminate option A as we have considered both cases where two patterned faces contact each other. Let us skip option choice B and try option C next.

Option C

Similar to option choice A, option choice C shows two patterned faces in contact with each other. From our work on option choice A, we know this only occurs with faces 1 and 2, and 2 and 3.

Also from our previous work, we know how these faces will be oriented towards one another. Therefore, we simply have to compare what we have already done, to the option choice to see whether the two are compatible. Let us look at the combination of faces 1 and 2 first. We can see that the bottom right face in option C shows the black rectangular pattern to be on the right. This contradicts the unfolded pattern which displays the black rectangular pattern to be on the left (face 2)

And now let us look at the combination of faces 2 and 3. We can see that the top face in option C shows the black rectangular pattern to be away from the black rectangular pattern seen on the bottom right face. This contradicts the unfolded pattern which displays the two black rectangular patterns in contact with one another. Let us now consider option choice D

Option D

This option choice is unique in that it displays all three patterned faces to be contacting one another. From our previous work, we know that faces 1 and 3 will never be in contact with one another and therefore, we can eliminate option choice D. And finally, let us consider option choice B.

Option B

By process of elimination, we can conclude that option choice B is correct. However, let us validate this claim and determine which squares give rise to the pattern by using the cut and paste technique. It is important to note that as you become comfortable with rotating specific faces to adjacent areas, you will no longer have to think about the actual process of cutting out the piece, separating it from the overall pattern, rotating it, and finally placing it back where it belongs. Instead, you will simply topple the face in question onto an adjacent face and visualize how it’s pattern will be oriented in relation to the rest of the faces.

We must now figure out which face is being presented in option choice B. By analyzing the pattern we can see that face 2 will not give rise to the pattern seen in option B as it’s horizontal length as seen in the pattern will never be in contact with a white face. Or more simply, if we rotate face 2 90 degrees clockwise and attach it to the left adjacent white face, we will not get option B once folded.

Similarly, face 3 will also not give rise to the pattern seen in option B as it’s black rectangular pattern is located on the bottom of the face rather than the top as shown in the option choice. This is better visualized when the top white face on the unfolded pattern is moved using the cut and paste technique so it matches the two white faces on option choice B.

Now that the top white square is where we would like it to be, we can focus on the segment and compare it to what is presented in option choice B. Again, we can see that once folded, the black rectangular pattern on face 3 will be positioned on the bottom of the face instead of the top as shown in the option choice. Therefore, option choice B is not compatible with face 2, or 3!

The only face left to analyze is face 1 which to most of us would have been the obvious answer from the beginning but it is beneficial to go through all the possibilities as to why the other faces are incorrect. No cutting or pasting is required to analyze this one as all the faces are already in their proper spots!

There you have it, we have successfully completed a pattern folding question without ever folding the unfolded pattern. All you need to keep in mind is the relative orientation of the shapes and how they will contact one another. This skill can easily be acquired and mastered with continual practice of the technique.

 

With that said, we have reached the conclusion of  the paper-folding guide. You should now have an idea of how you can approach and excel at this section. Although you may be slow at first, with practice this section becomes one of the easiest sections should you use these techniques to your advantage. And remember, paper-folding is definitely one of the sections of the DAT that can be easily aced with efficient and proper practice! Click below to head over to the Classroom and start practicing:

PAT Practice Tests