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Multiple People Movement Puzzle

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Introduction

Multiple People Movement puzzles give several persons individual movement instructions (different directions/lengths or turn sequences). Your task is to compare their final positions or facings - for example, who is farthest from a point, who is to the left of whom, or the relative directions between them.

This pattern is important because many competitive exams and aptitude tests use multi-agent movement to test spatial reasoning, vector addition, and relative-position inference.

Pattern: Multiple People Movement Puzzle

Pattern: Multiple People Movement Puzzle

Key concept: Treat each person’s path as a vector (or a facing-change sequence), compute their final coordinates/facings, then compare pairwise to answer relative queries.

Practical rules

  • Convert each person’s movements into coordinate changes on a plane: North = +y, South = -y, East = +x, West = -x.
  • Add/subtract components separately (sum all x-components, sum all y-components) to get final positions.
  • Use Pythagoras for straight-line distances and sign of components to determine direction (NE/SE/NW/SW or cardinal).
  • For facing questions, track turns sequentially (Step-by-step facing updates) rather than deducing from displacement.
  • When multiple people move, align all results on the same coordinate system before comparing - draw a quick sketch if uncertain.

Step-by-Step Example

Question

A walks 5 m North then 4 m East. B walks 3 m East then 6 m North. C walks 6 m North then 1 m East. Who is farthest from the starting point?

Solution

  1. Step 1: Set a coordinate system

    Let the starting point be (0,0). Use (x, y) with East = +x and North = +y.

  2. Step 2: Convert A’s movements to coordinates

    A: 5 m North → y = +5; then 4 m East → x = +4. Final A = (4, 5).

  3. Step 3: Convert B’s movements to coordinates

    B: 3 m East → x = +3; then 6 m North → y = +6. Final B = (3, 6).

  4. Step 4: Convert C’s movements to coordinates

    C: 6 m North → y = +6; then 1 m East → x = +1. Final C = (1, 6).

  5. Step 5: Compute straight-line distances from origin

    A distance = √(4² + 5²) = √(16 + 25) = √41 ≈ 6.40.
    B distance = √(3² + 6²) = √(9 + 36) = √45 ≈ 6.71.
    C distance = √(1² + 6²) = √(1 + 36) = √37 ≈ 6.08.

  6. Step 6: Compare distances

    B (≈6.71) > A (≈6.40) > C (≈6.08). So B is farthest from the start.

  7. Final Answer:

    B is farthest from the starting point.

  8. Quick Check:

    B has the largest y-component (6) and decent x (3), giving largest hypotenuse → B ✅

Quick Variations

1. Compare who is to the left/right of another after movements (use relative vectors).

2. Determine final facing when different persons follow turn sequences instead of straight moves.

3. Mixed problems: some people move, others turn - combine displacement + facing logic.

4. Problems that ask for who crosses paths or who meets whom - solve by equating parametric positions if timings are given.

Trick to Always Use

  • Step 1: Convert each person’s movement into (Δx, Δy) immediately.
  • Step 2: Sum components and compute distances with √(Δx² + Δy²) only once per person.
  • Step 3: For comparisons, you can compare squared distances (Δx² + Δy²) to avoid square roots.

Summary

  • Convert each person’s movement into coordinate components (Δx, Δy).
  • Add all x- and y-components to find each person’s final position.
  • Use √(Δx² + Δy²) or compare squared distances to identify who is farthest.
  • Draw a quick sketch to visualize relative positions and confirm direction logic.

Example to remember:
A: (4, 5), B: (3, 6), C: (1, 6) → B is farthest (largest √(Δx² + Δy²)).

Practice

(1/5)
1. A walks 5 m North, B walks 3 m East, and C walks 4 m South. Who is farthest from the starting point?
easy
A. A
B. B
C. C
D. All are equally distant

Solution

  1. Step 1: Convert to coordinates

    A(0,5), B(3,0), C(0,-4).

  2. Step 2: Compute distances

    A = √(0² + 5²) = 5; B = √(3² + 0²) = 3; C = √(0² + (-4)²) = 4.

  3. Step 3: Compare

    A has the largest distance.

  4. Final Answer:

    A → Option A

  5. Quick Check:

    Largest magnitude 5 → A ✅
Hint: Convert to coordinate form (x, y) and apply √(x² + y²).
Common Mistakes: Comparing vertical or horizontal moves instead of diagonal displacement.
2. P walks 6 m East and 8 m North, Q walks 5 m East and 10 m North. Who is farther from the start?
easy
A. P
B. Q
C. Both equally distant
D. Cannot be determined

Solution

  1. Step 1: Compute coordinates

    P(6,8), Q(5,10).

  2. Step 2: Distances

    P = √(6² + 8²) = 10; Q = √(5² + 10²) = √125 ≈ 11.18.

  3. Step 3: Compare

    Q farther from start.

  4. Final Answer:

    Q → Option B

  5. Quick Check:

    Q has higher y-component (10) → farther ✅
Hint: Compare squared distances to avoid calculating square roots.
Common Mistakes: Forgetting to square each component before adding.
3. T walks 7 m North and 4 m East, while U walks 5 m North and 6 m East. Who is to the North-East of the other?
medium
A. T is to the North-East of U
B. U is to the North-East of T
C. Both at same point
D. None of these

Solution

  1. Step 1: Represent movements as coordinates

    T = (4, 7), U = (6, 5) where coordinates are (x = East, y = North).

  2. Step 2: Compute relative vector (U - T)

    (6 - 4, 5 - 7) = (2, -2).

  3. Step 3: Interpret the vector

    Δx = +2 (East), Δy = -2 (South) → from T to U is East and South → South-East. So U is South-East of T, and T is North-West of U.

  4. Final Answer:

    None of these → Option D

  5. Quick Check:

    U has larger x but smaller y than T (x↑ → East, y↓ → South) ⇒ SE relation, not NE. ✅
Hint: Compute (other - reference) as (Δx, Δy): +x = East, -x = West; +y = North, -y = South. Both positive → NE; mixed signs → intercardinal accordingly.
Common Mistakes: Mixing up the sign of Δy (treating a negative Δy as North instead of South) or reversing the subtraction order (T-U instead of U-T).
4. A walks 6 m North, B walks 8 m South, and C walks 5 m North. Who is farthest from B?
medium
A. A
B. B
C. Both equally distant
D. Cannot be determined

Solution

  1. Step 1: Positions

    A at +6 m (North), B at -8 m (South), C at +5 m (North) on the same vertical axis.

  2. Step 2: Distance from B

    Distance A-B = |6 - (-8)| = 14 m. Distance C-B = |5 - (-8)| = 13 m.

  3. Step 3: Compare

    A is farther from B since 14 > 13.

    4. Final Answer:

    A → Option A
Hint: When movement is along one axis, take absolute differences to compare distances.
Common Mistakes: Forgetting to take absolute difference (sign) when comparing distances along the same axis.
5. P walks 8 m East and 6 m North, Q walks 10 m East and 2 m North. How far and in which direction is Q from P?
medium
A. 2√5 m South-East
B. 5 m South
C. 3 m North-East
D. 4 m East

Solution

  1. Step 1: Coordinates

    P(8,6), Q(10,2).

  2. Step 2: Relative vector (Q-P)

    (10-8, 2-6) = (2, -4).

  3. Step 3: Distance

    √(2² + (-4)²) = √20 = 2√5.

  4. Step 4: Direction

    +x, -y → South-East.

  5. Final Answer:

    2√5 m South-East → Option A

  6. Quick Check:

    Δx=2, Δy=-4 → SE direction ✅
Hint: For relative direction, subtract coordinates (Q-P) and interpret sign pattern.
Common Mistakes: Reversing the subtraction order (P-Q instead of Q-P).