CAT 2024 Slot 2 · DILR · Set 4: Numbers 1 to 10 in a Staircase Grid

Solving the set

The grid has Row 1: Columns 1–4; Row 2: Columns 2–4; Row 3: Columns 3–4; Row 4: Column 4 only. Write (r, c) for the slot in Row r, Column c.

Step 1 – Place 10. Every slot (r, c) is smaller than (r, 4) (rows increase to the right), and (r, 4) is smaller than (1, 4) (columns decrease downwards). So (1, 4) is the largest slot: 10 is at (1, 4).

Step 2 – Place 2 and 3. They are not in Row 1 or Column 4 (fact 4), so they lie in (2, 2), (2, 3) or (3, 3). Now (2, 3) is larger than both (2, 2) and (3, 3). If 3 were at (2, 3), the other two slots would need numbers below 3, i.e., 1 and 2, but 1 must be in Row 1 or Column 4 (fact 3). So 2 and 3 are at (2, 2) and (3, 3) in some order.

Step 3 – Place 9. Only (1, 3) and (2, 4) have no larger slot other than (1, 4), so 9 is at one of them.

• If 9 is at (2, 4): 7 and 8 cannot be in Row 2 or Column 4 (fact 5), so they must be in Row 1, Columns 1–3. The numbers left (1, 4, 5, 6) go into one free Row 1 slot, (2, 3), (3, 4) and (4, 4): no row has two free slots, so 4 and 6 cannot share a row (fact 6). Not possible.

• So 9 is at (1, 3). Then 7 and 8 are not in Row 1 or Column 3, so they are in Column 4, rows 2–4.

Step 4 – Place 1. The smallest number must be at the bottom of a column, i.e., (1, 1), (2, 2), (3, 3) or (4, 4); with fact 3 it is (1, 1) or (4, 4).

• If 1 is at (1, 1): 4 and 6 need a row with two free slots. Row 1 has only (1, 2) free, so they go to Row 2: (2, 3) = 4 and (2, 4) = 6. But 7 and 8 must then be below 6 in Column 4, which breaks the decreasing order. Not possible.

• So 1 is at (4, 4). Then 7 and 8 are at (3, 4) and (2, 4); decreasing order gives (2, 4) = 8, (3, 4) = 7.

Step 5 – The numbers 4, 5, 6 remain for (1, 1), (1, 2) and (2, 3). 4 and 6 are in the same row, so they are (1, 1) = 4 and (1, 2) = 6, and (2, 3) = 5. All conditions are satisfied.

	Column 1	Column 2	Column 3	Column 4
Row 1	4	6	9	10
Row 2	—	2 or 3	5	8
Row 3	—	—	3 or 2	7
Row 4	—	—	—	1

This question

Row sums from the final grid: Row 1 = 4 + 6 + 9 + 10 = 29; Row 2 = (2 or 3) + 5 + 8 = 15 or 16; Row 3 = (3 or 2) + 7 = 10 or 9; Row 4 = 1.

Row 4 has the least sum in every case.

Hence, the answer is 4.

Solving the set

The grid has Row 1: Columns 1–4; Row 2: Columns 2–4; Row 3: Columns 3–4; Row 4: Column 4 only. Write (r, c) for the slot in Row r, Column c.

Step 1 – Place 10. Every slot (r, c) is smaller than (r, 4) (rows increase to the right), and (r, 4) is smaller than (1, 4) (columns decrease downwards). So (1, 4) is the largest slot: 10 is at (1, 4).

Step 2 – Place 2 and 3. They are not in Row 1 or Column 4 (fact 4), so they lie in (2, 2), (2, 3) or (3, 3). Now (2, 3) is larger than both (2, 2) and (3, 3). If 3 were at (2, 3), the other two slots would need numbers below 3, i.e., 1 and 2, but 1 must be in Row 1 or Column 4 (fact 3). So 2 and 3 are at (2, 2) and (3, 3) in some order.

Step 3 – Place 9. Only (1, 3) and (2, 4) have no larger slot other than (1, 4), so 9 is at one of them.

• If 9 is at (2, 4): 7 and 8 cannot be in Row 2 or Column 4 (fact 5), so they must be in Row 1, Columns 1–3. The numbers left (1, 4, 5, 6) go into one free Row 1 slot, (2, 3), (3, 4) and (4, 4): no row has two free slots, so 4 and 6 cannot share a row (fact 6). Not possible.

• So 9 is at (1, 3). Then 7 and 8 are not in Row 1 or Column 3, so they are in Column 4, rows 2–4.

Step 4 – Place 1. The smallest number must be at the bottom of a column, i.e., (1, 1), (2, 2), (3, 3) or (4, 4); with fact 3 it is (1, 1) or (4, 4).

• If 1 is at (1, 1): 4 and 6 need a row with two free slots. Row 1 has only (1, 2) free, so they go to Row 2: (2, 3) = 4 and (2, 4) = 6. But 7 and 8 must then be below 6 in Column 4, which breaks the decreasing order. Not possible.

• So 1 is at (4, 4). Then 7 and 8 are at (3, 4) and (2, 4); decreasing order gives (2, 4) = 8, (3, 4) = 7.

Step 5 – The numbers 4, 5, 6 remain for (1, 1), (1, 2) and (2, 3). 4 and 6 are in the same row, so they are (1, 1) = 4 and (1, 2) = 6, and (2, 3) = 5. All conditions are satisfied.

	Column 1	Column 2	Column 3	Column 4
Row 1	4	6	9	10
Row 2	—	2 or 3	5	8
Row 3	—	—	3 or 2	7
Row 4	—	—	—	1

This question

From Step 1, 10 is at (1, 4), so I is true. From Step 4, 1 is at (4, 4), so II is true.

Hence, the answer is option 2.

Solving the set

The grid has Row 1: Columns 1–4; Row 2: Columns 2–4; Row 3: Columns 3–4; Row 4: Column 4 only. Write (r, c) for the slot in Row r, Column c.

Step 1 – Place 10. Every slot (r, c) is smaller than (r, 4) (rows increase to the right), and (r, 4) is smaller than (1, 4) (columns decrease downwards). So (1, 4) is the largest slot: 10 is at (1, 4).

Step 2 – Place 2 and 3. They are not in Row 1 or Column 4 (fact 4), so they lie in (2, 2), (2, 3) or (3, 3). Now (2, 3) is larger than both (2, 2) and (3, 3). If 3 were at (2, 3), the other two slots would need numbers below 3, i.e., 1 and 2, but 1 must be in Row 1 or Column 4 (fact 3). So 2 and 3 are at (2, 2) and (3, 3) in some order.

Step 3 – Place 9. Only (1, 3) and (2, 4) have no larger slot other than (1, 4), so 9 is at one of them.

• If 9 is at (2, 4): 7 and 8 cannot be in Row 2 or Column 4 (fact 5), so they must be in Row 1, Columns 1–3. The numbers left (1, 4, 5, 6) go into one free Row 1 slot, (2, 3), (3, 4) and (4, 4): no row has two free slots, so 4 and 6 cannot share a row (fact 6). Not possible.

• So 9 is at (1, 3). Then 7 and 8 are not in Row 1 or Column 3, so they are in Column 4, rows 2–4.

Step 4 – Place 1. The smallest number must be at the bottom of a column, i.e., (1, 1), (2, 2), (3, 3) or (4, 4); with fact 3 it is (1, 1) or (4, 4).

• If 1 is at (1, 1): 4 and 6 need a row with two free slots. Row 1 has only (1, 2) free, so they go to Row 2: (2, 3) = 4 and (2, 4) = 6. But 7 and 8 must then be below 6 in Column 4, which breaks the decreasing order. Not possible.

• So 1 is at (4, 4). Then 7 and 8 are at (3, 4) and (2, 4); decreasing order gives (2, 4) = 8, (3, 4) = 7.

Step 5 – The numbers 4, 5, 6 remain for (1, 1), (1, 2) and (2, 3). 4 and 6 are in the same row, so they are (1, 1) = 4 and (1, 2) = 6, and (2, 3) = 5. All conditions are satisfied.

	Column 1	Column 2	Column 3	Column 4
Row 1	4	6	9	10
Row 2	—	2 or 3	5	8
Row 3	—	—	3 or 2	7
Row 4	—	—	—	1

This question

From Step 2, 2 and 3 are at (2, 2) and (3, 3), but which is where cannot be decided. So 2 may be in Column 3 and 3 may be in Column 2.

Neither statement must be true.

Hence, the answer is option 1.

Solving the set

The grid has Row 1: Columns 1–4; Row 2: Columns 2–4; Row 3: Columns 3–4; Row 4: Column 4 only. Write (r, c) for the slot in Row r, Column c.

Step 1 – Place 10. Every slot (r, c) is smaller than (r, 4) (rows increase to the right), and (r, 4) is smaller than (1, 4) (columns decrease downwards). So (1, 4) is the largest slot: 10 is at (1, 4).

Step 2 – Place 2 and 3. They are not in Row 1 or Column 4 (fact 4), so they lie in (2, 2), (2, 3) or (3, 3). Now (2, 3) is larger than both (2, 2) and (3, 3). If 3 were at (2, 3), the other two slots would need numbers below 3, i.e., 1 and 2, but 1 must be in Row 1 or Column 4 (fact 3). So 2 and 3 are at (2, 2) and (3, 3) in some order.

Step 3 – Place 9. Only (1, 3) and (2, 4) have no larger slot other than (1, 4), so 9 is at one of them.

• If 9 is at (2, 4): 7 and 8 cannot be in Row 2 or Column 4 (fact 5), so they must be in Row 1, Columns 1–3. The numbers left (1, 4, 5, 6) go into one free Row 1 slot, (2, 3), (3, 4) and (4, 4): no row has two free slots, so 4 and 6 cannot share a row (fact 6). Not possible.

• So 9 is at (1, 3). Then 7 and 8 are not in Row 1 or Column 3, so they are in Column 4, rows 2–4.

Step 4 – Place 1. The smallest number must be at the bottom of a column, i.e., (1, 1), (2, 2), (3, 3) or (4, 4); with fact 3 it is (1, 1) or (4, 4).

• If 1 is at (1, 1): 4 and 6 need a row with two free slots. Row 1 has only (1, 2) free, so they go to Row 2: (2, 3) = 4 and (2, 4) = 6. But 7 and 8 must then be below 6 in Column 4, which breaks the decreasing order. Not possible.

• So 1 is at (4, 4). Then 7 and 8 are at (3, 4) and (2, 4); decreasing order gives (2, 4) = 8, (3, 4) = 7.

Step 5 – The numbers 4, 5, 6 remain for (1, 1), (1, 2) and (2, 3). 4 and 6 are in the same row, so they are (1, 1) = 4 and (1, 2) = 6, and (2, 3) = 5. All conditions are satisfied.

	Column 1	Column 2	Column 3	Column 4
Row 1	4	6	9	10
Row 2	—	2 or 3	5	8
Row 3	—	—	3 or 2	7
Row 4	—	—	—	1

This question

Only the two slots holding 2 and 3 ((2, 2) and (3, 3)) are undetermined. The other 8 slots are fixed.

Hence, the answer is 2.

Solving the set

The grid has Row 1: Columns 1–4; Row 2: Columns 2–4; Row 3: Columns 3–4; Row 4: Column 4 only. Write (r, c) for the slot in Row r, Column c.

Step 1 – Place 10. Every slot (r, c) is smaller than (r, 4) (rows increase to the right), and (r, 4) is smaller than (1, 4) (columns decrease downwards). So (1, 4) is the largest slot: 10 is at (1, 4).

Step 2 – Place 2 and 3. They are not in Row 1 or Column 4 (fact 4), so they lie in (2, 2), (2, 3) or (3, 3). Now (2, 3) is larger than both (2, 2) and (3, 3). If 3 were at (2, 3), the other two slots would need numbers below 3, i.e., 1 and 2, but 1 must be in Row 1 or Column 4 (fact 3). So 2 and 3 are at (2, 2) and (3, 3) in some order.

Step 3 – Place 9. Only (1, 3) and (2, 4) have no larger slot other than (1, 4), so 9 is at one of them.

• If 9 is at (2, 4): 7 and 8 cannot be in Row 2 or Column 4 (fact 5), so they must be in Row 1, Columns 1–3. The numbers left (1, 4, 5, 6) go into one free Row 1 slot, (2, 3), (3, 4) and (4, 4): no row has two free slots, so 4 and 6 cannot share a row (fact 6). Not possible.

• So 9 is at (1, 3). Then 7 and 8 are not in Row 1 or Column 3, so they are in Column 4, rows 2–4.

Step 4 – Place 1. The smallest number must be at the bottom of a column, i.e., (1, 1), (2, 2), (3, 3) or (4, 4); with fact 3 it is (1, 1) or (4, 4).

• If 1 is at (1, 1): 4 and 6 need a row with two free slots. Row 1 has only (1, 2) free, so they go to Row 2: (2, 3) = 4 and (2, 4) = 6. But 7 and 8 must then be below 6 in Column 4, which breaks the decreasing order. Not possible.

• So 1 is at (4, 4). Then 7 and 8 are at (3, 4) and (2, 4); decreasing order gives (2, 4) = 8, (3, 4) = 7.

Step 5 – The numbers 4, 5, 6 remain for (1, 1), (1, 2) and (2, 3). 4 and 6 are in the same row, so they are (1, 1) = 4 and (1, 2) = 6, and (2, 3) = 5. All conditions are satisfied.

	Column 1	Column 2	Column 3	Column 4
Row 1	4	6	9	10
Row 2	—	2 or 3	5	8
Row 3	—	—	3 or 2	7
Row 4	—	—	—	1

This question

Column 4 from the final grid: 10 + 8 + 7 + 1 = 26.

Hence, the answer is 26.