訓練數學感 32 ─ 面積怎麼變大了？ (Banach-Tarski paradox)

http://4rdp.blogspot.com/2014/09/32-banach-tarski-paradox.html?m=0

這是有名的數學悖論題目，原來 8 x 8 正方面積切成四份後重新組合排列，竟變成 5 x 13 的矩形面積！聰明的讀者應該可以看出問題在哪裡。

下面是這題目的英文原文：
This puzzle is based on the Banach-Tarski paradox.
The story of this puzzle involves buying and selling gold.
The gold is purchased in the shape of a square, eight units (e.g. inches) on a side.
The area of square is 64 square units (inches).
The square is cut into four pieces as shown in illustration on the right.
The four pieces are the re-arranged as shown on the left to form a 65 square-unit rectangle.
The 65 square-unit rectangle of gold is re-sold for the same price per unit area, and the profit is 1 square unit (inch) of gold.
How do you explain this?

同樣地，我在想有沒有類似的整數解切割重組，使得正方形面積變小，或是其它變大的例子？(進階題)
上面的正方形，以何種比例切割，重組後面積不變？(加分題)