# How many boxes do you see?

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

There is only one question: How many boxes do you see? We were not given any other information and it has been bothering me for the past few months.

Help will be appreciated. Thank you!

• 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... – BmyGuest May 21 '17 at 12:58
• 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! – IanF1 May 21 '17 at 13:59
• @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. – Rubio May 21 '17 at 15:13

### 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 ...

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.