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This problem is even simpler if you notice that the rectangle made up of yellow + green + pink is just 3x the area of the orange rectangle, by definition.

Since these two rectangles share the same width, the combined rectangle must have 3x the height of the orange one, making the height of the yellow rectangle 9.

This gives a side length of 12 and an area of 144.

5 comments

If you notice some implicit relationship, you're already solving the puzzle. The most assistance a computer (or human) could provide was by accepting the original description. Every inference you add to the original problem is part of the solution.

This observation is pretty nice, though. Of course technically it is part of doing the puzzle, but it skirts doing most of the arithmetic.

Note that this argument does not depend on the fact that the blue rectangle also has equal area.

This argument thus teaches us even more, namely: the configuration of the four equal-area rectangles orange, yellow, green, pink has the special property that if you widen it to get a square, the extra piece you add also has equal area.

I spent some time this morning failing to solve this via an algebraic relationship between the orange and blue rectangles. I came home just now, opened my computer, and your insight above jumped out at me immediately. Brains are funny.

Very elegant! I like puzzles like these - the solutions provide a kind of satisfaction.

And good sources of such puzzles. I am hoping to get my school age kid interested in these.

By which definition? I think you're confusing deduction with definition. You could likewise say the area is 144 by definition.

I believe the parent was referring to line from the article which said "...square that gets partitioned into rectangles of equal area" - i.e. area of yellow, green, pink and orange is the same, so the area of the big rectangle containing all those four colours must be 4 times the height of orange rectangle (since their widths are the same).