CAT — Grid / Matrix Arrangement
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The schematic diagram below shows 12 rectangular houses in a housing complex. House numbers are mentioned in the rectangles representing the houses. The houses are located in six columns – Column-A through Column-F, and two rows – Row-1 and Row-2. The houses are divided into two blocks - Block XX and Block YY. The diagram also shows two roads, one passing in front of the houses in Row-2 and another between the two blocks.
Some of the houses are occupied. The remaining ones are vacant and are the only ones available for sale.
The road adjacency value of a house is the number of its sides adjacent to a road. For example, the road adjacency values of C2, F2, and B1 are 2, 1, and 0, respectively. The neighbour count of a house is the number of sides of that house adjacent to occupied houses in the same block. For example, E1 and C1 can have the maximum possible neighbour counts of 3 and 2, respectively.
The base price of a vacant house is Rs. 10 lakhs if the house does not have a parking space, and Rs. 12 lakhs if it does. The quoted price (in lakhs of Rs.) of a vacant house is calculated as (base price) + 5 × (road adjacency value) + 3 × (neighbour count).
The following information is also known.
1. The maximum quoted price of a house in Block XX is Rs. 24 lakhs. The minimum quoted price of a house in block YY is Rs. 15 lakhs, and one such house is in Column-E.
2. Row-1 has two occupied houses, one in each block.
3. Both houses in Column-E are vacant. Each of Column-D and Column-F has at least one occupied house.
4. There is only one house with parking space in Block YY.
How many houses are vacant in Block XX?
Step 1:
The quoted price (in lakhs of rupees) is calculated as:
Quoted Price = Base Price + 5 × (Number of adjacent roads) + 3 × (Number of neighbouring occupied houses)
Thus,
QP = 10/12 + 5(0, 1, 2) + 3(0, 1, 2, 3)
Using conditions (1) and (2),
- 24 = 10 + 5 × 1 + 3 × 3
- 15 = 10 + 5 × 1 + 3 × 0 or 15 = 12 + 5 × 0 + 3 × 1
In Block XX, the highest quoted price is Rs. 24 lakh.
Hence, B1, A2, and C2 are occupied, while A1, B2, and C1 are vacant (i.e., available for sale).
Step 2:
From condition (3), E1 and E2 are vacant.
Also, from conditions (2) and (3), exactly one of D1 and F1 is occupied.
For E1,
Quoted Price = 15
= 12 + 5 × 0 + 3 × 1
Therefore, by condition (4), E1 is the only house in Block YY with a parking space.
Further, from condition (3), each of Column D and Column F contains at least one occupied house.
This gives two possibilities:
- If D1 is occupied, then F2 must also be occupied, while D2 may be either occupied or vacant.
- If F1 is occupied, then D2 must be occupied, while F2 may be either occupied or vacant.
Three houses A1, C1 and B2 are vacant in Block XX.
Which of the following houses is definitely occupied?
B1
A1
D2
F2
B1
Step 1:
The quoted price (in lakhs of rupees) is calculated as:
Quoted Price = Base Price + 5 × (Number of adjacent roads) + 3 × (Number of neighbouring occupied houses)
Thus,
QP = 10/12 + 5(0, 1, 2) + 3(0, 1, 2, 3)
Using conditions (1) and (2),
- 24 = 10 + 5 × 1 + 3 × 3
- 15 = 10 + 5 × 1 + 3 × 0 or 15 = 12 + 5 × 0 + 3 × 1
In Block XX, the highest quoted price is Rs. 24 lakh.
Hence, B1, A2, and C2 are occupied, while A1, B2, and C1 are vacant (i.e., available for sale).
Step 2:
From condition (3), E1 and E2 are vacant.
Also, from conditions (2) and (3), exactly one of D1 and F1 is occupied.
For E1,
Quoted Price = 15
= 12 + 5 × 0 + 3 × 1
Therefore, by condition (4), E1 is the only house in Block YY with a parking space.
Further, from condition (3), each of Column D and Column F contains at least one occupied house.
This gives two possibilities:
- If D1 is occupied, then F2 must also be occupied, while D2 may be either occupied or vacant.
- If F1 is occupied, then D2 must be occupied, while F2 may be either occupied or vacant.
House B1 is definitely occupied.
Which of the following options best describes the number of vacant houses in Row-2?
Exactly 3
Either 2 or 3
Either 3 or 4
Exactly 2
Either 2 or 3
Step 1:
The quoted price (in lakhs of rupees) is calculated as:
Quoted Price = Base Price + 5 × (Number of adjacent roads) + 3 × (Number of neighbouring occupied houses)
Thus,
QP = 10/12 + 5(0, 1, 2) + 3(0, 1, 2, 3)
Using conditions (1) and (2),
- 24 = 10 + 5 × 1 + 3 × 3
- 15 = 10 + 5 × 1 + 3 × 0 or 15 = 12 + 5 × 0 + 3 × 1
In Block XX, the highest quoted price is Rs. 24 lakh.
Hence, B1, A2, and C2 are occupied, while A1, B2, and C1 are vacant (i.e., available for sale).
Step 2:
From condition (3), E1 and E2 are vacant.
Also, from conditions (2) and (3), exactly one of D1 and F1 is occupied.
For E1,
Quoted Price = 15
= 12 + 5 × 0 + 3 × 1
Therefore, by condition (4), E1 is the only house in Block YY with a parking space.
Further, from condition (3), each of Column D and Column F contains at least one occupied house.
This gives two possibilities:
- If D1 is occupied, then F2 must also be occupied, while D2 may be either occupied or vacant.
- If F1 is occupied, then D2 must be occupied, while F2 may be either occupied or vacant.
The number of vacant houses in Row-2 is either 2 or 3.
What is the maximum possible quoted price (in lakhs of Rs.) for a vacant house in Column-E?
Step 1:
The quoted price (in lakhs of rupees) is calculated as:
Quoted Price = Base Price + 5 × (Number of adjacent roads) + 3 × (Number of neighbouring occupied houses)
Thus,
QP = 10/12 + 5(0, 1, 2) + 3(0, 1, 2, 3)
Using conditions (1) and (2),
- 24 = 10 + 5 × 1 + 3 × 3
- 15 = 10 + 5 × 1 + 3 × 0 or 15 = 12 + 5 × 0 + 3 × 1
In Block XX, the highest quoted price is Rs. 24 lakh.
Hence, B1, A2, and C2 are occupied, while A1, B2, and C1 are vacant (i.e., available for sale).
Step 2:
From condition (3), E1 and E2 are vacant.
Also, from conditions (2) and (3), exactly one of D1 and F1 is occupied.
For E1,
Quoted Price = 15
= 12 + 5 × 0 + 3 × 1
Therefore, by condition (4), E1 is the only house in Block YY with a parking space.
Further, from condition (3), each of Column D and Column F contains at least one occupied house.
This gives two possibilities:
- If D1 is occupied, then F2 must also be occupied, while D2 may be either occupied or vacant.
- If F1 is occupied, then D2 must be occupied, while F2 may be either occupied or vacant.
The maximum possible quoted price for a vacant house i.e., E2 in Column-E = 10 + 5 × 1 + 3 × 2 = Rs.21 lakhs.
Which house in Block YY has parking space?
E1
F1
F2
E2
E1
Step 1:
The quoted price (in lakhs of rupees) is calculated as:
Quoted Price = Base Price + 5 × (Number of adjacent roads) + 3 × (Number of neighbouring occupied houses)
Thus,
QP = 10/12 + 5(0, 1, 2) + 3(0, 1, 2, 3)
Using conditions (1) and (2),
- 24 = 10 + 5 × 1 + 3 × 3
- 15 = 10 + 5 × 1 + 3 × 0 or 15 = 12 + 5 × 0 + 3 × 1
In Block XX, the highest quoted price is Rs. 24 lakh.
Hence, B1, A2, and C2 are occupied, while A1, B2, and C1 are vacant (i.e., available for sale).
Step 2:
From condition (3), E1 and E2 are vacant.
Also, from conditions (2) and (3), exactly one of D1 and F1 is occupied.
For E1,
Quoted Price = 15
= 12 + 5 × 0 + 3 × 1
Therefore, by condition (4), E1 is the only house in Block YY with a parking space.
Further, from condition (3), each of Column D and Column F contains at least one occupied house.
This gives two possibilities:
- If D1 is occupied, then F2 must also be occupied, while D2 may be either occupied or vacant.
- If F1 is occupied, then D2 must be occupied, while F2 may be either occupied or vacant.
House E1 in Block YY has parking space.
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.
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?
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
Which of the following statements MUST be true?
I. 10 is placed in a slot in Row 1.
II. 1 is placed in a slot in Row 4.
Only I
Both I and II
Only II
Neither I nor II
Both I and II
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
Which of the following statements MUST be true?
I. 2 is placed in a slot in Column 2.
II. 3 is placed in a slot in Column 3.
Both I and II
Neither I nor II
Only II
Only I
Neither I nor II
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
For how many slots in the grid, placement of numbers CANNOT be determined with certainty?
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
What is the sum of the numbers placed in Column 4?
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
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