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Hole Punching (HP) Questions

What is hole punching?

Hole punching problems are the fourth group of problems that appear on the perceptual ability test (PAT) of the dental admissions test (DAT) and comprise Items 46-60. In these problems, a flat square sheet of paper is folded one or more times. After the last paper fold, one or more holes are punched in the paper. Your task is to mentally unfold the paper and determine where the hole or holes would be in the unfolded paper.

4 - Hole punching 1 1.svg
Here are the rules:

A dotted line allows you to keep track of the original position of the unfolded sheet of paper, which is always a square.


The folded paper always remains within the edges of the original square, and the paper is never flipped or twisted. 


Each image leading up to the last image shows a consecutive paper fold.


The last image shows the position of the hole punch (or hole punches), which is indicated by a white circle. Each hole punch extends through all layers of paper located at the position of the hole punch.


In the answer options, the pattern of black circles indicates where holes would end up in the unfolded paper. Try mentally unfolding to paper, as below, to envision where the hole punches will end up:

4 - Hole punching 1 2 (2).svg

Note: Directions given on test day may be formatted differently, but objectives are the same.

More information about the paper folds

During the folding steps, the paper may be folded horizontally, diagonally, or vertically.

Frame 1301.svgFrame 1302.svgFrame 1303.svg

The paper can fold in the same direction, or fold back on itself.

Frame 1304 (1).svgFrame 1305 (1).svg

In some problems, the paper will fold past the layers underneath; in other problems, the top layer of the paper may fold back on itself.

Frame 1307.svgFrame 1309.svg

Use the folding maps in our explanations to better understand confusing folds and where the layers end up. Note that the layers of paper a hole goes through will determine how many holes it forms on unfolding. (So the answer here must be E.)

b - HP Partial horizontal fold, diagonal fold, partial vertical fold  - Question 3.svgFrame 1310.svg

Example problem

Here is a typical hole punching problem you’d encounter on your DAT, along with an Erudition-style explanation to help you understand how to solve it.

b - HP Full horizontal fold, diagonal twist - Question.svg

Notice the paper is folded horizontally, then diagonally; so, it is unfolded across a diagonal axis and then a horizontal axis.

c - HP Full horizontal fold, diagonal twist - Overview.svg
Step by step unfolding:

Notice how many layers are formed during the folding steps; notice how the bottom of the paper folds up such that the top area has only two layers.

d - HP Full horizontal fold, diagonal twist - Layers.svg

During the first unfolding step, the hole in the top left is moved to the bottom right (its layers do not separate). The hole originally in the bottom left, however, is reflected across the diagonal axis of the paper fold, as its layers separate.

HP Full horizontal fold, diagonal twist - Analysis 2.svg

The second unfolding step creates three more holes. Notice how the first three holes are reflected across the horizontal axis of the paper fold.

HP Full horizontal fold, diagonal twist - Analysis 2 (1).svg

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HP tips & strategies!

Familiarize yourself with the hole punching grid. Remember that holes are always reflected across the axis of the paper fold such that they end up in a symmetrical location on this hole punching grid.

Group 1320.svg

When you get to a hole punching problem, make sure you understand how it’s folding first.

HP 1 - Summary - Section 4 - 2.svg

Then, track the unfolding steps in your head to see where the hole would end up.

HP 1 - Summary - Section 4 - 7.svg

You can use your scratch paper to draw a hole punching grid if it helps you to keep track of holes.

HP 1 - Summary - Section 4 - 12.svg

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