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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.

Squares image

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?

daOnlyBG
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user1092719
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    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. – Bill Dubuque Mar 01 '14 at 23:28
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    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. – Asaf Karagila Mar 01 '14 at 23:29
  • @AsafKaragila it probably is, I didn't make the image I'm just providing the original in its entirety. – user1092719 Mar 01 '14 at 23:35
  • @BillDubuque I don't know how the original source of the image defines what a square is, but I define squares as shapes with 4 sides all of equal length. Is that what you mean? (Apologies if it's a bad question/I'm not being clear enough). – user1092719 Mar 01 '14 at 23:40
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    @user1092719 You need to say precisely what it means for a square to be in the picture. – Bill Dubuque Mar 01 '14 at 23:42
  • Right now, I would define it as all the shapes in the picture that don't have any intersections. However I'm not particularly mathematically inclined, so I'm interested to see answers that are more well explained than you'd see on places like Facebook. – user1092719 Mar 01 '14 at 23:55
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    @user1092719: That would include rectangles. I'd say all shapes in the figure which can be formed by choosing line segments which are part of the picture and whose endpoints are chosen from the various points of intersection that are squares. That's what immediately comes to mind when I see the picture and the question. – The_Sympathizer Mar 02 '14 at 03:15
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    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. – keshlam Mar 02 '14 at 04:05
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    related: http://math.stackexchange.com/a/623307/105069 – Mohammad Mar 02 '14 at 19:10
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    I find it amusing that the OP, who obviously studied the puzzle in some depth, did fail the simple test. – Mr Lister Mar 03 '14 at 07:38
  • @AsafKaragila, How can someone tell from the number 92 that is is made up? What percentage would be more convincing? – jwg Mar 03 '14 at 09:29
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    I'm surprised OP doesn't include 42 as a possible answer. The answer is always 42. – ike Mar 03 '14 at 15:12
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    To clarify, there are 40 enumerated in the answers, and an additional 2 for the dots underneath the question and exclamation marks. (If you include the dots above the "i"s , your total comes to 47). – ike Mar 03 '14 at 15:15
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    In fact, there are infinite squares. The problem is they are white over a white background. – jinawee Mar 03 '14 at 15:17
  • I will add a link to a nice animation: http://media-geeks.com/culture/culture-special-feature/how-many-squares-indeed/ – Martin Sleziak Jul 02 '15 at 10:59
  • Well it's 40, isn't it? Apart from dealing with the smaller squares and the two "unit" squares they produce, everything else is standard. (Also this is the most viewed Math StackExchange question :D) – Prometheus Nov 25 '21 at 08:47
  • First question to 700k views... – mathlander Dec 17 '22 at 19:43

5 Answers5

140

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:

40 squares

A.L
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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.

Ross Millikan
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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$

5

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.

Using Occam's Razor, we can state the following "truths" about this image:

  • 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.
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    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. – Dan Rust Mar 02 '14 at 19:04
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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

b.sahu
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