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Counting those Pesky Rectangles.
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.
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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.