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

Let's deduce the vacant houses in Block XX step-by-step.
1. The houses are divided into two blocks - Block XX (Columns A, B, C) and Block YY (Columns D, E, F).
2. Block XX: Maximum quoted price = ₹24 lakhs.
3. Block YY: Minimum quoted price = ₹15 lakhs with presence in Column-E, and only Column-E houses are vacant.
4. Row-1 has two occupied houses, one in each block.
5. Column-D and Column-F each have at least one occupied house, indicating the presence of at least one occupied house in Block YY by default.
6. Let's denote vacant houses with base price without parking (₹10 lakhs) and with parking if specified (₹12 lakhs).

Column	House	Occupied
A	A1, A2	No occupancy info
B	B1, B2	No occupancy info
C	C1, C2	No occupancy info
D	D1, D2	At least one
E	E1, E2	Both vacant
F	F1, F2	At least one

7. Analyze potential vacant houses in Block XX satisfying the price constraints. Assume maximum road adjacency and neighbor counts to find highest possible quotes in Block XX.
8. If Block XX houses have a maximum occupancy and prices match, infer the occupancy of others as per vacancy indications: C1 or C2 may maximize the adjacency, leading to those vacated.
9. Thus, vacant houses in Block XX: A1, B1, C1
Therefore, Block XX contains 3 vacant 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 spot.
So, the price of a vacant house is calculated as: 
Price = Base price + 5 × (road adjacency value) + 3 × (neighbor count).  

Now calculate for the maximum price of a house in Block XX.

Case 1: House with parking lot
Let road adjacency value be $a$, and neighbor count be $b$. 
Then, 
$12 + 5a + 3b = 24$
$\Rightarrow 5a + 3b = 12$
Try integer solutions:
If $a = 0$, then $b = 4$, which is invalid since the maximum number of neighbors is 3.
So, this case is not possible.

Case 2: House without parking lot
$10 + 5a + 3b = 24$
$\Rightarrow 5a + 3b = 14$
Try integer values: if $(a, b) = (1, 3)$, it satisfies the equation.
So the house must have 3 neighbors and 1 road adjacent.
The only house in Block XX satisfying this is B2.
Its neighbors: B1, A2, and C2 must be occupied.
Now, we know that Row 1 has two occupied houses, one in each block.
Since B1 is occupied, the other occupied house in Row 1 must be in Block YY.
Hence, A1 and C1 must be vacant.

So, the total occupied houses in Block XX are: B1, A2, C2 → 3 houses.

Correct option: (C) 3