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. 

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? [This question was asked as TITA]

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is C

The correct answer is 3 houses.

Answer 2

Given:
Some houses are already taken, but others are empty and available for sale. The base price for an empty house is either Rs. 10 lakhs if it doesn't have a parking spot and Rs. 12 lakhs if it does have a parking lot. 
So the price of a vacant house is calculated as = Base price+5(road adjacency value)+3(neighbor count). We know the most expensive house in Block XX costs Rs. 24 lakhs.

Now calculate for the maximum price of a house in block XX.
1. House with parking lot;
road adjacency value = a, neighbor count = b
12+5a+3b=24
5a+3b=12
The equation only works if a equals 0 and b equals 4. However, b can't be 4 because the most neighbors a house can have is 3.
So, it is invalid.

2. House without parking lot:
10+5a+3b=24
5a+3b=14 , (a,b)=(1,3)
So, the house must have 3 neighbors and 1 road connected to it. Therefore, the only possible case is B2. Thus, the neighboring houses of B2, which are B1, A2, and C2, are also occupied. Since we know that Row 1 has two occupied houses, with one in each block, and B1 is already occupied, it implies that A1 and C1 are vacant.
That means 3 houses are occupied in Block XX.

So, the correct option is (C): 3.