This pretty proof of the Pythagorean theorem is attributed to Leonardo da Vinci. Draw a right triangle and construct a square on each side, and make a copy of the original triangle and add it to the bottom of the hypotenuse square as shown. Now the shaded hexagon in the first figure can be rotated 90 degrees clockwise around the indicated point to occupy the position shown in the second figure. The orange and green quadrilaterals in the second figure are seen to be congruent to those in the first figure: The three shortest sides of the orange quadrilateral in the second figure correspond to their counterparts in the first, and the angles between them are assembled from the same constituents. The same is true of the green quadrilaterals. In each figure the shaded hexagon contains two instances of the original right triangle; remove these and we can see that the two squares in the first figure equal the large square in the second figure, proving Pythagoras.

10/10/2021 UPDATE: A number of readers point out that only the orange quadrilateral here can properly be said to turn; in the second diagram the green quadrilateral has been reflected as well. (Thanks, Mark and Bill.)

In a December 1985 letter to the Mathematical Gazette, Middlesex Polytechnic mathematician Ivor Grattan-Guinness writes that Astronomer Royal George Biddell Airy “would sometimes go around the Observatory, and on finding an empty box, insert a piece of paper saying ‘Empty box’ and thereby falsify its description! This last achievement deserves, in my proposal, the name of ‘Airy’s paradox’.”

Which of the two shaded areas is larger, the central disc or the outer ring?

Surprisingly, they’re equal. Each of the concentric circles has a radius 1 unit larger than the last. So the area of the central disc is π × 3² square units, and the area of the outer ring is π × 5² – π × 4² = π × 3² square units. So the two areas are the same.

A brewery stored its beer in a cellar some distance from the bottling plant. The cellar was cooled by pipes that circulated a saline solution from a central cooling unit. The main pipe that connected this cooling unit and the cellar happened to pass near the cellar of a retailer.

The brewery’s owner eventually discovered that the retailer was using the saline solution to cool his own cellar. He sued the retailer for theft, but the judge ruled, “In accordance with Article 242 of the Criminal Code, theft is the unlawful appropriation of commodities belonging to another party. In the present case no theft has been committed, since the saline solution was not misappropriated; rather, it was returned in its entirety to the brewery’s main pipe.”

The brewery owner appealed the case, arguing, “The issue is not the theft of saline solution but the theft of energy. If the saline solution is used to cool the defendant’s cellar in addition to my own, I have to pay more for electricity to operate the central cooling unit.”

The court of appeals ruled: “The saline solution acquires heat from the retailer’s cellar; therefore, energy belonging to the brewery is not being stolen. On the contrary, the brewery is receiving gratuitous energy from the retailer.”

This story appeared in a German scientific monograph, “Questions of Thermodynamical Analysis,” by P. Grassman. In propounding it in May 1990, Quantum added, “We all agree the judge was wrong, but not everyone can correctly explain his error. Can you?”