Mirror images, paper folding and figure counting for RRB Group D

Non-verbal questions show pictures instead of words, and many candidates skip them because they "can't see" the answer. That is a mistake. Each type follows one fixed rule, and once you apply the rule you are checking options, not imagining shapes. These questions are also quick: most take 30 to 45 seconds when the rule is clear.

The site has no images, so every figure below is described in words or on a labelled grid. Draw each one on rough paper as you read. That is exactly what you should do in the exam too.

Mirror images

A mirror placed to the right or left of a figure flips it left to right. Top stays top and bottom stays bottom.

• Capital letters that look the same in such a mirror: A, H, I, M, O, T, U, V, W, X, Y. Each has a vertical line of symmetry.
• A word is reversed, and each letter is flipped. "RAIL" appears as "LIAR" with every letter back to front.
• A word made only of the letters above, which also reads the same backwards, looks unchanged: "MOM", "WOW", "TOOT".

If the question says the mirror is placed above or below the figure, treat it like a water image instead.

Water images

A reflection in water flips a figure top to bottom. Left and right stay the same, and the order of letters in a word does not change.

• Capital letters that look the same in water: B, C, D, E, H, I, K, O, X. Each has a horizontal line of symmetry.
• So "BOX" and "CHECK" look unchanged in water, because every letter in them is on this list.
• Letters on both lists: H, I, O, X.

Clock mirror images use a formula instead of drawing: see calendars and clocks.

Paper folding and punching

Two rules do all the work:

• Each fold doubles the layers, so one punch through paper folded once makes 2 holes, folded twice 4 holes, folded three times 8 holes.
• When you unfold, each hole appears again as a mirror image across the fold line.

Worked example on a grid

Picture a square sheet divided into a 4 × 4 grid. Label the rows 1 to 4 from top to bottom and the columns A to D from left to right.

1. Fold the left half over onto the right half. The fold line runs between columns B and C. The folded paper now covers columns C and D.
2. Fold the top half down onto the bottom half. The fold line runs between rows 2 and 3. The folded paper now covers only rows 3–4 of columns C–D.
3. Punch one hole in cell 4C.

Now unfold in reverse order:

• Undo the top-to-bottom fold. Row 4 mirrors to row 1, so holes are at 4C and 1C.
• Undo the left-to-right fold. Column C mirrors to column B, so each hole gets a partner: 4C, 1C, 4B, 1B.

Four holes, as the doubling rule predicts: two folds, one punch, 2 × 2 = 4.

Counting squares

In a square grid of n × n small squares, count every size separately and add. The total is 1² + 2² + … + n².

For a rectangular grid, count sizes one by one. A grid 2 squares tall and 3 wide has 6 small squares and 2 squares of size 2 × 2 (one on the left, one on the right), so 8 in all.

Counting triangles

Lines from one corner. Draw a triangle and then draw lines from the top corner down to the base. If you draw n such lines, the base is cut into n + 1 pieces and the number of triangles is:

Each triangle is formed by choosing any two of the slanting lines (including the two sides), so the pattern goes 3, 6, 10, 15.

Add a horizontal line. If one horizontal line crosses all these lines, it makes a second "base". The count doubles. For example, 2 lines from the top corner plus 1 horizontal line give 6 × 2 = 12 triangles.

A square with both diagonals. There are 4 small triangles meeting at the centre, plus 4 larger ones (each diagonal cuts the square into 2 halves). Total 8.

Dice

• On a standard die, opposite faces add up to 7: 1–6, 2–5 and 3–4.
• Two faces that appear together in any picture of the die are next to each other, so they cannot be opposite.
• To find the face opposite a number, list every number ever seen beside it. The one number never seen beside it is opposite.

Worked example. A die (not necessarily standard) is shown in two positions. In the first, the faces 1, 2 and 3 are visible. In the second, the faces 1, 5 and 6 are visible. Which number is opposite 1?

• Seen next to 1: 2, 3, 5 and 6.
• Never seen next to 1: 4. So 4 is opposite 1.

This die is not standard (1 and 6 are not opposite), which is why you should use the "add up to 7" rule only when the question says the die is standard.

Figure series

A figure series changes one feature step by step: position, rotation, number of lines, or shading. Describe the change in words and continue it.

Worked example. In figure 1 an arrow points up. In each next figure it turns 45° clockwise. Where does it point in figure 5?

• Figure 2: up-right. Figure 3: right. Figure 4: down-right. Figure 5: down.
• Check: four turns of 45° is 180°, and up turned 180° is down.

Practice set

1. How many squares are there in a 3 × 3 grid?
2. How many squares are there in a 4 × 4 grid?
3. A sheet is folded three times and punched once. How many holes appear when it is opened?
4. Which of these letters look the same in a side mirror: H, N, O, R?
5. Which word looks unchanged in water: BOX, BAT or CUP?
6. On a standard die, which number is opposite 3?
7. Three lines are drawn from the top corner of a triangle to its base. How many triangles are there?
8. An arrow points up in figure 1 and turns 90° clockwise in each next figure. Where does it point in figure 7?

Answers:

1. 14. 9 + 4 + 1.
2. 30. 16 + 9 + 4 + 1.
3. 8. 2 × 2 × 2.
4. H and O. N and R are not symmetric about a vertical line.
5. BOX. A, T, U and P are not symmetric about a horizontal line.
6. 4. 3 + 4 = 7.
7. 10. The pattern 3, 6, 10.
8. Down. Six turns of 90° is 540°, which is 360° + 180°.

What to do next

• Memorise the two letter lists: side mirror and water.
• Learn the square counts (5, 14, 30) and the triangle pattern (3, 6, 10, 15).
• Practise ten figure questions a day for a week, drawing every figure on rough paper.
• Revise the full set of rules in the reasoning revision sheet.