Miscellaneous Logical Puzzles Questions and Answers

Q: Five cubes of identical size and another smaller cube are assembled as shown in Figure A. If viewed from direction X, the planar image of the assembly appears as Figure B.If viewed from direction Y, the planar image of the assembly (Figure A) will appear as

The view from Y is (A).Seen from direction Y, the column holding the upper cube appears as two squares stacked on the right, and the other column as a single square on the left. The small cube rests on the lower layer directly in front of the upper cube, on the side nearer the viewer, so it hides part of the upper cube's face and shows as a small square at the bottom of the upper square.In Figure A the small cube sits against the half of the upper cube that is on the right as seen from Y (Figure B confirms it occupies only a corner of its quarter), so the small square sits at the lower right of the upper square, as in (A). (C) puts it on the wrong side, (D) puts it outside the upper cube, and (B) shows a full-height dividing line that no edge of the assembly produces.

Q: Visualize a cube that is held with one of the four body diagonals aligned to the vertical axis. Rotate the cube about this axis such that its view remains unchanged. The magnitude of the minimum angle of rotation is

The minimum angle is 120°.A body diagonal of a cube passes through two opposite vertices. At each of those vertices three edges and three faces meet, arranged symmetrically around the diagonal. A rotation of one third of a turn, 120°, carries each of these edges and faces onto the next, so the cube looks the same.A 60° rotation would need six-fold symmetry about the diagonal, which the three edges at a vertex do not have. 90° is the symmetry about an axis through the centres of opposite faces, and 180° is not the smallest such rotation.

Q: Consider an art gallery whose walkways are shown as lines in the diagram. A black dot represents a junction of two walkways. A guard may be placed at a junction to watch over the walkways that join at that junction. The minimum number of guards needed to watch all the walkways is ________.(The diagram shows five junctions joined by five walkways into a single closed loop, drawn as the outline of a house: a square with a triangular roof.)

The minimum number of guards is 3.The walkways form a single loop of 5 junctions and 5 walkways. A guard at a junction watches exactly the 2 walkways meeting there, so 2 guards can watch at most 4 walkways, which is not enough for 5. At least 3 guards are needed.Three are enough. Label the roof apex A, the eave corners B and C, and the floor corners D and E. Guards at A, D and E watch AB and AC (from A), BD and DE (from D), and CE (from E): all five walkways. So 2 is too few, and 4 or 5 are more than the minimum.

Q: A block with a trapezoidal cross-section is placed over a block with rectangular cross section as shown above.Which one of the following is the correct drawing of the view of the 3D object as viewed in the direction indicated by an arrow in the above figure?

The correct view is drawing (A).Looking along the arrow, at the end of the long lower block, that block shows as a plain rectangle: its end face is flat, and the upper block sits on top of it, so no line of the upper block can cross this rectangle. The upper block appears above it in its trapezoidal cross-section. In the picture its face towards the viewer has a sloping edge on the near-left side and an almost vertical edge on the far side, so in the view the trapezoid has its sloping side on the left and its vertical side on the right, standing towards the right of the rectangle.Drawings (B) and (D) carry a sloping line across the lower rectangle, which the flat end face cannot show. Drawing (C) puts the sloping side of the trapezoid on the right, which is the mirror image of the correct view.

Q: Four cities P, Q, R and S are connected through one-way routes as shown in the figure. The travel time between any two connected cities is one hour. The boxes beside each city name describe the starting time of first train of the day and their frequency of operation. For example, from city P, the first trains of the day start at 8 AM with a frequency of 90 minutes to each of R and S. A person does not spend additional time at any city other than the waiting time for the next connecting train.If the person starts from R at 7 AM and is required to visit S and return to R, what is the minimum time required?

The minimum time is 6 hours 30 minutes.The arrows leave little choice. From R the only route is to Q; from Q the only route is to P; from P the person can go straight to S; and from S there is a direct route back to R. So the journey is R, Q, P, S, R, and the time is set by the waits for connecting trains:R: first train at 7 AM, leave 7:00, reach Q at 8:00.Q: trains at 5, 7, 9 AM (every 120 min); wait for 9:00, reach P at 10:00.P: trains at 8:00, 9:30, 11:00 (every 90 min); wait for 11:00, reach S at 12:00.S: trains from 8:00 every 45 min, so 11:45 then 12:30; leave 12:30, reach R at 1:30 PM.From 7:00 AM to 1:30 PM is 6 hours 30 minutes. The shorter options count only the four hours of travel plus some of the waits; every connection here forces a wait, and none can be skipped because each city has only one useful outgoing route.

Q: An opaque pyramid (shown below), with a square base and isosceles faces, is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The pyramid can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible?

The shadow that is NOT possible is Q.A parallel beam casts a shadow that is the parallel projection of the solid onto the screen. A pyramid is a convex solid, and the projection of a convex solid is always a convex region: any straight segment between two points of the shadow stays inside the shadow. So every possible shadow of the pyramid is a convex polygon.Shadow Q has a notch cut into its upper edge, a re-entrant corner, so it is not convex and cannot be cast. The others can: looking straight along the axis gives the square S, and tilted orientations give other convex outlines, five-sided like P or four-sided like R, depending on which vertices end up on the boundary of the shadow.

Q: Consider an art gallery whose walkways are shown as lines in the diagram. A black dot represents a junction of two walkways. A guard may be placed at a junction to watch over the walkways that join at that junction. The minimum number of guards needed to watch all the walkways is ___.

The minimum number of guards is 3.The walkways form a single closed loop of 5 junctions and 5 walkways. Each guard covers at most the 2 walkways that meet at his junction, so 2 guards can watch at most 4 walkways, which is not enough for 5.Three guards suffice: number the junctions 1 to 5 round the loop and place guards at 1, 3 and 4. Guard 1 covers walkways 5-1 and 1-2, guard 3 covers 2-3 and 3-4, and guard 4 covers 4-5.4 and 5 guards also work, but they are not the minimum.

Q: How many triangles are there in the given figure ?(The figure is a rectangle divided by a vertical line into two squares, with both diagonals drawn in each square and a horizontal line drawn across the whole rectangle through the two points where the diagonals cross.)

There are 28 triangles in the figure.Take each square as 2 units wide, so the whole figure is 4 by 2. Count by size:TriangleCountSmallest: halves of the left and right quarters of each square, cut by the horizontal line8Top and bottom quarters of each square4Whole left and right quarters of each square4Triangles across the middle line, on the horizontal line with apex at the top or bottom middle2Half-squares cut off by one diagonal8Large triangles with the whole long side as base and apex at the middle of the opposite side2\[ 8 + 4 + 4 + 2 + 8 + 2 = 28 \]Counting only the small pieces gives 12, and missing the triangles that span both squares gives 18 or less.

Q: 31 children are standing in a circle and one of them (say child-1) has a ring. The ring is passed clockwise. Child-1 passes it on to child-2, child-2 passes it on to child-4, child-4 passes it on to child-7 and so on. After how many minimum such passes (including the first pass from child-1) will the ring come in the hands of child-1 again?

The ring returns to child-1 after 30 passes.The k-th pass moves the ring k places (1 to 2 is one place, 2 to 4 is two, 4 to 7 is three). After n passes it has moved \(1 + 2 + \dots + n = n(n+1)/2\) places, and it is back with child-1 when that total is a multiple of 31.\[ \begin{aligned} \frac{n(n+1)}{2} &\equiv 0 \pmod{31} \\ n = 30 &: \frac{30 \times 31}{2} = 465 = 15 \times 31 \end{aligned} \]Since 31 is prime, it must divide n or n + 1, so the smallest n is 30. n = 31 also works (31 × 32/2 = 496 = 16 × 31), but 30 comes first, and 28, 29 and 32 do not give a multiple of 31.

Q: How many triangles are there in the given figure?

The figure holds 28 triangles.Each of the three cells is a rectangle cut by both diagonals. Inside one cell there are 4 small triangles (the four parts the diagonals make) and 4 larger ones, each made of two small parts, with a side of the cell as the base. That is 8 per cell.\[ 3 \times 8 = 24 \]Triangles also span two neighbouring cells. Take the top edge of two adjacent cells as the base, with the apex at the bottom point where the diagonals meet on the shared line. The same works with the bottom edge and the apex at the top. That is 2 for each pair of neighbouring cells, and there are 2 such pairs.\[ 24 + 2 \times 2 = 28 \]No triangle spans all three cells, since the outer diagonals do not meet over a common apex.

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31.

Five cubes of identical size and another smaller cube are assembled as shown in Figure A. If viewed from direction X, the planar image of the assembly appears as Figure B.

Figure A: cube assembly with view directions X and Y. Figure B: view from X

If viewed from direction Y, the planar image of the assembly (Figure A) will appear as

Exam Year(s)

GATE Mathematics - 2024,

Answer: A

The view from Y is (A).

Seen from direction Y, the column holding the upper cube appears as two squares stacked on the right, and the other column as a single square on the left. The small cube rests on the lower layer directly in front of the upper cube, on the side nearer the viewer, so it hides part of the upper cube's face and shows as a small square at the bottom of the upper square.

In Figure A the small cube sits against the half of the upper cube that is on the right as seen from Y (Figure B confirms it occupies only a corner of its quarter), so the small square sits at the lower right of the upper square, as in (A). (C) puts it on the wrong side, (D) puts it outside the upper cube, and (B) shows a full-height dividing line that no edge of the assembly produces.

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32.

Visualize a cube that is held with one of the four body diagonals aligned to the vertical axis. Rotate the cube about this axis such that its view remains unchanged. The magnitude of the minimum angle of rotation is

A.120°
B.60°
C.90°
D.180°

Exam Year(s)

GATE Mathematics - 2024,

Answer: A

The minimum angle is 120°.

A body diagonal of a cube passes through two opposite vertices. At each of those vertices three edges and three faces meet, arranged symmetrically around the diagonal. A rotation of one third of a turn, 120°, carries each of these edges and faces onto the next, so the cube looks the same.

A 60° rotation would need six-fold symmetry about the diagonal, which the three edges at a vertex do not have. 90° is the symmetry about an axis through the centres of opposite faces, and 180° is not the smallest such rotation.

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33.

Consider an art gallery whose walkways are shown as lines in the diagram. A black dot represents a junction of two walkways. A guard may be placed at a junction to watch over the walkways that join at that junction. The minimum number of guards needed to watch all the walkways is ________.

(The diagram shows five junctions joined by five walkways into a single closed loop, drawn as the outline of a house: a square with a triangular roof.)

Exam Year(s)

GATE Mathematics - 2026, GATE Life Sciences - 2026,

Answer: B

The minimum number of guards is 3.

The walkways form a single loop of 5 junctions and 5 walkways. A guard at a junction watches exactly the 2 walkways meeting there, so 2 guards can watch at most 4 walkways, which is not enough for 5. At least 3 guards are needed.

Three are enough. Label the roof apex A, the eave corners B and C, and the floor corners D and E. Guards at A, D and E watch AB and AC (from A), BD and DE (from D), and CE (from E): all five walkways. So 2 is too few, and 4 or 5 are more than the minimum.

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34.

A long rectangular block with a smaller block of trapezoidal cross-section on top of it, and an arrow pointing at the right-hand end face

A block with a trapezoidal cross-section is placed over a block with rectangular cross section as shown above.

Which one of the following is the correct drawing of the view of the 3D object as viewed in the direction indicated by an arrow in the above figure?

Exam Year(s)

GATE ME - 2022,

Answer: A

The correct view is drawing (A).

Looking along the arrow, at the end of the long lower block, that block shows as a plain rectangle: its end face is flat, and the upper block sits on top of it, so no line of the upper block can cross this rectangle. The upper block appears above it in its trapezoidal cross-section. In the picture its face towards the viewer has a sloping edge on the near-left side and an almost vertical edge on the far side, so in the view the trapezoid has its sloping side on the left and its vertical side on the right, standing towards the right of the rectangle.

Drawings (B) and (D) carry a sloping line across the lower rectangle, which the flat end face cannot show. Drawing (C) puts the sloping side of the trapezoid on the right, which is the mirror image of the correct view.

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35.

Four cities P, Q, R and S are connected through one-way routes as shown in the figure. The travel time between any two connected cities is one hour. The boxes beside each city name describe the starting time of first train of the day and their frequency of operation. For example, from city P, the first trains of the day start at 8 AM with a frequency of 90 minutes to each of R and S. A person does not spend additional time at any city other than the waiting time for the next connecting train.

If the person starts from R at 7 AM and is required to visit S and return to R, what is the minimum time required?

Square of four cities with one-way arrows Q to P, P to R, P to S, S to Q, S to R and R to Q. First train and frequency: P 8 AM, 90 min; Q 5 AM, 120 min; R 7 AM, 60 min; S 8 AM, 45 min

A.6 hours 30 minutes
B.3 hours 45 minutes
C.4 hours 30 minutes
D.5 hours 15 minutes

Exam Year(s)

GATE ME - 2022,

Answer: A

The minimum time is 6 hours 30 minutes.

The arrows leave little choice. From R the only route is to Q; from Q the only route is to P; from P the person can go straight to S; and from S there is a direct route back to R. So the journey is R, Q, P, S, R, and the time is set by the waits for connecting trains:

R: first train at 7 AM, leave 7:00, reach Q at 8:00.
Q: trains at 5, 7, 9 AM (every 120 min); wait for 9:00, reach P at 10:00.
P: trains at 8:00, 9:30, 11:00 (every 90 min); wait for 11:00, reach S at 12:00.
S: trains from 8:00 every 45 min, so 11:45 then 12:30; leave 12:30, reach R at 1:30 PM.

From 7:00 AM to 1:30 PM is 6 hours 30 minutes. The shorter options count only the four hours of travel plus some of the waits; every connection here forces a wait, and none can be skipped because each city has only one useful outgoing route.

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36.

An opaque pyramid (shown below), with a square base and isosceles faces, is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The pyramid can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible?

An opaque square-based pyramid, and four grey shadow outlines labelled P, Q, R and S

Exam Year(s)

GATE ME - 2023,

Answer: B

The shadow that is NOT possible is Q.

A parallel beam casts a shadow that is the parallel projection of the solid onto the screen. A pyramid is a convex solid, and the projection of a convex solid is always a convex region: any straight segment between two points of the shadow stays inside the shadow. So every possible shadow of the pyramid is a convex polygon.

Shadow Q has a notch cut into its upper edge, a re-entrant corner, so it is not convex and cannot be cast. The others can: looking straight along the axis gives the square S, and tilted orientations give other convex outlines, five-sided like P or four-sided like R, depending on which vertices end up on the boundary of the shadow.

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37.

House outline of five junctions (black dots) joined by five straight walkways in one closed loop: a square with a triangular roof

Exam Year(s)

GATE XE - 2026,

Answer: B

The minimum number of guards is 3.

The walkways form a single closed loop of 5 junctions and 5 walkways. Each guard covers at most the 2 walkways that meet at his junction, so 2 guards can watch at most 4 walkways, which is not enough for 5.

Three guards suffice: number the junctions 1 to 5 round the loop and place guards at 1, 3 and 4. Guard 1 covers walkways 5-1 and 1-2, guard 3 covers 2-3 and 3-4, and guard 4 covers 4-5.

4 and 5 guards also work, but they are not the minimum.

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38.

How many triangles are there in the given figure ?

(The figure is a rectangle divided by a vertical line into two squares, with both diagonals drawn in each square and a horizontal line drawn across the whole rectangle through the two points where the diagonals cross.)

A.10
B.12
C.18
D.28

Exam Year(s)

HPPSC HPAS - 2025,

Answer: D

There are 28 triangles in the figure.

Take each square as 2 units wide, so the whole figure is 4 by 2. Count by size:

Triangle	Count
Smallest: halves of the left and right quarters of each square, cut by the horizontal line	8
Top and bottom quarters of each square	4
Whole left and right quarters of each square	4
Triangles across the middle line, on the horizontal line with apex at the top or bottom middle	2
Half-squares cut off by one diagonal	8
Large triangles with the whole long side as base and apex at the middle of the opposite side	2

\[ 8 + 4 + 4 + 2 + 8 + 2 = 28 \]

Counting only the small pieces gives 12, and missing the triangles that span both squares gives 18 or less.

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39.

31 children are standing in a circle and one of them (say child-1) has a ring. The ring is passed clockwise. Child-1 passes it on to child-2, child-2 passes it on to child-4, child-4 passes it on to child-7 and so on. After how many minimum such passes (including the first pass from child-1) will the ring come in the hands of child-1 again?

A.28
B.29
C.30
D.32

Exam Year(s)

Haryana Civil Services - 2026, HPSC - 2026,

Answer: C

The ring returns to child-1 after 30 passes.

The k-th pass moves the ring k places (1 to 2 is one place, 2 to 4 is two, 4 to 7 is three). After n passes it has moved \(1 + 2 + \dots + n = n(n+1)/2\) places, and it is back with child-1 when that total is a multiple of 31.

\[ \begin{aligned} \frac{n(n+1)}{2} &\equiv 0 \pmod{31} \\ n = 30 &: \frac{30 \times 31}{2} = 465 = 15 \times 31 \end{aligned} \]

Since 31 is prime, it must divide n or n + 1, so the smallest n is 30. n = 31 also works (31 × 32/2 = 496 = 16 × 31), but 30 comes first, and 28, 29 and 32 do not give a multiple of 31.

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40.

How many triangles are there in the given figure?

A long rectangle divided by two vertical lines into three equal cells, each cell crossed by both of its diagonals

A.12
B.24
C.26
D.28

Exam Year(s)

ISRO Assistant - 2023,

Answer: D

The figure holds 28 triangles.

Each of the three cells is a rectangle cut by both diagonals. Inside one cell there are 4 small triangles (the four parts the diagonals make) and 4 larger ones, each made of two small parts, with a side of the cell as the base. That is 8 per cell.

\[ 3 \times 8 = 24 \]

Triangles also span two neighbouring cells. Take the top edge of two adjacent cells as the base, with the apex at the bottom point where the diagonals meet on the shared line. The same works with the bottom edge and the apex at the top. That is 2 for each pair of neighbouring cells, and there are 2 such pairs.

\[ 24 + 2 \times 2 = 28 \]

No triangle spans all three cells, since the outer diagonals do not meet over a common apex.

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