( See also, The Triangles Puzzle …
)
A number of my Facebook friends have been posting a puzzle based on the image below.
How many squares can you count in the above image. I count 40. I’ve included several sequences of the puzzle with squares colored in red and yellow to illustrate the ones I see.
First, let’s just count the basic squares across and down.
That gives us 16.
Next, let’s count the little interior squares.
There are 8 of these, bringing us to 24.
There are 2 larger interior squares.
This brings the count to 26.
There are 4 corner squares.
The count is now 30.
The entire puzzle is a square.
Adding 1 to the count, we now have 31 squares.
There are two squares in the middle across the horizontal axis.
This brings the count to 33.
There are two middle squares across the vertical axis.
This brings the count to 35.
There’s 1 square, dead center.
The count is now 36.
The remainder of squares are comprised of nine of the smaller squares. See the four pics below.
This brings the count to 40. I don’t see any others.
my boyfriend thinks ur retarded because 23out of 40 of ur squares the inner squares cut into the larger ones causing them to not have 4 equal sides!!!
That’s an interesting point. If the puzzle had been to print a paper copy of the topmost image and cut it into the maximum number of squares with no lines in the interior, I would (mostly) agree with you. I think I’d end up with 16 squares … 8 of which would be the much smaller squares.
However, the puzzle asked how many squares I saw. I see 40 4-sided shapes that are equilateral and equiangular.
Your boyfriend doesn’t sound bright enough to call anyone retarded.
I see several more than you do. The hardest ones to see are the ones that kinda look like picture frames. Take a look at the 2×2 squares that have 3 mini’s in the center. think of those cut out and look at the thick frame around the mini’s. That is a square just thicker. If you can see that there are those two and another to be found when you look for the similar frame cut out by looking at the 4×4 square and cutting out the inner 2×2. There are also the picture frames of the 3×3′s with a single taken out.
This is my full list. From largest to smallest:
1 big one 4×4
1 picture frame 4×4 with the inner 2×2 cut out
4 3×3
4 picture frame 3×3′s with the inner 1×1 cut out of each.
9 2×2
2 cut out 2×2
18 1×1
8 minis
for a total of 47.
I have seen people who count up to 63. In the original picture there are faint little tiny tiny squares on the interior corners. I see those so faintly, and I also see faint outlines which would make extra squares almost throughout but 47 seems enough till I look with a better resolution computer or iPad.
These picture frame ones don’t count. The sides of a square are made out of lines, not shapes. If you’re counting the outside of the “picture frame,” it’s already been counted. If you’re counting the inside, that’s already heed counted as well. You cant make up lines and laws that don’t exist so you can pretend you know better than everyone else.
I am just counting the frame itself, as a white frame, not black lines at all. Like thick wall, thick square drawn in white.. On a different point do you see the tini tiny shadow lines? They seem equally distant to the outer frame that surrounds the largest divided square. I can not see them well enough, to count them. nor the shadow squares on the interior, but there they are none the less. Some of interior shadow look made up of many littler squares. My point is that the shadows do not need to be there. Precision drawing with pen does not cause shadows as this appears to be. Or even created computer drawings do not generate shadows, yet there they are. They are part of the drawing on the top of this page. They are found on the one that has circulated on Facebook, also. Is this picture/drawing a poorly scanned hand drawn drawing or is it a computer generated. I think we shall never know.
By definition in geometry, a line has no width. A square is a geometric figure made up of four lines of equal length and connected with four equal angles. What you are describing as “picture frame” squares are more properly defined as the area of a larger square less the area of a concentric smaller square contained within the larger one. As Luke has already pointed out, both the squares contained in this figure have already been counted.
The “shadow” squares you describe are an optical illusion created by the flicker of your display’s refresh rate and are not visible if you print the figure in high enough resolution.
A square is not defined by the thickness of its lines, which as someone else points out theoretically have no width at all. It is defined by its outermost edges. This picture, whilst it does clearly have lines with a width, is meant to represent a theoretical problem, therefore your picture frame lines don’t count.
i believe the replies have added a few more squares to the total! but hey, its hip to be square LOL
Depends how you count them. If you are counting the white blocks (uninterrupted by black lines), then it is 16 squares. But if you are using the black lines to define the border, then technically there is an infinite amount of squares – because if you zoom really close to the black borders so the width of the lines are the size of the computer screen, you can technically divide that up as many times as you like to define the square border – so there can be infinite amount of squares – but ill stick to my 16 square answer
Jared, this is obviously a theoretical problem and as such the lines, by definition, should be assumed to have no width at all. And the question is “how many squares can you count?” the definition of a square is four equal lines connected by four equal angles. There is no qualification that another line (or curve) can’t cross the square. Otherwise it would be impossible to construct a square with a compass and straight edge as we all learned to do in high school geometry. So the question is really “how many sets of four equal lines connected by four equal angles can you count in the figure?” Since all of the angles in the figure are right angles the question can be reduced to, “How many sets of four equal lines can you count?”
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