500 men, all of different heights, are arranged in a grid of 10 rows and 50…
2025
500 men, all of different heights, are arranged in a grid of 10 rows and 50 columns so that height increases row by row, left to right — from the shortest man in row 1 to the tallest man in row 10. In each row, the tallest man is asked to step out; the shortest among these 10 men is called A. The men then resume their original positions, and in each column, the shortest man is asked to step out; the tallest among these 50 men is called B. Who is taller, A or B?
- A.
A and B are the same person
- B.
A and B are not the same person
- C.
None of these
- D.
Cannot be determined
Show answer & explanation
Correct answer: A
Concept: For any grid of distinct values arranged in rows and columns, taking the maximum of each row and then the smallest of those row-maxima always gives a value greater than or equal to taking the minimum of each column and then the largest of those column-minima. Equality happens exactly when one cell is simultaneously the maximum of its own row and the minimum of its own column — that cell is called a saddle point of the grid.
Application: Rank the 500 men from 1 (shortest) to 500 (tallest) and fill the grid row by row in increasing rank order.
Row 1 gets ranks 1–50, row 2 gets ranks 51–100, and so on up to row 10, which gets ranks 451–500, with rank increasing left to right across the 50 columns.
The maximum of row i is its right-most (highest-rank) entry, so the ten row-maxima are 50, 100, 150, …, 500. The smallest of these is 50 — so A is the man ranked 50, sitting in row 1, column 50.
The minimum of column j is its row-1 entry, because row 1 holds the 50 smallest ranks in the whole grid (1 to 50), which are smaller than every other row's entry in that same column. So the fifty column-minima are exactly the ranks 1, 2, …, 50, one per column. The largest of these is 50 — so B is also the man ranked 50, sitting in row 1, column 50.
Row | Ranks in row | Row maximum |
|---|---|---|
1 | 1–50 | 50 |
2 | 51–100 | 100 |
3 | 101–150 | 150 |
4 | 151–200 | 200 |
5 | 201–250 | 250 |
6 | 251–300 | 300 |
7 | 301–350 | 350 |
8 | 351–400 | 400 |
9 | 401–450 | 450 |
10 | 451–500 | 500 |
Cross-check: Row 1, column 50 is simultaneously the maximum of row 1 (the largest rank in that row) and the minimum of column 50 (the smallest rank in that column) — exactly the saddle-point condition from the concept above. Since A and B both resolve to this one cell, they are not merely equal in height; they are the same person.