Reasoning · Lesson 5

Mirror, Water & Hidden Figures

Learn exactly what a mirror and a pond do to letters, numbers, clocks and pictures — and how to spot figures hiding inside busy drawings.

~35 min read7 quick checksTest: 20 questions

1What is a mirror image?

Stand in front of a mirror and raise your right hand. Your reflection raises the hand on its left! A mirror swaps left and right but does not turn you upside down.

In olympiad questions the mirror is usually drawn as a vertical line (standing straight up) on the right side of the figure. To draw the mirror image:

Left ↔ right swap. Whatever was on the left is now on the right. Arrows pointing right now point left.
Top and bottom stay. Whatever was at the top stays at the top.
Distance stays the same. A part close to the mirror stays close to the mirror in the image; a part far away stays far away.
Size and shape stay. Nothing gets bigger, smaller or turned.

Vertical mirror: left and right swap, top and bottom stay. The flag now points left; the dot near the mirror stays near the mirror.

💡 Tip

Mirror on the left or mirror on the right? For a vertical mirror the answer picture is the same — only its position on the page changes. And a mirror placed below (or above) a figure behaves like water: it swaps top and bottom.

✏️ Worked example: the arrow and the dot

A figure has an arrow pointing to the right at the top and a black dot at the bottom-left. A mirror is placed on its right. What does the mirror image look like?

Top stays top: the arrow is still at the top.
Left ↔ right: the arrow now points left.
The dot was bottom-left, so it moves to bottom-right (it stays at the bottom).

Answer: Arrow at the top pointing left, dot at the bottom-right.

📌 Remember

The mirror image of a mirror image is the original figure. So if a question shows you the mirror image and asks for the original, just take the mirror image again!

2Mirror images of letters, words and numbers

Some capital letters are symmetric about a standing (vertical) line: their left half matches their right half. These letters look exactly the same in a mirror. There are 11 of them:

A H I M O T U V W X Y

Blue dotted line = the letter matches itself in a mirror. Green dotted line = it matches itself in water. H, I, O, X have both.

Remember: Say them in alphabet order: A · H I · M · O · T U V W X Y. Almost the whole end of the alphabet from T to Y is mirror-friendly (but not Z!), plus A, H, I, M, O.

Whole words: two changes at once

When a word is seen in a mirror, two things happen:

The order of the letters reverses — the last letter comes first.
Each letter is flipped left-to-right.

In a mirror the order is reversed (the last letter comes first) and every letter is flipped.

✏️ Worked example: MOM and TOOT

Which words look exactly the same in a mirror: MOM, NOON, TOOT, BOB?

For a word to stay the same, every letter must be mirror-friendly and the word must read the same backwards.
MOM → M, O, M are all mirror-friendly and MOM backwards is MOM. ✔
NOON reads the same backwards, but N flips into a back-to-front N. ✘
TOOT → T, O, O, T all mirror-friendly, and TOOT backwards is TOOT. ✔
BOB → B flips (its bumps point left). ✘

Answer: MOM and TOOT. Other examples: WOW, TOT, OTTO, AHA.

Small letters and digits

In a mirror…	becomes…
b	d (and d becomes b)
p	q (and q becomes p)
digits 0 and 8	look the same
digit 1 (a plain stick)	looks the same
2, 3, 4, 5, 6, 7, 9	look back-to-front

⚠️ Olympiad trap

For a mirror image of a word, students often just write the letters backwards (CAT → TAC) and forget to flip each letter. Or they flip each letter but keep the order. Both changes are needed. In the options, look first at which letter is on the left — it must be the flipped last letter.

Which capital letter looks exactly the same in a mirror placed on its right?

U has a standing line of symmetry, so its left half matches its right half. E is symmetric top-to-bottom (it matches in water, not a mirror). N and S are not symmetric at all.

What is the mirror image of the small letter q?

A mirror swaps left and right. q has its round part on the left and tail going down; flipped, the round part is on the right with the tail going down — that is p.

3Water images

Look at a tree standing beside a still pond. Its reflection hangs upside down in the water. The water surface is a flat (horizontal) mirror.

Water image: top and bottom swap, left and right stay. The flag still points right, but hangs upside down.

Top ↔ bottom swap. The top of the object is now at the bottom of the image.
Left and right stay. A flag pointing right still points right.
In a word or number, the order stays the same; each letter or digit is turned upside down (flipped top-to-bottom).

Capital letters that are symmetric about a lying-down (horizontal) line look the same in water. There are 9 of them:

B C D E H I K O X

Remember: “BOX HIDE CK” (sounds like “box hide check”) — B O X H I D E C K stay the same in water. And H I O X stay the same everywhere — mirror and water!

In a water image the order stays the same; each letter just turns upside down. B and E look unchanged, C looks unchanged, but U does not.

In water…	becomes…
b	p (and p becomes b)
d	q (and q becomes d)
M	looks like W
digits 0, 8 (and 3 in most fonts)	look the same
digits 6 and 9	do NOT simply swap — a water image of 6 is a back-to-front 9 shape

✏️ Worked example: which word stays the same in water?

Which word looks exactly the same in its water image: CUBE, BOOK, BEAD, HOPE?

In water the order does not change, so we only need every letter to be water-friendly (B C D E H I K O X).
CUBE → U is not water-friendly (it becomes ∩). ✘
BOOK → B, O, O, K are all water-friendly. ✔
BEAD → A turns upside down. ✘
HOPE → P turns into a b-like shape. ✘

Answer: BOOK

Mirror image, water image and half turn are three different things. A half turn (upside down and back to front) is not a water image.

⚠️ Olympiad trap

A water image is not the same as turning the picture upside down. Turning a picture half a turn (180°) swaps top–bottom and left–right. Look at the J in the table: its hook is on the left in the water image but on the right after a half turn.

⚠️ Olympiad trap

Letters like A, M, T, U, V, W, Y stay the same in a mirror, not in water. Letters like B, C, D, E, K stay the same in water, not in a mirror. Read the question twice: mirror or water?

Which word looks exactly the same in its water image?

TOOK
CODE
MOON
DUCK

C, O, D, E are all symmetric top-to-bottom, and a water image keeps the order. TOOK has T, MOON has M and N, DUCK has U — these change.

4Clocks in a mirror

A clock seen in a mirror shows a wrong-looking time. There is a neat shortcut to find the real time.

The minute hand at 9 appears at 3; the hour hand between 7 and 8 appears between 5 and 4. 7:45 + 4:15 = 12:00.

Remember: Real time = 11:60 − time seen in the mirror. (11:60 is just 12:00 written so that we can subtract minutes easily.)

✏️ Worked example: the 11:60 rule

A clock seen in a mirror shows 2:25. What is the real time?

Write 12:00 as 11:60.
Subtract hours and minutes separately: 11 − 2 = 9 and 60 − 25 = 35.
Check with the picture idea: the minute hand at 5 (25 min) appears at 7 (35 min) in the mirror. ✔

Answer: 9:35

✏️ Worked example: exact hours

The mirror image of a clock shows 3:00. What is the real time?

For exact hours, subtract from 12:00: 12 − 3 = 9.
Or use 11:60 − 3:00 = 8:60, which is the same as 9:00.

Answer: 9:00 — the hour hand at 3 looks like it is at 9.

✏️ Worked example: a tricky one near 12

The mirror shows 12:20. What is the real time?

11:60 − 12:20 does not work (12 is bigger than 11). Use 23:60 instead: 23 − 12 = 11 and 60 − 20 = 40.
Picture check: the minute hand at 4 (20 min) appears at 8 (40 min); the hour hand just after 12 appears just before 12.

Answer: 11:40

Seen in mirror	Real time	Check: add them
4:15	7:45	4:15 + 7:45 = 12:00
2:25	9:35	= 12:00
10:50	1:10	= 12:00
9:15	2:45	= 12:00
6:00 or 12:00	the same time!	hands on the 12–6 line

💡 Tip

Quick check: the mirror time and the real time always add up to 12:00 — or to 24:00 when one of them is 11-something or 12-something. The times 12:00 and 6:00 look the same in a mirror because both hands lie on the up–down line.

⚠️ Olympiad trap

The time you see in the mirror is not the answer. Always ask: is the question giving me the mirror time and asking for the real time? Then use 11:60 −. (The same rule works the other way round too: real 7:45 looks like 4:15.)

Neha sees a clock in a mirror. It shows 5:10. What is the actual time?

6:50
7:50
6:10
5:50

11:60 − 5:10 = 6:50. Check: 5:10 + 6:50 = 12:00.

5Hidden (embedded) figures

In an embedded figure question, a small figure is hidden inside a big, busy figure — like a picture hidden in a puzzle. You must find which option hides it.

The hidden figure must be the same size.
It must be the same way up — not turned, not flipped (a mirror copy does not count).
Every line of the small figure must be there in the big figure (extra lines around it are fine).

Find the small triangle (same size, same way up) inside the big figure.

Found it twice! Top-left and bottom-right. Other triangles are turned the wrong way.

✏️ Worked example: finding the triangle

Where is the small triangle (square corner at bottom-left, slanting side going down from left to right “\”) hidden?

Notice the special parts: a square corner at the bottom-left and a slanting side like \. Its sides are half as long as the big square.
Look for half-size triangles made by the middle lines and the diagonals.
Top-left quarter: left side + middle flat line + the \ diagonal. Square corner at bottom-left ✔.
Bottom-right quarter: middle standing line + bottom side + the \ diagonal. Square corner at bottom-left ✔.
Triangles using the / diagonal lean the wrong way ✘. The big triangle (whole side) is too large ✘.

Answer: It is hidden twice.

💡 Tip

How to search: pick the most unusual part of the small figure (a slant, a curve, a sharp point), find it in the big figure first, then trace the rest with your finger or pencil tip.

⚠️ Olympiad trap

An option that has the figure turned or flipped is a trap. A Z-shape and a back-to-front Z (like S made with straight lines) are different figures. So are ▷ and ◁.

6Completing figures and patterns

In figure completion questions a part of a picture is missing. You choose the piece that finishes it. First find the rule the design follows.

🪞 Mirror pattern

The design is the same on both sides of a dotted line. Across a standing line the piece is a mirror image; across a lying line it is a water image.

🌀 Turning (pinwheel) pattern

Each quarter is the previous quarter turned 90° (a quarter turn). Nothing is flipped.

🧩 Jigsaw pattern

A shape is cut out; the missing piece must have exactly the right edges, lines and shading to fit.

Symmetric across both dotted lines. What goes in the “?”

Answer: the top-left piece turned upside down and back to front (a half turn).

Turning (pinwheel) pattern: each quarter is the one before it turned 90° clockwise — nothing is flipped.

✏️ Worked example: the missing quarter

In the first design, the top-left piece has a pink triangle in its outer corner, a blue ring near the centre and a slanting line. It is symmetric across both dotted lines. What goes in the bottom-right?

Bottom-right is across the standing line from bottom-left, so it is the mirror image of bottom-left.
Bottom-left is the water image of top-left: pink triangle in the bottom-left corner, ring near the centre.
Mirror that: pink triangle goes to the bottom-right corner, the ring stays next to the centre, the slanting line tilts the other way.

Answer: The top-left piece turned a half turn: pink triangle in the outer (bottom-right) corner, ring next to the centre.

✏️ Worked example: the pinwheel

Each quarter is the one before it turned a quarter turn clockwise: top-left → top-right → bottom-right → bottom-left. In the top-left the arrow points right. Where does it point in the bottom-left?

Top-left → top-right: one quarter turn clockwise: right → down.
Top-right → bottom-right: another turn: down → left.
Bottom-right → bottom-left: another turn: left → up.

Answer: Up.

⚠️ Olympiad trap

In a mirror pattern, copying the neighbouring piece exactly as it is (without flipping) breaks the symmetry. In a turning pattern, a flipped piece never appears. Decide which kind of pattern it is before looking at the options.

A design is symmetric across a standing dotted line. On the left of the line there is an arrow pointing left, near the top. What is on the right side?

An arrow pointing left, near the bottom
An arrow pointing right, near the top
An arrow pointing right, near the bottom
An arrow pointing left, near the top

A standing line works like a mirror: left and right swap (the arrow now points right), but top stays top.

7Mixed practice and top traps

✏️ Worked example: mirror or water?

The number 1082 is written on a card. (a) What does it look like in a mirror on its right? (b) What does it look like in water below it?

(a) Mirror: the order reverses → 2, 8, 0, 1, and every digit is flipped left-to-right. 8, 0 and 1 look the same, but 2 is back-to-front.
(b) Water: the order stays 1, 0, 8, 2, and every digit is flipped top-to-bottom. 1, 0 and 8 look the same; 2 is upside down.

Answer: (a) the reversed order with a back-to-front 2 on the left; (b) same order with an upside-down 2 at the end.

✏️ Worked example: find the original

A word seen in a mirror reads “TAHW” with every letter looking normal. What was the real word?

The mirror image of a mirror image is the original, so reverse the order again: TAHW → WHAT.
Check the letters: W, H, A, T are all mirror-friendly, so none of them looks flipped. ✔

Answer: WHAT

Question type	What changes	What stays
Mirror (standing mirror)	left ↔ right, order of letters	top/bottom, size
Water (or mirror below)	top ↔ bottom	left/right, order of letters
Mirror clock	time → 11:60 − time	6:00 and 12:00
Hidden figure	nothing! same size, same way up	everything
Pattern completion	follow the pattern's rule (mirror or turn)	—

⚠️ Olympiad trap

Check every part. Many wrong options flip most of the figure but forget one small part — an arrow still pointing the old way, or a dot on the wrong side. Tick off each part one by one.

A clock in a mirror shows 12:40. What is the real time?

11:20
12:20
11:40
1:20

Use 23:60 − 12:40 = 11:20. Check: the minute hand at 8 appears at 4 (20 min), and the hour hand between 12 and 1 appears between 12 and 11.

Which of these letters looks the same in a mirror AND in water?

X is symmetric about both a standing and a lying line. A and M only match in a mirror; E only matches in water.

8Explore more

9Quick recap

A standing mirror swaps left ↔ right; top and bottom stay.
A water surface (or a mirror below) swaps top ↔ bottom; left and right stay.
Mirror image of a word: the order reverses and every letter flips.
Water image of a word: the order stays; every letter turns upside down.
Mirror-friendly capitals (11): A H I M O T U V W X Y.
Water-friendly capitals (9): B C D E H I K O X. H, I, O, X work for both.
Mirror: b ↔ d, p ↔ q. Water: b ↔ p, d ↔ q.
Mirror clock: real time = 11:60 − mirror time (use 12:00 for exact hours, 23:60 for 12-something).
A water image is not a half turn — a half turn also swaps left and right.
The mirror image of a mirror image is the original figure.
Hidden figures: same size, same way up, no flipping; extra lines are allowed around it.
Pattern completion: decide if it is a mirror pattern or a turning (pinwheel) pattern first.

Flip cards

QuestionMirror image of b?

Answerd

QuestionWater image of b?

Answerp

QuestionMirror shows 3:40. Real time?

Answer8:20 (11:60 − 3:40)

QuestionWhich capitals look the same in mirror AND water?

AnswerH, I, O, X

QuestionDoes a mirror reverse the order of letters in a word?

AnswerYes — the last letter comes first.

QuestionDoes water reverse the order of letters?

AnswerNo — the order stays, each letter turns upside down.

QuestionName a word that looks the same in a mirror.

AnswerMOM, WOW, TOOT, OTTO…

QuestionCan a hidden figure be turned?

AnswerNo — same size and same way up.