Question

What is the row number which has the least sum of numbers placed in that row?

• A. Only II
• B. Both I and II
• C. Neither I nor II
• D. Only I
• A. Only II
• B. Neither I nor II
• C. Only I
• D. Both I and II

Answer 1

Correct Answer: 4

Here's the rephrased text, maintaining all HTML tags and LaTeX:

1	2	3	4
5	6	7	8
9	10	Nbsp;	Nbsp;

To solve this, we'll follow these logical steps based on the given conditions:

1. Analyze Conditions:
   • Numbers in each row must increase from left to right.
   • Numbers in each column must decrease from top to bottom.
   • 1 and 10 must share either a row or a column.
   • 10 cannot be in the same row or column as 2 or 3.
   • 9 cannot be in the same row or column as 7 or 8.
   • 4 and 6 must be in the same row.
2. Place 1 and 10: Because 1 and 10 must be in the same row or column, they are placed at opposite corners. We'll put 9 in the third row, first column, and 10 in the first column, third row.
3. Place 4 and 6: Given the increasing order requirement for rows, 4 and 6 are placed in the first row as 4, 6.
4. Place Remaining Numbers:
   • 2 and 3 must not be with 10. Place 2 and 3 in the first row, before 4, as 2, 3.
   • 7 and 8 must not be with 9. Place them in the second row, after 6, as 7, 8.
5. Calculate Row Sums:
   • First row sum: 2 + 3 + 4 + 6 = 15
   • Second row sum: 5 + 7 + 8 + 9 = 29
   • Third row sum: 1 + 10 = 11
6. Conclusion: The first row has the smallest sum, which is 15.

After calculating the row sums, the smallest sum is 11. This value falls within the expected range of (4, 4). Therefore, the third row has the smallest sum while satisfying all conditions.

Answer 2

Based on the given conditions, we can deduce the following placements:

• Condition 1: Numbers in each row must increase from left to right.
• Condition 2: Numbers in each column must decrease from top to bottom.
• Condition 3: The number 1 must share a row or column with the number 10. Specifically, if 10 is in Row 1, then 1 must also be in Row 1.
• When all conditions are applied, placing 10 in Row 1 and 1 in Row 4 satisfies all requirements.

Therefore, both statements (I) and (II) are true.

Answer 3

Based on the conditions:

• Condition 3 states that 2 cannot be in the same row or column as 10. Therefore, 2 is not necessarily in Column 2.
• Condition 4 indicates that 3 is not required to be in Column 3. Its placement is contingent on other conditions not met here.

Consequently, neither statement I nor II must be true. The correct option is 2: Neither I nor II.

Answer 4

Based on the given constraints, we can deduce the following:
(a) 1 and 10:Nbsp;
• Must be in the same row or column.
• Because rows increase and columns decrease, they must be in opposite corners.
• Possible placements:
– 1 in Row 1, Column 1 and 10 in Row 4, Column 1.
Nbsp; – 1 in Row 4, Column 4 and 10 in Row 1, Column 4.

(b) 4 and 6:
• Must be in the same row.
• Cannot be in Row 1 or Row 4 (due to 1 and 10).
• Therefore, they must be in either Row 2 or Row 3.

(c) 2, 3, 7, and 8:
• Their positions are limited by where 1, 10, 4, and 6 are placed.

(d) Uncertain Slots:

• Due to these restrictions, we cannot definitively place numbers in the following two slots:
Nbsp;– The slot in Row 4, Column 2 or Column 3: This slot cannot be filled with 1, 2, 3, 4, 6, 7, 8, or 10.
Nbsp;– The other slot in Row 4: This slot also cannot be filled with 1, 2, 3, 4, 6, 7, 8, or 10.

Therefore, the answer to the question, "For how many slots in the grid can the placement of numbers NOT be determined with certainty?" is 2.

Column 1	Column 2	Column 3	Column 4
Row 1	1	2	3
Row 2	4	5	6
Row 3	7	8	9
Row 4	10	Nbsp;	Nbsp;

Note: While other valid arrangements might exist, the number of uncertain slots will remain the same.

Answer 5

Given the constraints, we can infer the following:
(a) Numbers 1 and 10:
• They must occupy the same row or column.
• Because rows increase and columns decrease, they must be placed in opposite corners.
• Possible arrangements:
– 1 in Row 1, Column 1 and 10 in Row 4, Column 1.
Nbsp; – 1 in Row 4, Column 4 and 10 in Row 1, Column 4.

(b) Numbers 4 and 6:
• They must be in the same row.
• They cannot be in Row 1 or Row 4 (due to the placement of 1 and 10).
• Therefore, they must be in either Row 2 or Row 3.

(c) Numbers 2, 3, 7, and 8:
• Their positions are limited by where 1, 10, 4, and 6 are placed.

Calculating the Sum of Column 4:
Based on the constraints and possible placements, we can determine that:
Nbsp;- Column 4 must include the numbers 1, 9, and 10.
The sum of the numbers in Column 4 is therefore:
1 + 9 + 10 = 20

The final answer is 20.