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

To determine the row with the least sum of numbers, we need to carefully analyze the
placement of the numbers in the grid based on the given conditions.
(a) From Condition 1, numbers in rows must increase from left to right, and from Condition 2, numbers in columns must decrease from top to bottom. This restricts the placement of higher numbers like 9 and 10.
(b) From Condition 3, the placement of 1 must coincide with either the row or column containing 10.
(c) From Condition 4, neither 2 nor 3 can appear in the same row or column as 10.
(d) Condition 5 eliminates 7 and 8 from the row or column containing 9.
(e) Condition 6 requires 4 and 6 to be in the same row.

Based on these constraints, we evaluate each row’s possible sum. After placing the num-
bers, the row with the least sum is Row 4.

Thus, the correct answer is: 4 .

Answer 2

Using the given conditions, we can determine the following placements:

• From Condition 1, numbers in rows must increase from left to right.
• From Condition 2, numbers in columns must decrease from top to bottom.
• Condition 3 ensures that 1 is in the same row or column as 10. If 10 is in Row 1, then 1 must be in Row 1.
• Combining all constraints, the placement of 10 in Row 1 and 1 in Row 4 fulfills the required criteria.

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

Answer 3

Based on the conditions:

• From Condition 3, 2 cannot be in the same row or column as 10. Thus, it cannot
  necessarily occupy Column 2.
• From Condition 4, 3 is not required to be placed in Column 3. Its placement depends
  on other conditions that are not fulfilled here.

As a result, neither statement I nor II must be true, making the correct option 2. Neither I nor II.

Answer 4

Based on the given constraints, we can deduce the following:
(a) 1 and 10:
• Must be in the same row or column.
• Due to increasing rows and decreasing columns, they must be placed in opposite corners.
• Possible placements:
– 1 in Row 1, Column 1 and 10 in Row 4, Column 1.
  – 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).
• So, they must be in either Row 2 or Row 3.

(c) 2, 3, 7, and 8:
• Their placements are restricted by the placements of 1, 10, 4, and 6.

(d) Uncertain Slots:

• Due to these constraints, we cannot definitively determine the placement of num-
bers in the following two slots:
 – 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.
 – 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, placement of
numbers CANNOT 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

Note: Other valid configurations may exist, but the number of uncertain slots remains
the same.

Answer 5

Based on the given constraints, we can deduce the following:
(a) 1 and 10:
• Must be in the same row or column.
• Due to increasing rows and decreasing columns, they must be placed in opposite corners.
• Possible placements:
– 1 in Row 1, Column 1 and 10 in Row 4, Column 1.
  – 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).
• So, they must be in either Row 2 or Row 3.

(c) 2, 3, 7, and 8:
• Their placements are restricted by the placements of 1, 10, 4, and 6.

Determining the Sum of Column 4:
Considering the constraints and the possible placements, we can deduce that:
 - Column 4 must contain the numbers 1, 9, and 10.
Therefore, the sum of the numbers in Column 4 is:
1 + 9 + 10 = 20

Thus, the final answer is 20.