# How many squares actually ARE in this picture? Is this a trick question with no right answer?

This is one of those popular pictures on sites like Facebook. I always see a huge variation of answers such as $8, 9, 16, 17, 24, 28, 30, 40, 41, 52,$ etc., yet I've never seen a definitive answer on any website. I counted $16$ whole squares ($4$ going down on left side, $4$ going down on right side, $8$ small squares in the centre). With this method the answer could also be $17$ if you want to count the entire shape as one big square.

I can also see the answer being $24$ if you count the smaller squares separately, as if they were floating above the main grid instead of intersecting with it.

Is there a right answer, or is it possible that there is no 'right' answer and it just depends on your counting method?

• For this to be a well-posed mathematical question you need to define precisely what you mean by the term "a square in the picture". Otherwise it is not mathematics. Mar 1 '14 at 23:28
• Not to take anything away from this puzzle; but $92%$ sounds made up, it doesn't help that it says "UR" instead of "Your" either. Mar 1 '14 at 23:29
• @user1092719 You need to say precisely what it means for a square to be in the picture. Mar 1 '14 at 23:42
• This is a classic case of "infectious meme pretending to be a puzzle." The 92% figure IS made up to try to make folks feel motivated to (a) play with it and (b) inflict it on others. As such, you can think of it as a wetware-mediated network worm. ANYTHING THAT EXPLICITLY SAYS "SHARE ME" SHOULD NOT BE SHARED. PERIOD. Mar 2 '14 at 4:05
• I find it amusing that the OP, who obviously studied the puzzle in some depth, did fail the simple test. Mar 3 '14 at 7:38

Thanks to Ross Millikan for helping me to find the last squares. Here is an GIF animation (344 kb) showing 40 squares from biggest to smallest: – qwr
Mar 2 '14 at 2:49
• How did you make this gif? (unrelated to math I know but...) I've been always curious to learn how to make gifs! Mar 2 '14 at 4:46
• @frogeyedpeas: I used the GIMP software, there is one layer for each red square(s), displayed every second. This is a good tool but the interface is ... weird.
– A.L
Mar 2 '14 at 5:01
• But the numeral 1 is also a square, making 41. :D Mar 3 '14 at 5:50
• @JonathanVanMatre: What about the numeral 9, and the word "square"? Mar 3 '14 at 8:54

Ignoring the two strange squares in the middle, there are $16\ 1 \times 1$ squares, $9\ 2 \times 2,\ 4\ 3 \times 3,$ and $1\ 4 \times 4$ for a total of $30$ Each of the two strange squares has four small and one large square, five each, ten total.

The grand total is then $40$ squares.

• Of course, what else. I wonder how this couldn't be a definite answer, a square is a square. Mar 2 '14 at 2:20
• What about the $\sqrt{5} \times \sqrt{5}$ square created by the points $(1,0), (0, 2), (2, 3), (3, 1)$? (I'm looking at the larger squares only and considering the intersections as lattice points.) Am I missing something? Mar 2 '14 at 2:58
• there are no lines formed there so those i don't think are counted Mar 2 '14 at 4:48
• +1 Now try counting the squares in the background image of this SE site. Mar 2 '14 at 23:04

To find total squares of $n \times n$ squares, the total no. of squares in it is given by: $$n^2+(n-1)^2+(n-2)^2\ldots1^2$$ In the box above of $4 \times 4$, the total no. of squares are $$4^2+3^2+2^2+1^2=30$$

Additionally, there are $2$ small squares in the middle along with 4 even smaller squares each. They total to $10$ squares.

$\therefore$ The total squares is $30+10=40$

• Total squares of size k units in an nxn unit grid is (n+1-k)^2 ;k≤n.
– ARi
Jan 28 '16 at 13:34

Generally speaking, most puzzles that are written are intended to be answered using the minimum number of assumptions possible to arrive at the correct solution. William of Ockham first stated this principle as a guideline, commonly known as Occam's Razor. The exceptions to this principle are those that are trick questions, which are usually worded in a way that suggest that the immediately obvious answer isn't correct, or appear as an optical illusion of some sort.

Even questions that are trick questions generally have one correct answer; we can arrive at the suitable solution by way of Occam's Razor as well. The types of problems that we are given in school textbooks, for example, all abide by Occam's Razor; the intent is to prove critical thinking and analysis skills, not "thinking outside the box" skills (a sad truth, really, as we'd do much better as a society if the latter were true).

This puzzle, and others along the same vein, are generally meant to invoke critical thinking; find the correct answer with the minimum number of assumptions necessary to solve the problem. The question is asking is to find the number of squares, a square being defined as a closed object with four sides, with each side meeting at 90 degree angles, and each side being the same length.

For example, this Stock Photo shows a square window; it's mentioned as such in the description. The fact that it contains four smaller squares is irrelevant here; most people will naturally state that this is a square window (note, I didn't actually measure it, so it may not be perfectly square, but it appears as such, and we'll take it at face value for purposes of this discussion). This window actually has five squares; the frame plus the four smaller squares from the lattice that reinforces the glass.

Since the question doesn't state that any of the squares are floating or are in any other way special, and since the question also doesn't appear to be an optical illusion, we can apply Occam's Razor to come up with the obvious solution.

• Each angle in the image appears to be 90 degrees.
• Each line in the image appears to be unbroken, and can thus participate as a side in a closed object that has four corners connected at 90 degree angles and may either be a rectangle or a square.
• The lines for the offset squares in the middle of the grid appear to be the same length as any of the other 1x1 squares in this grid, and their vertices appear to land exactly in the middle of the squares in the grid.

Assuming that this question isn't an optical illusion, which it clearly does not appear to be, the obvious solution to the question is 40. We arrive at this answer by enumerating all of the squares of various sizes and adding them together.

• 1 square for the outside border of the square grid $$(4\times 4)$$
• 4 squares that are 9/16th the entire size of the square grid $$(3\times 3)$$
• 9 squares that are 1/4th of the entire size of the square grid $$(2\times 2)$$
• 18 squares that are 1/16th of the entire size of the square grid $$(1\times 1)$$; the 16 aligned in a grid plus the 2 that are overlaying the grid.
• 8 squares that are 1/64th of the entire size of the square grid $$(0.5\times 0.5)$$, caused by the offset squares being bisected by the grid lines.
• Am I the only one that still thinks invoking Occam's razor here is a misuse of Occam's razor which is a general 'rule of thumb'-type principal applied to explanations of phenomena, as opposed to calculations? This answers appears to do nothing more than obfuscate the fairly simple problem of finding all embedded geometric squares in the image, which is a well defined problem. Mar 2 '14 at 19:04

Taking the side of each small square as ' one ' ( excluding the two squares in the middle ) ; the number of squares = 16 (1x1 squares)+9 (2x2 squares)+4(3x3 squares) +1(4x4 square) = 30 ................................The two square in the middle have small sqares , with each side having a value of 1/2 . So the total no. of squares of these two squares = 2x [4(1/2x1/2 squares)+1 (1x1 square)] =10 .......Thus the total number of squares , are 30+10 =40