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.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743430001/aptidude_questions/zs3sax8pq1mucrm2j3z9.png)

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.

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How many houses are vacant in Block XX?

Question

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.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743430001/aptidude_questions/zs3sax8pq1mucrm2j3z9.png)

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.

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How many houses are vacant in Block XX?

Accepted Answer

#### **Key Information:**
1. **Block and Row Structure:**  
   - Block XX contains houses in Columns A, B, and C.  
   - Block YY contains houses in Columns D, E, and F.  
   - Each block has houses in two rows (Row-1 and Row-2).

2. **Pricing Formula for Vacant Houses:**  
   Quoted price = (Base price) + $$ 5 	imes 	ext{Road adjacency value} $$ + $$ 3 	imes 	ext{Neighbour count} $$.  
   - Base price: Rs. 10 lakhs (without parking) or Rs. 12 lakhs (with parking).

3. **Known Conditions:**  
   - Maximum quoted price in Block XX: Rs. 24 lakhs.  
   - Minimum quoted price in Block YY: Rs. 15 lakhs (one such house is in Column-E).  
   - Row-1 has two occupied houses, one in each block.  
   - Both houses in Column-E are vacant.  
   - Each of Columns D and F has at least one occupied house.  
   - Only one house with parking space exists in Block YY.

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#### **Step-by-Step Deduction:**

1. **Occupied Houses in Row-1:**  
   - Row-1 has two occupied houses, one in each block.  
   - In Block XX, Row-1 must have one occupied house (either A1, B1, or C1).  
   - In Block YY, Row-1 must have one occupied house (either D1, E1, or F1).

2. **Vacant Houses in Column-E:**  
   - Both houses in Column-E (E1 and E2) are vacant.

3. **Maximum Quoted Price in Block XX (Rs. 24 lakhs):**  
   - To achieve this price:  
     $$ 	ext{Base price} = Rs. 12 $$ lakhs (with parking).  
     $$ 	ext{Road adjacency value} = 2 $$.  
     $$ 	ext{Neighbour count} = 2 $$.  
     Calculation: $$ 12 + (5 	imes 2) + (3 	imes 2) = 24 $$.  
     This suggests the house with maximum quoted price is C2.

4. **Occupied Houses in Columns D and F:**  
   - At least one house is occupied in each of Columns D and F.

5. **Vacant Houses in Block XX:**  
   - Based on the above deductions:  
     - A2, B2, and C2 are vacant houses in Block XX.

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#### **Conclusion:**
There are **3 vacant houses** in Block XX: A2, B2, and C2. Thus, the correct answer is **3**.

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