Unfortunately, at least according to the preface of the Ramble House edition of this book, things did not go as planned. To make a long story short (see the preface for the long version), two of the four mysteries were pulled out as stand alone novels (they became The Barking Clock and The Murdered Mathematician). Then, the number of mysteries Brown had to solve was reduced from 4 to 3, and the third mystery was one written by Keeler's wife Hazel Goodwin Keeler to which the giant mathematician and mathematics were only added as an afterthought. The uneven combination of three mysteries this process produced was not published during the authors' lifetimes and only appeared in print for the first time in 2001.

The plot of the book as eventually published by Ramble House is as follows:

A legal document filled out by Quiribus Brown is altered by a clerk so as to indicate that he will solve three mysteries using mathematics. He consequently has to do so or face charges of perjury. In the first mystery he works on, he is hired by the owner of a restaurant in Chicago's Chinatown who has lost all of his money in an unusual game of chance. He selects a number of small stones to hold in his hand and the other man, who was running the game, got to guess whether it was even or odd. Quiribus "shows" (see note below) that this game was unfair. In the second, he identifies a spy, a former Nazi mathematician passing military secrets to the Russians. And, in the third (titular) mystery, he aids another detective in figuring out how the hands and fingerprints of a dead man appear to be responsible for the attempted murder of an old woman.

One problem with the book is that the last of the mysteries is not only written with a different style but with a different purpose and so undermines the whole idea that Quiribus must solve three mysteries mathematically. Some may also be bothered by the book's attitude about race and ethnicity (which, for example, makes Chinese people appear to be technologically backward and mentally deficient), that may have seemed more appropriate in the 1940s when it was written. But, the worst thing about it was the mathematics! I am amazed at how bad Keeler's understanding of mathematics is considering that he chooses to write about it frequently!

Spoiler Warning: In the remainder of this review, I am going to address the mathematical details as well as the resolutions of the mysteries. Do not read on if you intend to read the book for pleasure.

One key point of Quiribus' evidence that the game with the stones was "unfair" was that, since there were 11 stones to choose from, there were 6 possible ways to have an odd number of stones in hand but only 5 ways to have an even number. There are two things wrong with this logic. A minor point is that one could have zero stones, which would be a 6th way to have an even number of stones. But, the real point is that the number of stones in his hand was completely under the control of the "victim". He could have selected an odd or even number at each round of the game, regardless of the number of possibilities for each. (And, in any case, there is no reason to assume that each of the numbers of stones is equally likely to be selected!)

But, even that is not the real problem with the mathematics in this case. If we assume, as the author seems to, that an odd number of stones would appear with probability 6/11 and that an even number would appear with probability of only 5/11, this does not mean (as Quiribus claims) that the optimal strategy for the other player is to guess "odd" 6 out of every 11 times! The false logic of this is immediately obvious if one considers a single round of play: if you are asked to guess whether the number of stones in the person's hand is odd or even, and you know it is more likely to be odd than even, why would you ever guess even?! A more rigorous analysis of the situation reveals that if the probability of it being odd is p and you select "odd" with probability q then the probability that you win is (2p-1)q+(1-p). Note first that if p is 1/2 then it doesn't matter what q is as your probability of winning is 1/2. But, if p is not 1/2, then you can maximize your chances of winning by guessing for whichever is more likely. In particular, if the probability of odd is 6/11 and you guess odd your probability of winning is (of course) 6/11, but if you guess odd with a probability of 6/11 and even with a probability of 5/11 your probability of winning goes down to 61/121. (Note: My former college roommate, Ned Welch, says that the error made by Keeler here is well-known and called the Probability Matching Fallacy.)

In the second mystery, Quiribus goes to a weapons facility to find the Nazi-Russian-Mathematician-Spy. One obvious candidate is ruled out because he was a student of Quiribus' father. (A typical Keeler coincidence.) But, the reader is not supposed to pay much attention to the "Negress" performer who says things like "Yassuh, Mist' Soop'tenden'. Yassuh." I might, at this point, refer you to the note above about racial stereotypes, except that this one is broken when it turns out that she is the Nazi-Russian-Mathematician-Spy! I am quite skeptical of the notion that a black woman would have been trained in advanced mathematics in Nazi Germany, despite Keeler's claim "no, they don't have the color-antipathy, or color-feeling over in Europe as we do here". But, I am even more unimpressed with Quiribus' explanation of how he recognized her for what she was. One of the things she mentioned in her fake dialect was a tale of the descendants of 8 people after 6 generations on an island. According to Quiribus/Keeler, only a mathematician could figure out that the expected number of people alive on the island at the end of this would be 8 (3/2)⁶ which is 91.02. Even putting aside the question of why she would mention this or whether this is the correct computation to do in order to determine the number of people on the island, there are two serious problems. One is that I think many non-mathematicians ought to be able to work this out without resorting to the use of logarithms as the book suggests. In particular, since 2³=8, this is the same as 27²/8. Since 27=24+3, 27/8=24/8+3/8=3+3/8. Then, multiplying by 27 we get 81+81/8=91+1/8. So, it is pretty easy to see that it is a bit bigger than 91. The other problem is that 91+1/8 is 91.125 not 91.02 at all!

Hazel Goodwin Keeler's part of the story does a pretty good job of including some mathematical terminology in the discussion, but the math does not turn out to be related to the mystery at all. Rather, the mystery is solved using conventional, non-mathematical techniques (including private detectives getting away with violations of civil rights that I would not think even an official police investigation could). Quiribus' problem with perjury charges is resolved in an unsatisfactory way when the detective under whom he is working is able to demonstrate the invalidity of the documents with which he was charged, rendering the first two mysteries superfluous.

I can only see this book being of interest to die-hard fans of Harry Stephen Keeler and those interested in mathematical errors made by authors of mathematical fiction. I do think it could have been far better if it had been carried out as originally planned, but as it is, I think that The Murdered Mathematician is a better vehicle for Quiribus Brown and his mathematical methods.