Six people U, V, W, X, Y, and Z, are standing at different intersections. No two people are standing at the same intersection.
The following additional facts are known.
1. X, U, and Z are standing at the three corners of a triangle formed by three street segments.
2. X can see only U and Z.
3. Y can see only U and W.
4. U sees V standing in the next intersection behind Z.
5. W cannot see V or Z.
6. No one among the six is standing at intersection d.
Who is standing at intersection a?
Who is standing at intersection a?
- No one
- V
- W
- Y
The Correct Option is A
Solution and Explanation
Let's analyze the provided facts:
- X, U, and Z form a triangle where they are the only people at the three corners.
- X can see only U and Z, implying X must be on an edge, with U and Z on direct lines.
- Y can see only U and W, restricting Y's position to a line with U and W, but not with Z or V.
- U sees V behind Z, suggesting V is on a line further from U's perspective behind Z.
- W can't see V or Z, so W mustn't be on interconnected lines with them.
- No one is at intersection d.
- X, U, and Z form a triangle; possible corners: X at f, U at g, Z at e. X can see U (g) and Z (e).
- V needs placement where U can see it behind Z. Possible placement: V at l, seen from g through e.
- Y sees only U and W: potential positions are i for Y, seeing U at g and W at j.
- W at j fulfills not seeing V or Z.
- a: No one remains unplaced among U, V, W, X, Y, and Z.
Who can V see?
- U only
- U, W and Z only
U and Z only
- Z only
The Correct Option is C
Solution and Explanation
The problem involves identifying who individual V can see given specific constraints in a street intersection map. Let's resolve this step by step using the provided conditions:
1. X, U, and Z form the corners of a triangle on the street map, indicating they are connected by street segments.
2. X can see only U and Z, suggesting that X is possibly at a corner with unobstructed sight to intersections U and Z exclusively.
3. Y's sight is limited to U and W, indicating that Y is at an intersection in line with U and W but not further lines of sight such as V or Z.
4. U has a direct line of sight to V, positioned behind Z relative to U. This implies Z is between U and V along a straight line.
5. W cannot see V or Z, meaning W's position intercepts lines leading to those points.
6. No one is located at intersection d.
Given the constraints above, for V's placement: V is positioned directly in line with U and Z such that U sees Z and V behind Z.
| Intersections | Sighted Individuals |
| X | U, Z |
| Y | U, W |
| U | V (behind Z) |
| W | No V or Z |
From the above, V can see U and Z directly because they share a linear alignment, confirming the correct answer: "U and Z only".
What is the minimum number of street segments that X must cross to reach Y?
- 2
- 3
- 1
- 4
The Correct Option is A
Solution and Explanation
The problem asks us to determine the minimum number of street segments that X must cross to reach Y, based on provided constraints.
Below is the reasoning:
- People are standing at different intersections with no two people standing at the same intersection.
- X, U, and Z form a triangle with three street segments, meaning they are positioned at three intersections that create a triangular path.
- X can only see U and Z, indicating X is at one corner of the triangle.
- Y can only see U and W, suggesting Y is positioned in line with these intersections.
- U sees V in the next intersection behind Z, aligning V in the visual path behind Z from U's perspective.
- W cannot see V or Z, limiting W's line of sight to other specific intersections.
To deduce the street segments X must cross to reach Y, consider:
- The triangle comprises the intersections of X, U, and Z exclusively. Since X can only see U and Z, X’s movement to any other intersection must involve crossing a segment not part of the triangle.
- To move toward Y, X must cross either directly to U or Z first, then navigate through additional segments to align with Y’s line of visibility, incorporating intersections that allow visual contact with U and W.
Considering these deductions, the minimum street segments X must cross to align the path toward Y, taking optimal routes and respecting line of sight constraints, is 2.
| Options | Intersection Strategy |
|---|---|
| 1 | Requires an unobstructed line sight which isn't feasible here. |
| 2 | Optimal path as explained above. |
| 3 | Requires crossing one extra, unnecessary segment. |
| 4 | Exceeds the needed path and involves indirect navigation. |
Hence, the minimum number of street segments X must cross to reach Y is 2.
Should a new person stand at intersection d, who among the six would she see?
- U and Z only
V and X only
- W and X only
- U and W only
The Correct Option is C
Solution and Explanation
Valid Configuration:
Case (v) is the only valid one.
Explanation
- U is at g
- Z is at f
- V is at e
This satisfies the condition: \( U \rightarrow Z \rightarrow V \) are in a straight line.
X must be at b because X can see only U and Z (positions g and f).
Hence, no one is at positions: a, c, j.
W cannot be at h or i because W cannot see Z or V.
Thus, W must be at I.
Y must see both U and W. The only valid position for Y to see both g and I is from k.
✅ Final Positions
| Person | Position |
|---|---|
| X | b |
| U | g |
| Z | f |
| V | e |
| W | I |
| Y | k |
All conditions are satisfied in this configuration.
If a new person is standing at d, that person can see W and X. Answer: (W and X only)
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