Non-Verbal Reasoning (figures)

Snapshot

• Non-Verbal Reasoning tests figures instead of words: series and analogies on shapes, odd-one-out, mirror and water images, paper folding and punching, counting figures, cubes and dice.
• You cannot read these, you must watch what changes from one figure to the next. Every question hides one or two simple rules: a rotation, a count, a shading flip, a movement round the corners.
• The idea that cracks it: track one feature at a time. Rotation first, then count, then shading, then position. Do not try to take in the whole figure.
• Marking is +5 / −1. If two options both look right, you have missed a feature: compare them again feature by feature.

Part 1 — Figure series: what can change

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Prepared for the CUET UG General Test.

A figure series shows three or four figures and asks for the next one. The figures differ in a small number of ways, and each way follows its own pattern. Always list the features before you look for rules.

In the series above, the only feature that changes is the position of the dot. It goes top-left, top-right, bottom-right, so it moves one corner clockwise each step and the next position is the bottom-left. A figure analogy ("A is to B as C is to ?") applies exactly the same change that turns the first figure into the second to the third figure.

Part 2 — The method, step by step

1. List the features of the first figure: shapes, count of parts, shading, position of marks, direction of lines.
2. Compare figure 1 to figure 2 one feature at a time, and write the change.
3. Check the same change from figure 2 to figure 3. If it does not repeat, the feature may follow a count pattern (1, 2, 3) or alternate between two rules.
4. Look at alternate figures (1st, 3rd, 5th) if the steps do not agree. Many CUET series have two interleaved patterns.
5. Predict the next figure feature by feature, then pick the option that matches every feature.
6. Eliminate by one feature at a time. Often a single feature (shading, count) removes two options.

Part 3 — The full menu of question types

Part 4 — Odd one out, analogy and matrices

Odd one out. Four of the five figures share one property, and one does not. Typical properties: number of sides, closed or open, symmetrical or not, shading, number of marks inside. Test one property across all five. If two figures break the rule, change the property.

Analogy. Describe the change from the first figure to the second in words ("rotates $90^\circ$ clockwise and becomes shaded"). Apply the same words to the third figure.

Pattern grids (matrices). In a 3 by 3 grid, find what each row has in common and what each column has in common. Common patterns are: each row and column contains each of three shapes exactly once, a row's third figure is the first and second combined, or shading increases from left to right. The missing cell is the one that makes every row and column obey the rule.

Part 5 — Mirror and water images

A mirror image is the reflection when a mirror stands on one side of the figure: left and right swap, top and bottom stay. A water image is the reflection in water below the figure: top and bottom swap, left and right stay.

For a word, a mirror reverses the order of the letters and flips each one, while water keeps the order and flips each letter upside down. A single letter such as b becomes d in a mirror and p in water.

For a clock in a mirror (the mirror placed to the side), the reflected time is

$$12{:}00 - t$$

where $t$ is the time shown. So 4:20 reflects to $12{:}00-4{:}20=7{:}40$, and 3:00 reflects to 9:00. Water images of clocks are asked rarely. If one appears, flip each hand directly about the horizontal line instead of using a formula.

Part 6 — Paper folding and punching

When a sheet is folded, each fold doubles the layers. A single punch goes through every layer, so one punched hole creates one hole per layer after unfolding. Unfold one fold at a time, and reflect every hole in the fold line you are undoing.

Two special cases: a hole punched on a fold line is cut across both halves, so it becomes a single hole sitting on the fold line when opened. For a diagonal fold, reflect each hole across the diagonal line instead of a horizontal or vertical one.

Part 7 — Counting figures

Count systematically, never by eye. Label the lines or corners with letters, list shapes by size, and tick them off.

If $k$ lines are drawn from the apex of a triangle to its base, the base is split into $n = k+1$ segments, and the number of triangles is

$$\frac{n(n+1)}{2}$$

In an $n \times n$ grid of small squares, the number of squares of all sizes is $\dfrac{n(n+1)(2n+1)}{6}$ and the number of rectangles of all sizes (squares included) is $\left(\dfrac{n(n+1)}{2}\right)^2$. For a 3 by 3 grid that is 14 squares and 36 rectangles; for a 4 by 4 grid, 30 squares and 100 rectangles.

Part 8 — Cubes and dice

Cutting a painted cube. A cube painted on all faces is cut into $n^3$ equal small cubes. Their paint counts are:

Dice with two views. Two views of the same die show three faces each. If the two views share two faces, then the third faces in the two views each touch both shared faces from opposite ends of the same edge, so they are opposite each other. Never assume that opposite faces add to 7 unless the question says it is a standard die.

Part 9 — Worked examples

Example 1. A rotation series. An arrow points North, then North-East, then East, then South-East. What is next?

It turns $45^\circ$ clockwise each step, so the next direction is South.

Example 2. A counting series. Figures contain 1, 2, 4, 7 dots. How many dots are in the next figure?

The differences are $+1, +2, +3$, so the next is $+4$: $7+4=$ 11 dots.

Example 3. A position series. A dot sits at the top-left, then top-right, then bottom-right of a square. Where is it next?

It moves one corner clockwise each step, so it goes to the bottom-left corner, as in the figure of Part 1.

Example 4. Two features at once. The figures are a black triangle, a white square, a black pentagon, a white hexagon. What comes next?

The number of sides rises by one each time, and the shading alternates black, white. The fifth figure has 7 sides and is black: a black heptagon.

Example 5. Odd one out. Which is the odd one: square, hexagon, octagon, pentagon?

Square (4), hexagon (6) and octagon (8) have an even number of sides. The pentagon has 5. The odd one is the pentagon.

Example 6. A figure analogy. An upward arrow inside a circle becomes a rightward arrow inside a circle. A downward arrow inside a square becomes what?

The arrow turns $90^\circ$ clockwise. Down turned $90^\circ$ clockwise is West (left). The answer is a leftward arrow inside the square.

Example 7. Mirror and water images of letters. (a) What is the mirror image of the letters H O T, with the mirror on the right? (b) What is the water image of the word BOX? (c) What does the letter b become in each?

(a) A mirror reverses the order, and H, O, T are all symmetric, so it reads T O H. (b) B, O and X are symmetric about the horizontal line and water keeps the order, so it reads BOX upside down. (c) b becomes d in a mirror and p in water.

Example 8. A clock in a mirror. What time does a mirror show for 4:20, and for 8:15?

$12{:}00 - 4{:}20 =$ 7:40. And $12{:}00 - 8{:}15 =$ 3:45.

Example 9. Folding and punching. A square sheet is folded in half, folded in half again, and a single hole is punched through all layers away from the folds. How many holes appear when the sheet is opened? What changes if the sheet is folded in half once and the hole is punched on the fold line?

Two folds give four layers, so there are 4 holes, one in each quarter and symmetric about the fold lines. On a single fold, a hole centred on the fold line cuts both halves at the same spot and opens as one hole on the fold line.

Example 10. Counting triangles. A triangle has two lines drawn from the apex to the base. How many triangles are there?

The base has 3 segments, so there are $\dfrac{3 \times 4}{2} =$ 6 triangles. (Three small, two made of two small ones, and the whole triangle.)

Example 11. Squares and rectangles in a grid. How many squares and how many rectangles are in a 4 by 4 grid?

Squares $=1^2+2^2+3^2+4^2=$ 30. Rectangles $=\left(\dfrac{4 \times 5}{2}\right)^2 = 10^2 =$ 100 (squares included).

Example 12. Cutting a painted cube. A cube painted on all faces is cut into 64 equal small cubes. How many have exactly two painted faces, and how many have none?

$64 = 4^3$, so $n=4$. Two painted faces: $12(4-2) =$ 24. No paint: $(4-2)^3=$ 8.

Example 13. Two views of a die. In the first view the visible faces are 1, 2 and 3. In the second view of the same die the visible faces are 1, 2 and 4. Which number is opposite 3?

Faces 1 and 2 are common to both views. Faces 3 and 4 each touch both faces 1 and 2, from opposite ends of their common edge. So 4 is opposite 3.

Example 14. A 3 by 3 grid. The rows are: circle, square, triangle; square, triangle, circle; triangle, circle, ?. What is the missing figure?

Each row and column contains each shape once. The third column has triangle and circle, so the missing figure is a square.

Part 10 — Common traps

Part 11 — Rapid-fire practice with answers

Part 12 — A timed-practice routine

In the exam, do the figure questions that need one feature first. Leave embedded-figure and complicated folding questions for the end. Use a pencil on rough paper to redraw the first figure when you are unsure about a rotation.

Part 13 — Where this connects

Sequence thinking carries over to Series, where the same "find the rule" habit works on numbers and letters. Figure analogies follow the logic of Analogy, and odd-one-out is the visual cousin of Classification. Counting formulas and cube cutting link with Mensuration & Basic Geometry, and rotation by angles uses the same bearings as Direction Sense.

One-line revision: list the features and track one at a time, mirror swaps left and right while water swaps top and bottom, each fold doubles the layers a punch passes through, and count figures by size rather than by eye.