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### Sid Kemp says

I see 80. That includes all squares, which are a special case of rectangles, but clearly a type of rectangle. It also includes the entire figure itself, and the larger rectangles in the diagram which are made up of smaller squares and/or rectangles. I found them all by first counting the 21 smallest squares or rectangles, then returning to each upper-left corner and finding all larger rectangles starting from that upper left corner. So, for example, there are 8 rectangles that have an upper left corner matching the upper left corner of the entire figure. The smallest is green, plus the white lower right corner. More grown downwards, sideways, and both. The largest is the whole figure.

### LetitiaFT says

I counted 32 (assuming squares could be counted as part of rectangles but not as rectangles themselves)

Squares are a specific type of rectangle, so they also count as rectangles. You can adjust your answer if you'd like. :)

### Aficionada says

Calculus-geometry, I had a specific question about this puzzle that I wanted to ask you privately, but I did not find an e-mail link on your profile. Is there some other way I could contact you? If not, I'll ask the question here. :) Thanks!

Incidentally, I believe I found 108 rectangles. But -- I have counted several times and organized the answer several times, and I'm not absolutely sure that 108 was my most authoritative result.

You can ask here. I removed the email link because of spamming :/

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### TR Smith says

So the answer is 106, which Aficionada was the first to get. The image above shows a systematic way to count all the rectangles in stages by looking at three sets of lines, similar to how Fitnezz Jim counted them.

If you look at the top rectangle in the figure, before any lines are added to the basic 3x3 grid, there are 36 rectangles.

If you look at the middle rectangle where I have added additional lines in blue, you get 35 new rectangles inside the blue border, and 8 new rectangles exterior to the blue border, for a total of 43 additional rectangles.

If you look at the bottom rectanle where I have added the final lines in red, you get 17 new rectangles inside the red border and 10 new rectangles outside the border, for a total of 27 additional rectangles.

Adding 36+43+27 you get 106 rectangles in the entire figure. Thanks to everyone who took the time to solve this puzzle. :)

Wow. Sure enough it is 106, or at least 106 are plain as day now. I'll update my Hub to add the extra six when I figure out which are the six I missed,

### TimHL says

For those who had trouble counting, try the following method. At every intersection of lines, place a dot in the corner that could be the top-left of a rectangle. There are 22 of them. Now place an x in the corners that could be the lower-right of a rectangle, there are 23. For each dot, find all the x's that make a complete rectangle with the dotted corner, and write the number down next to the dot. In this puzzle the top left corner of the figure pairs with 10 x's that make complete rectangles.

With so many lines, even counting with this method can be confusing. To help with that, find the x's by using the current top line as a guide. At each vertical line that intersects the top line, proceed down and count each x you find, but only if the horizontal directly below the x extends all the way to left edge to complete the rectangle.

The counts for each top-left corner are below, as read from left to right and grouped by the horizontal line they share:

10, 8, 3

10, 7, 3

10, 8, 7, 4, 1

5, 4, 3

4, 4, 6, 4, 2

2, 1, 0

Summing each row and adding them together: 21 + 20 + 30 + 12 + 20 + 3 = 106

### Counting those Pesky Rectangles.

FitnezzJim, I will go back to your Hub and see if I can understand your labeling and counting system now. I did finish my video and it is now live on YouTube with the name "How Many Rectangles"? Thanks again to calculus-geometry. I love this puzzle!

### FitnezzJim says

I must be counting the wrong way, I got ninety-three so far and am still not sure I got them all.

### naturalife says

This is good and different from the normal questions asked on HP. It got me confused to only 17! Hahaha. I will follow your questions and probably get it right the next time.