Hello fellow café denizens.

I thought I would share a mathemagic puzzle I invented for my pi-day shows. It is a way to get students to think about how math can deceive them if they aren't careful. As my father always said, "Figures don't lie, but liars can figure!" It works well for advanced middle school as well as high school and college level students. I show them the swindle as an extra in the discussion period after the main show and encourage teachers to assign explaining it for extra credit (for middle and high school students). It is a lot of fun seeing their puzzled faces when you interactively walk them through it on the board.

Please let me know if you find any errors as I quickly typed it up from my notes.

Have a happy pi-day!


The Sibling Pi Swindle by Frank Findley (copyright 2023)

Two siblings inherited the family land on which each had built their homes. The plot was 2 units wide by pi units long for a total area of 2*pi sq. units or about 6.28 sq. units.



The siblings didn't always get along so they decided to divide the land between them. One of them was a dastardly mathematician who plotted to swindle the other. That sibling bought some fencing to encircle their house saying that since the radius of the circle was 1 unit, the area enclosed was pi square units (A=pi*r^2 = pi ~3.14), perfectly half of the lot.



The other sibling, not trusting the first, examined the proposal intently. They noted that using a circle cut off access to the southernmost corners. This divided the land in an unfair manner as there was no access to those pieces, effectively reducing that sibling's share!



So, the shortchanged sibling argued that using a circle was inherently unfair. The dastardly mathematician conceded the point and explained that this was completely unintentional (yeah, right). To alleviate any ill-will between them, he offered the first sibling to take his 2pi of linear fencing as a gift and enclose their house by a rectangle of any dimensions that sibling desired. The first sibling would then abide by that distribution of land. The second sibling instantly agreed!

Wanting to maximize space the second sibling enclosed their house with a square of sides 1/2pi, using up every bit of the provided 2pi of fencing. And with great dismay discovered that they had been swindled even worse!



How did this happen?