I was searching through my (physical) files when I came across this letter I found in a library book twenty years or so ago. It's not a complete letter—it appears to be from the second page and it included a hand drawn diagram.
On its surface, the [Banach-Tarski] paradox resembles the well-known puzzle involving tangrams, little pieces of paper cut into simple geometric shapes. Four such shapes can be assembled into a square that has an area of 64 square inches. Yet the very same pieces can also be assembled into a rectangle that has a greater area—65 square inches, to be precise. If you do not believe such a thing is possible, try cutting up the pieces as shown in my drawing.
If the little pieces of paper were instead pieces of gold, there would seem to be an automatic increase in wealth in going from the square to the rectangular configuration. Start with a square of gold, say, eight inches on a side and an inch thick. Then cut it according to the first figure. If the pieces are reassembled according to the second figure, an extra cubic inch of gold will have appeared. The extra cube would weigh approximately 4.3 ounces and at current prices would be worth about $160.
It goes on from there in increasingly dense prose and mathematics and it's signed “Arlo Lipof,” whom I never heard of, nor could I find any information about him at the time (nor can I even today).
Like I said, I found this some twenty years ago but there's no telling how old the actual letter is. I mean, I could figure out when gold was last $35 an ounce, but that still wouldn't tell me anything about the letter.
I remember my amazement at the time, to find such an incredible means of generating wealth, but being unable to even act upon it. I mean, here I am, a teenager. What was I going to do? Go to my parents? “Yeah, can I borrow a fifty thousand dollars? I want to buy 64 ounces of gold.” Like that would happen. Besides, where would one even buy bars of gold? Can one even buy gold bullion?
So the paper was filed away and forgotten.
But now that I'm an adult, it should be easier for me to get gold. Let's see … 64 cubic inches … 4.3 ounces per cubic inch … $580 an ounce … $159,616.00! Ouch!
Wonder if I can get a second mortgage on Casa New Jersey?