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My professor gave us this puzzle at the beginning of the semester but would not reveal the answer:

a two-by-two tiled square, which then has 1-by-2 or 2-by-1 rectangles along each edge, and in between just two of the rectangles is a 1-by-1 square filling up a "corner"

There is only one question: How many boxes do you see? We were not given any other information.

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    $\begingroup$ Can it be assumed that boxes is synonym to "rectangles" ? Then Rubio's answer is surely correct. But in case your teacher is a trickster, you may also consider nets of (3D) boxes, i.e. "cutting" (some) of the lines and folding the paper on some other lines... $\endgroup$
    – BmyGuest
    Commented May 21, 2017 at 12:58
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    $\begingroup$ The main problem is that the word "boxes" is ambiguous. I read it as "squares" in which case the answer is 11; Rubio's answer below interprets it as "rectangles"; and BmyGuest suggests nets of 3-dimensional shapes. Without a clear definition there can be no definitive answer. Which may well be the point your professor was trying to make! $\endgroup$
    – IanF1
    Commented May 21, 2017 at 13:59
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    $\begingroup$ @ffao raised similar doubts earlier, in a (now deleted) comment on my answer. My response (also now deleted) was, in essence: Then the "puzzle" isn't really one. In any case, a box can be a simple 2D rectangle, as found in the common usage phrase "Check the box". Your point, though, about the ambiguity perhaps being the point, is well-made. $\endgroup$
    – Rubio
    Commented May 21, 2017 at 15:13
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    $\begingroup$ It is well possible that the teacher wants to see how many students will think of a rectangle and how many will think of a carboard box and argue about who is correct. And whether there is anyone to respond with: "Define 'box'.". Ironically a "think outside of the box" question. $\endgroup$
    – Florian F
    Commented Oct 25, 2021 at 19:31

3 Answers 3

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Number of boxes is ...

27

They break down thusly:

4x2 = 2
3x3 = 1
3x2 = 4
3x1 = 2
2x2 = 5
2x1 = 8
1x1 = 5
Total = 27.

Drawing below - it gets a little hard to see in places but you should be able to trace the colors ...

enter image description here

There is no place remaining where lines in the original diagram exist, where another box of each indicated size could still be drawn. That should suffice to show that all possible boxes of each size are counted. It should be clear on inspection as well that only the box sizes listed are possible.

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Here is a trick how to solve this kind of problems.

You simply start with any corner and see how many boxes you can do from that corner.
Then you discard the corner and repeat witth the rest.
See below how easy the counting becomes. If you choose the corners well you can do it in your head.

Illustration how to count boxes in shrinking patterns

And indeed, the answer is ...

27

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There are two boxes. First one is carton box with opened flaps, like this
Second one is box depicted as a small square in upper right corner.

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