Counting those Pesky Rectangles.

Puzzle Figure

A puzzle from Hubber Calculus-Geometry.
A puzzle from Hubber Calculus-Geometry. | Source


Writer Calculus-Geometry posed the question: ‘How many rectangles are contained in this figure?’ A quick examination of the responses indicates those who answered have different counts for these rectangles. This note proposes an answer to the question, and describes a method to use to label and count the number of rectangles that appear in the figure.

First, note that you can think of this figure as broken into three parts. There is the outer rectangle, there is an inner rectangle, and there is a third rectangle (it looks like a square) in the lower left hand corner of the outer rectangle. For convenience, label each of these rectangles with Roman Numerals, I, II, and III.

Second, consider any row of each rectangle labeled with a Roman numeral. Assign labels to each of these rows. The individual rows in the figure can be labeled I.1, I.2, I.3, II.1, II.2, II.3, III.1, and III.2. Denote rows combined from other rows similar to I.1&2, I.2&3, and I.1&2&3.

Third, note that each row in the figure is separated by vertical lines into individual rectangles. For example, the upper row of the outermost rectangle (rectangle I) can be labeled with the individual rectangles I.1.A, I.II.B, and I.II.C.

Fourth, note that the total number of rectangles contained in a row relates to the total number of rectangles for the row through a function. For convenience, label the function RC, dependent upon the number of distinct individual rectangles N, as RC(N). Then note that RC(1)=1, RC(2)=3, RC(3)=6, RC(4)=10, RC(5)=15, RC(6)=21, and so on. You can verify this by construction. Draw a rectangle, add vertical lines to divide the rectangle, count the total number of rectangles, then write down the value of the function. There is also a nice summation notation for this function, but writing that formula in this forum is beyond my writing skills.

Note the following:

  • RC for row I.1 is 6.
  • RC for row I.2 is 15
  • RC for row I.3 is 10
  • RC for row I.1&2 is 6
  • RC for row I.2&3 is 6
  • RC for row I.1&2&3 is 6.
  • RC for row II.1 is 6
  • RC for row II.2 has already been counted as part of I.2.
  • RC for row II.3 is 10.
  • RC for row II.1&2 is 6
  • RC for row II.2&3 is 6
  • RC for row II.1&2&3 is 6
  • RC for row III.1 is 6
  • RC for row III.2 has already been counted as part of I.3.
  • RC for row III1&2 is 3

Finally, the overlap of box II and III creates several rectangles that are not yet counted by this system. The row created by removing III from II, (call it I.2.i) creates three rectangles not yet counted. Another two are contained in the first column of rectangle II, with most counted already as part of the RC’s for I, II, and III. Similarly, there are two new rectangles from the second column of rectangle III. The row created by removing III from II, (call it I.2.i) creates two rectangles not yet counted. One more rectangle is formed in the upper right hand corner of rectangle III.

All in all, I counted 100 rectangles in this puzzle. If you see more, please refer to the system above to help describe where they are located.

I'd also like to thank writer Calculus-Geometry for this puzzle. I started dreaming about it because I went to sleep unsure of the number of rectangles in my first count, then woke up and wrote this article. It took three updates to this article to get to the final number. It's not quite as challenging as Krytpos, but it is one of the few that finagled its way into my sleep time.

Link to "The Answer"

POSTSCRIPT: There are more than 100

Hubbers Aficionada and calculus-geometry have both counted 106 rectangles. Calculus-geometry indicates 106 in an answer to the original question.

Aficionada calls out each one in a video (at right). The video highlights each and every rectangle found so far, and is complete.

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Comments 11 comments

Pete Maida profile image

Pete Maida 4 years ago

WOW! I wasn't even close. I was looking at about 22 rectangles.

FitnezzJim profile image

FitnezzJim 4 years ago from Fredericksburg, Virginia Author

Thanks. I know there is at least 100. The fun part was figuring out a methodical way to count them.

Aficionada profile image

Aficionada 4 years ago from Indiana, USA

I actually did find a few more, but I'm not sure I know how to describe them by your system. I have made them into a video, which I intend to post in the next day or so.

Aficionada profile image

Aficionada 4 years ago from Indiana, USA

I like your systematic work in naming the rectangles, but I tend to get bogged down in the system when the names have three component parts.

It's also sort of confusing to me to try to keep up with the count by individual rectangles. For me, it is easier to count by considering all the rectangles of one large row at a time, followed by overlapping rows, then the large columns and overlapping columns, and finally the combinations of the largest components. My final total (shown in my YouTube video) is 106.

calculus-geometry profile image

calculus-geometry 4 years ago from Germany

I like your method of counting them. I used a similar systematic approach that groups them according to region and boundary line. When first I made puzzle I didn't realize just how many there were. And then when I sat down and solved it I wondered if maybe I had made it too hard. You and Aficionada got a good handle on it though.

Aficionada profile image

Aficionada 4 years ago from Indiana, USA

On the question page, I saw calculus-geometry's pictures showing the three large rectangles. Seeing them made it easier for me to count the smaller rectangles in each section. I am working on a second video counting the rectangles in those groupings and in this sequence, and I will link to this Hub when I post it on YouTube.

FitnezzJim profile image

FitnezzJim 4 years ago from Fredericksburg, Virginia Author

I looked at both the video and at calculus-geomtetry's pictures, and sure enough it looks like one hundred and six. I'll update this Hub when I can figure out which are the six that I missed. Since my approach was 'by rows', i suspect I overlooked rectangle created in the columns. In any case, I'll add a link to your video.

Aficionada profile image

Aficionada 4 years ago from Indiana, USA

I finally posted the video (on YouTube) showing how to count by this method. There was also a slight goof in my first video, which a viewer pointed out to me, and I have now posted a corrected version of the first video.

Thanks for embedding it here - feel free to change to the corrected version and let me know if/when you do. I plan to remove the original one with the mistake after a few days. Or, you could embed the newer one that shows the counting by your method.

FitnezzJim profile image

FitnezzJim 4 years ago from Fredericksburg, Virginia Author


Thanks for the heads up, I modified the link to point to your new corrected version.

Eiddwen profile image

Eiddwen 4 years ago from Wales

Very interesting and I am returning to this one later when I have more time. I am intrigued!!!

Take care and enjoy your day.


FitnezzJim profile image

FitnezzJim 4 years ago from Fredericksburg, Virginia Author

Thank you for stopping by, Eiddwen.

Thank hubber Calculus-Geometry for the interesting problem, and hubber Aficionada for the answer.

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