IMAT Logical Reasoning · Spatial and Sequence Reasoning

Mirror Images, Half Turns and Folded Paper

A mirror or a half turn changes both each symbol and the order of the symbols; folding and punching paper is the same idea applied once per fold.

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Why this matters

The Cambridge papers asked about digital clock displays turned upside down (2013, 2021), a number plate seen in a mirror (2015), a word that reads the same from both sides of a sign (2021) and paper that is folded, cut and opened out (2016, 2020).

Concept 1 of 2: Letters and digits under mirrors and half turns

A mirror does two things at once: it flips each symbol, and it reverses the order of the symbols in a line. A half turn (turning the object upside down) also reverses the order. So read the transformed line from the other end, and then fix each symbol in turn.

Definition

A vertical mirror (a looking-glass beside the object) swaps left and right. It reverses the order of characters in a line.
A horizontal mirror (a reflection in still water) swaps top and bottom. The order of characters stays the same.
A half turn (rotation by 180°, "upside down") swaps both. It reverses the order too.
A symbol looks the same after the change only if it has the matching symmetry. On a seven-segment display (the digits of a digital clock), check each digit in the table.

Change	Order of characters	Capital letters that look the same	Seven-segment digits
Mirror on a vertical line	Reversed	A H I M O T U V W X Y	0, 1 and 8 stay; 2 and 5 swap; the others are not digits
Mirror on a horizontal line	Unchanged	B C D E H I K O X	0, 1, 3 and 8 stay; 2 and 5 swap; the others are not digits
Half turn (180°)	Reversed	H I N O S X Z	0, 1, 2, 5 and 8 stay; 6 and 9 swap; 3, 4 and 7 are not digits

Letters assume plain capitals with no serifs. A seven-segment 1 moves to the other side of its space but still reads as 1.

Practice this conceptself-check · 4 quick reps

A mirror reverses the order, not only the symbols

Mirroring each digit in place, without reading the line from the other end, gives a wrong answer that looks convincing. A vertical mirror and a half turn both reverse the order; only a horizontal mirror keeps it.

Concept 2 of 2: Folding, punching and unfolding paper

Each fold doubles the number of layers, so one punch through the folded paper makes one hole per layer. Unfolding is reflecting: each hole appears again as its mirror image across the fold line. Undo the folds in reverse order, reflecting every hole you have so far.

Definition

After $f$ folds in half, the paper has $2^f$ layers. A punch away from every fold line makes $2^f$ holes.
To unfold, reflect every hole across the last fold line first, then the one before, and so on.
Holes are symmetric about every fold line, so the unfolded pattern has a line of symmetry along each fold.
A cut on a fold line opens into a single hole, twice as wide, not two holes.

Holes after folding and punching

\text{holes} = \text{punches} \times 2^{f}

$f$ number of folds in half (punches not on a fold line)

Worked example

A square sheet 8 cm wide is folded in half by laying the right half over the left half. It is then folded again by laying the bottom half up over the top half. A hole is punched 1 cm from the left edge and 1 cm from the top edge of the folded square. Where are the holes after unfolding?

Practice this conceptself-check · 3 quick reps

A cut on the fold line makes one hole

When the cut touches the fold, the two halves of the hole are joined along that line and open into one shape, twice as wide. Counting it as two separate holes is wrong.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Folding, punching and unfolding paper
Holes after folding and punching
$\text{holes} = \text{punches} \times 2^{f}$

Reference tables (1)

Letters and digits under mirrors and half turns3 rows

Change	Order of characters	Capital letters that look the same	Seven-segment digits
Mirror on a vertical line	Reversed	A H I M O T U V W X Y	0, 1 and 8 stay; 2 and 5 swap; the others are not digits
Mirror on a horizontal line	Unchanged	B C D E H I K O X	0, 1, 3 and 8 stay; 2 and 5 swap; the others are not digits
Half turn (180°)	Reversed	H I N O S X Z	0, 1, 2, 5 and 8 stay; 6 and 9 swap; 3, 4 and 7 are not digits

Letters assume plain capitals with no serifs. A seven-segment 1 moves to the other side of its space but still reads as 1.

Watch out for (2)

A mirror reverses the order, not only the symbols
→ Letters and digits under mirrors and half turns
A cut on the fold line makes one hole
→ Folding, punching and unfolding paper

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