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.

Arbitrary Square Boundaries

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.

Eliminating (Ignoring) Non-Corner Inner Zones

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:

Eliminating (Ignoring) Corner Inner Zones

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.

2 * 2 Blocks

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

Four corners Corner 2x2s
Central Horizontal Horizontal 2x2s
Center Central 2x2
  • The count is up to 35.

3 * 3 Blocks

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

Left Right
Top Top left 3x3 Top right 3x3
Bottom Bottom left 3x3 Bottom right 3x3
  • The count is now 39, with the outer box yet to count.

4 * 4 Block

Outer square

Final Tally

Rule Contribution Running Tally
Flooding 16 16
Flooding, composition +2 18
Non-block merging: squares in corners +8 26
Non-block merging: inner contained zones (result 2*2) +2 28
Further 2*2 compositions +7 35
3*3 compositions +4 39
4*4 composition +1 40
square_counting.txt · Dernière modification: 2012/09/01 16:37 par suitable
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