Question 169
Camila and Lara each have a $4 \times 4$ board, initially both blank. With these boards they play a game as follows:
- Camila, hidden from Lara, paints some squares on her board black;
- Still on her board, Camila writes in each house the number of neighboring houses that are painted black (two different houses are neighbors if they have a side or a vertex in common);
- Camila copies the numbers written on her board onto Lara's board;
- Lara must guess, based on the numbers written on her board, how many black squares there are on Camila's board.
For example, if Camila painted her board like this
then she will place the numbers on Lara's board as follows:
| 1 | 1 | 3 | 1 |
| :--- | :--- | :--- | :--- |
| 2 | 3 | 2 | 2 |
| 1 | 3 | 1 | 1 |
| 1 | 2 | 0 | 0 |
If Lara's board has the numbers
| 1 | 2 | 1 | 1 |
| :--- | :--- | :--- | :--- |
| 0 | 2 | 1 | 2 |
| 2 | 3 | 3 | 1 |
| 1 | 0 | 2 | 1 |
How many houses did Camila paint?
A) 3
B) 4
C) 5
D) 6
E) 7
- Camila, hidden from Lara, paints some squares on her board black;
- Still on her board, Camila writes in each house the number of neighboring houses that are painted black (two different houses are neighbors if they have a side or a vertex in common);
- Camila copies the numbers written on her board onto Lara's board;
- Lara must guess, based on the numbers written on her board, how many black squares there are on Camila's board.
For example, if Camila painted her board like this
then she will place the numbers on Lara's board as follows:
| 1 | 1 | 3 | 1 |
| :--- | :--- | :--- | :--- |
| 2 | 3 | 2 | 2 |
| 1 | 3 | 1 | 1 |
| 1 | 2 | 0 | 0 |
If Lara's board has the numbers
| 1 | 2 | 1 | 1 |
| :--- | :--- | :--- | :--- |
| 0 | 2 | 1 | 2 |
| 2 | 3 | 3 | 1 |
| 1 | 0 | 2 | 1 |
How many houses did Camila paint?
A) 3
B) 4
C) 5
D) 6
E) 7