CAT 2024 Slot 2 — DILR Question 5
Answer the following questions based on the information given below.
The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are placed in ten slots of the following grid based on the conditions below.
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | □ | □ | □ | □ |
| Row 2 | □ | □ | □ | |
| Row 3 | □ | □ | ||
| Row 4 | □ |
(Row 1 has 4 slots in Columns 1–4; Row 2 has 3 slots in Columns 2–4; Row 3 has 2 slots in Columns 3–4; Row 4 has 1 slot in Column 4 — ten slots in all.)
1. Numbers in any row appear in an increasing order from left to right.
2. Numbers in any column appear in a decreasing order from top to bottom.
3. 1 is placed either in the same row or in the same column as 10.
4. Neither 2 nor 3 is placed in the same row or in the same column as 10.
5. Neither 7 nor 8 is placed in the same row or in the same column as 9.
6. 4 and 6 are placed in the same row.
What is the row number which has the least sum of numbers placed in that row?
Answer & solution
Answer: 4
Medium
Solve the grid first. Rows increase left-to-right, columns decrease top-to-bottom, so the global maximum 10 sits at the top-right of its row and column (Row 1, Column 4), and 1 sits at a bottom-left position. Pin 10, then 1, then the small numbers using the clues; only cells R2C2 and R3C3 (holding 2 and 3) stay interchangeable.
The fully solved grid (cells R2C2 and R3C3 hold {2, 3} in either order):
| Col 1 | Col 2 | Col 3 | Col 4 | |
|---|---|---|---|---|
| Row 1 | 4 | 6 | 9 | 10 |
| Row 2 | 2 or 3 | 5 | 8 | |
| Row 3 | 3 or 2 | 7 | ||
| Row 4 | 1 |
Place 10 and 1. 10 is the row-max and column-max ⇒ top-right cell R1C4. Clue 3 forces 1 into Row 1 or Column 4; combined with it being a row-min/column-min it lands at R4C4.
Compute the row sums. From the solved grid.