How to solve Paper Folding questions

Paper Folding questions ask you to imagine a folded sheet with holes punched through it, then choose how the holes look after the sheet opens. The official TACHS handbook includes this picture-question family in its Ability samples. The examples here are original teaching diagrams with explanations, not released official items.

The central idea is reflection. When you open a fold, a hole on one side produces a matching position across the crease if both layers were punched. That position must be the same perpendicular distance from the crease. It does not move by an arbitrary amount or turn into a different shape.

Read the fold before counting holes

Find the starting sheet, the crease, and the direction of movement. A dashed line marks a crease in these examples. The arrow shows which part moves. The open circles on the folded paper show punched holes, not printed dots that remain on only one layer.

Keep the page orientation fixed. If the right half folds onto the left half, the remaining folded rectangle occupies the left half of the original square. A hole close to its left edge will reflect to a position close to the right edge when the fold opens.

Do not assume that every punch always doubles at every fold. A hole on a crease or a punch through an unusual layer arrangement needs closer inspection. The examples below use punches away from creases and through all layers, so each unfolding creates distinct matching holes.

Worked example 1: One vertical fold

Fold the right half of the square onto the left half. Punch one round hole through both layers, halfway across the folded rectangle and one quarter of the way down. Which choice shows the opened sheet? The square does not turn in any of the drawings.

The answer is D. When you open the sheet, the hole appears again on the other side of the fold, like a mirror image. The result has two holes at the same height, each one quarter of the way in from a side edge.

A has only the original hole, so it fails to show the hole through the second layer. B puts the two holes on a diagonal, which changes height even though the fold was vertical. C gives four holes from a single punch through two layers. E puts two holes on the same side of the crease.

Only D preserves both the height and the equal distances on opposite sides. You can reject B without measuring the exact distance because its second hole has moved downward. A vertical reflection changes left and right, not up and down.

For a numerical check during learning, imagine the square has width one. A hole at one-quarter of the width reflects to three-quarters of the width. Its height remains one-quarter of the sheet. The two final positions are therefore one-quarter and three-quarters across, both one-quarter down.

Worked example 2: Two folds

First fold the right half onto the left. Then fold the bottom half of that rectangle upward. Punch one hole through all four layers at the center of the remaining upper-left quarter. Which option shows the sheet fully opened?

The answer is A. Undo the second fold first. The hole in the upper-left quarter gains a matching hole in the lower-left quarter. Undo the first fold next. Each of those two holes gains a matching hole on the right half.

The four final holes are at the centers of the four quarters. A shows exactly that arrangement. B has only two holes, so one unfolding has been omitted. C places holes in the far corners rather than the quarter centers. D leaves a single hole. E has four holes but moves the right-hand pair too close to the center crease.

The count helps you reject B and D, but it does not distinguish A from C or E. Location completes the proof. Each point must keep its distance from each crease as the folds open. The result must be symmetric about both center lines.

Using the same unit-square check, the punched location is one-quarter across and one-quarter down. Reflection across the horizontal midline adds one-quarter across and three-quarters down. Reflection across the vertical midline adds the two matching points at three-quarters across.

Avoid sliding and rotating the holes

A frequent mistake is to move the hole in the direction of the fold arrow when imagining the paper opening. The arrow describes the paper’s movement when folded. It does not tell you to slide a final hole along the page.

Another mistake is turning the entire sheet mentally and then reading answer choices as though the page had stayed still. Keep a fixed top edge. Describe holes as upper-left, upper-right, lower-left, or lower-right relative to that fixed orientation.

For diagonal folds, the same reflection principle applies, but horizontal and vertical positions can both change. Learn the orthogonal examples first, then follow the particular diagonal crease shown in a later question. Do not replace a diagonal reflection with a quarter-turn.

Practice with paper, then work mentally

At home, fold a square and mark a proposed punch location before making a hole. Predict the result first, unfold it, and compare the prediction with the actual locations. Keep scissors and hole punches under appropriate adult supervision.

Use that demonstration to learn the geometry, not as a required tool for every practice question. The official Parent Guide says scratch paper is allowed for Reading, Written Expression, and Mathematics and must be discarded after Mathematics. It is not a tool to rely on for the official Ability questions.

When reviewing mental practice, identify the exact unfolding where the error occurred. Check whether you forgot an existing hole, reflected across the wrong crease, or used unequal distances. Then solve a different configuration that targets that step.

TACHS Prep uses an estimated 16 Paper Folding questions in ten standard-time minutes. These are practice estimates based on public information. Read the six-part guide for the full practice model, and learn the distinct methods for Figure Matrices and Figure Classification as well.