A man asks an architect to build a square house with square windows, one on each side, so that each window faces south. The architect agrees, and promptly builds the house. How can this be?
Three men go to a hotel and take a suite for which the charge is $30. Each gives the bellhop $10, who takes the cash down to the manager. But the manager realizes that the actual cost of the suite is $25, so he sends the bellhop back with five $1. The bellhop realizes this will be hard to split, so he pockets two of the bills and returns $1 to each man. Now each man payed $9, so the total is $27. The bellhop snuck $2 in his pocket, for a total of $29. But the men originally handed over $30. What happened to the other dollar?
Want to know the answers? These and other puzzles come from Eugene P. Northrup’s book, Riddles in Mathematics.
In the introduction, Daniel Silver discusses the book’s originally proposed titles. Northrop wanted Two and Two Make Five, but the publishers were horrified by the idea of promoting such sacrilege and instead suggested a title that would never have made it through the gutter minds of today: Tricks with Figures. Eventually, they settled upon Riddles in Mathematics: A Book of Paradoxes.
To Northrop, a paradox is
“Anything which offhand appears to be true, but is actually false; or which is simply self-contradictory.”
This form of paradox is merely a misunderstanding, a mistake made by someone who is talking too fast and thinking too superficially. To me, a paradox is not simply a mistake in calculation, but something deeper. A perfect example of a “true” mathematical paradox, to my mind, is the probability of choosing any particular value from a distribution over the real numbers, a topic covered briefly by Northrop. If the distribution is real-valued, then the probability of choosing any particular value is 0, but somehow the integral of all these 0’s is 1.* For Northrop, on the other hand, a paradox is merely a fallacy caused by insufficient thought or insight, a student error that becomes obvious in hindsight.
Even so, the “paradoxes” that Northrup points out are certainly omnipresent. For example, one of the mistakes I find most frustrating is the general assumption that events with unknown probability are uniformly distributed. According to Northrup, this type of assumption actually distinguishes two schools of thought: the “cogent reasonists,” who assume that in absence of additional knowledge, two choices are equally likely, and the “insufficient reasonists,” who require some form of evidence before assuming a distribution. I’m definitely one of the latter. For example, say we don’t know whether or not it will rain tomorrow. To me, if you don’t know the probability, you don’t know it-- either do your best to estimate it or just state that you don’t know anything at all. A cogent reasonist would say that in absence of evidence, the chance is 50-50. Northrup goes on to point out the contradictions that can arise from such beliefs.
Riddles In Mathematics is enjoyable enough, although it’s a little hard to characterize. It starts out with word problems and brain teasers, where confusion is cultivated by simple phrasing. It then continues on to simple arithmetic, geometry, and algebraic fallacies. I must admit, Northrop demonstrates an impressive variety of ways in which people can be tripped up by division by zero. Despite the entertaining writing, all of these “paradoxes” will be painfully obvious to anyone with a basic mathematical background. From that point forward, however, things start getting technical, delving deep into complex geometric fallacies, discussions of infinity, problems with probability, lessons in logic, and conundrums in calculus. Personally, I’ve never been fond of geometry and have little interest in topology, but I thoroughly enjoyed the sections dealing with infinity and fun with contradictions in convergence. Northrup also writes an entertaining introduction to the subject of martingales. Considering the book is over half a century old, it has aged quite gracefully. Overall, I’d concur that while it is unlikely that someone will find all of the sections interesting, Riddles in Mathematics does indeed have something for everyone.
~~ I received this ebook from the publisher, Dover Publications, in exchange for my honest review. ~~
*As far as I understand it, this particular paradox caused the birth of measure theory, but that branch of mathematics is too advanced for me.