The novel is narrated by her boyfriend, another math grad student who coordinated that road trip. But, because Singer goes into hiding shortly afterwards, he doesn't see her for many years and doesn't even know that they had a child together. Much of the novel consists of him getting to know their teenage daughter as they search together for Crystal Singer, both in the sense of finding out where she is and also just to understand her.

The story takes place in an alternate universe where humans have been communicating with Martians for a long time. This communication takes the form of giant mathematical symbols illuminated on the surface of each planet which can be seen from the other planet. Moreover, it seems as if the Martians are posing mathematical problems for the humans, and they only respond once the "right answer" has been sent. The exchange began with some simple arithmetic problems, but soon became more advanced. Long before Singer carved her message in Arizona, Einstein carved one near that same spot concerning relativity which was deemed correct. The aliens then posted a new geometry question which apparently entails that the same two points can be either near to each other or far apart depending on how the distance is measured. Another character, a Bill Nye-like figure, famously failed to answer that question correctly, but Crystal Singer's revised concept of distance only understood by a small number of mathematicians proved to be correct. That resulted in the posing of a new Martian math question having something to do with entropy.

Another thing that seems different about the universe in this novel as compared to the one we live in is that sexism does not seem to be a problem there. Even though the whole part with Crystal Singer takes place during the 1960s and 1970s, a very high percentage of the mathematicians and physicists we meet are female. Unfortunately, that is not the way things were in this universe where I live. I'm not sure if the author intended us to interpret it that way -- it may be that we weren't intended to consciously notice the genders of these researchers at all -- but I'm thinking it as another aspect of this "alternate history" that differs from our own.

Mathematics is quite important to the story. Many of the characters are mathematicians and math professors. As in many works of science fiction, mathematics is used as a language to communicate with extra-terrestrial aliens. And, math is also seen to be a tool of scientific discovery. However, the specific mathematical ideas discussed are usually vague, and seem incorrect to me in the few instances when I could interpret what they meant. (For example, it is implied that the idea that there could be more than one concept of "distance" between the same two points would be new and surprising to mathematicians. That is not the case. As we would say in the jargon of math, mathematicians are very familiar with the idea that different metrics can be used on the same space. The definition of "Singer Distance" was never stated clearly, but it would just be one of many different concepts of distance in geometry and not the only alternative to a unique classical definition.)

Crystal Singer is obviously under a lot of pressure, as the most famous living mathematician whom many expect to be able to answer the Martian's riddle about entropy. So, it may not be surprising that she "cracks" and begins acting a bit crazy. She tries to commit suicide and ends up going many times to a psychiatric hospital for help. Even if this seems to make sense in the context of the story, it does play in to the very common stereotype linking mathematics and clinical mental disorders. I fear this common trope misleads readers of fiction like this into believing that such a link exists in reality. (I am not aware of any real evidence that such a link exists.)

For the last part of my review, I'm going to have to issue a SPOILER ALERT

---- SPOILER ALERT ---- do not read below if you want to be surprised by the ending

---- SPOILER ALERT ---- do not read below if you want to be surprised by the ending

---- SPOILER ALERT ---- do not read below if you want to be surprised by the ending

At the conclusion of the book, Singer (with the help of her boyfriend and daughter) again communicates successfully with Mars. But this time, instead of answering one of their questions, she uses mathematical notation to communicate with them about music. It is implied that with the help of a musicologist she has devised a better musical notation where there are no staves, where melodies are indicated by numerals representing intervals rather than individual notes, and these numerals themselves are drawn larger or wider to convey information about how loudly and long they are played.

I'm not sure this makes sense for a number of reasons. For one thing, I don't see how the Martians would recognize that this strange display of numbers represents music. Also, there is nothing universal about the notes in our scales (why are there no notes between E and F and only one note between F and G?) or about what makes music sound good to us. As I understand it, both of those things involve a combination of cultural preferences and the biology of our inner ears. So, even if they knew that it was music, I don't think they would know what it sounded like to us, and even if they knew what it sounded like there is no reason to think it would sound melodious to them.

For those reasons, I found the ending of the book to be a bit of a let down. Still, it is a nice work of mathematical fiction. I'm glad I read it and can happily recommend it to other readers.