Recognizing and Counting Squares

There is a puzzle figure that has been circulating on a social web site for a while, with the question “How many squares do you see?”

There are at least three answers which might be considered correct, depending on what rules are assumed.

1. Only “flooding” is allowed: one colors each zone up to its boundaries, and counts the zones that are squares.
2. Composition is allowed: k*k regions composed of identical squares can be counted.
3. Any square boundary is allowed; no matter what is inside, how the zone within the boundary is divided, the squareness of the boundary is all that matters.

The first solution is to count all the squares with nothing else in them. In the puzzle figure, we count 16:

If we allow composition by merging k*k blocks of squares, we can add two more to our count. Notice, however, that the area between the first and last columns (squares 1-8) are not composed of blocks of squares after this merger because the two additional squares are offset.

This allowance will enable us to remove the minisquares that we merged to form the two offset squares, producing eight more squares immediately. It also results in a 4*4 grid, which has many more possible k*k compositions.

Deriving from the previous figure, with squares 17 and 18 centered in larger (2*2) squares, eliminate those inner zones to clear the two larger (not yet counted) squares:

Deriving from the first figure, with squares 1 to 16, remove (ignore) mini-squares 9 to 16, clearing squares 21 to 28 in columns two and three:

• The count is now at least 28, with several more to come through composition.

The blocks 19 and 20 already “discovered” are the first two of nine 2*2 squares. Here are the seven others:

Four corners
Central Horizontal
Center

• The count is up to 35.

Four 3*3 blocks can be composed from the 4*4 grid,

	Left	Right
Top
Bottom

• The count is now 39, with the outer box yet to count.