Information About Neighbors

Logic Level 1

Some cells in a $3 \times 3$ grid are shaded.

As shown below, in each cell, we record the number of neighboring cells that are shaded, inclusive of diagonals but exclusive of the cell itself.

How many of the 9 cells are shaded?

1 2 3 4 5 6 7 8

2 solutions

Chung Kevin
Nov 10, 2016

Because the center cell is 3, this tells us that the number of shaded cells is either 3 or 4 (depending on whether the center cell is shaded or not).

Because the left center cell is 4, this tells us that the number of shaded cells is at least 4.

Hence, the number of shaded cells is exactly 4.

It remains to find the configuration that satisfies the conditions.
Since the left center cell is 4, we know that these 4 cells must be neighbours of it. Hence none of the right column is shaded.
Since the right center cell is 3, this tells us that the entire center column is shaded.
Since the bottom center cell is 1, this tells us that in the left column, the center and lower cells are not shaded.
Hence, the top left cell has to be shaded.

We can verify that this configuration leads to the numbers in the display.

Wow, the first part of the solution is ingenious! I never even thought of using the center cell to figure out how many cells were shaded (though it's obvious now)

Alex Li - 4 years, 7 months ago

Thanks. I didn't realize that initially. As the question setter, I just created the grid and then set the answer.

It was only when trying to understand @Sharky Kesa 's solution, that I chanced upon this observation :)

Chung Kevin - 4 years, 7 months ago

Sharky Kesa
Nov 9, 2016

In the following solution, corner squares refer to the squares at the corner of the grid, centre square refers to the square in the middle, and the other squares are edge squares.

Notice that all corner are squares have 2 inside them. Let's assume the centre square is not shaded. Then we must have that all the edge squares are shaded (since each corner square only is adjacent to 3 squares, one of which we have assumed to be not shaded). However, this implies that the centre square must contain a number that is at least 4, which is not true. Therefore, the centre square is shaded.

Since the bottom edge square has 1 inside it, and the centre square is shaded, we must have that the other adjacent squares of 1 must not be shaded. However, this implies that the left and right edge squares are blank. Since the bottom left and right corner squares contain 2, and each of these corner squares are adjacent to 3 squares, 1 of which is known to be blank, we must have the bottom edge square to be shaded.

Notice that the left edge square is adjacent to 5 squares, one of which is blank (from previously). Thus, the top left corner square and the top edge square are shaded.

We now check and notice all numbers satisfy this shading, so there are 4 shaded squares.

I liked that you explained how to determine the shaded configuration.

I have a faster way to explain why 4 squares are shaded, but it doesn't provide details for how to determine which squares these are.

Chung Kevin - 4 years, 7 months ago

Information About Neighbors

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