A Fun Fibonacci Puzzle

Here is a fun math trick! Simon and Neva have made a 8 x 8 cm square (with an area of 64 cm²) and cut it into four pieces, turning the square into a puzzle. Using the same four pieces, they built a 5 x 13 cm rectangle. But wait a minute! 5 x 13 equals 65, so the area of the rectangle is one cm² larger than that of the square!

They also made a similar puzzle using bigger pieces. A 13 x 13 = 169 cm² square turned into a 8 x 21 = 168 cm² rectangle! So now the area of the rectangle is one cm² smaller than that of the square! What’s gong on?

You have probably recognized the numbers in this trick: 5, 8, 13, 21… Those are Fibonacci numbers! Simon explains, that with Fibonacci numbers, the effect of the rectangle area being greater or smaller than the square area is alternating. Fibonacci have a converging ration to φ (Phi), but not φ. The pieces only look like they are golden ratio bigger/ smaller. In reality, there is a little gap between the pieces in the first rectangle and a little overlap in the second.

Simon has been inspired by Mathologer to build this.

the 8 x 8 square
the 5 x 13 rectangle
the 13 x 13 square
the 8 x 21 rectangle

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