The title says it in a nutshell: five famous math problems and how they were solved.

“Famous” is a relative term, of course. Fermat’s Last Theorem and the Four-Color Problem are almost certainly familiar to those who scan the science section of their local newspapers; those who took college math have probably run across Cantor’s continuum hypothesis. The other two seem clear enough once stated, but few other than mathematicians are likely to be aware of their importance. In fact, the Kepler conjecture, on the optimal packing of spherical objects in three-dimensional space, often seems trivial to those with practical experience in stacking cantaloupes without having to explain their methods in the rigorous terms serious math requires. But math professor Casti (*The Cambridge Quintet*, 1998) does a solid job of presenting the problems for a mathematically unsophisticated audience. He sets each problem in the context of mathematical history and offers glimpses of the personalities that were key to their formulation. Casti also makes an effort to show the wider implications of the problems—the fact, for example, that Diophantine equations are not only the basis for Fermat’s Theorem, but of general importance throughout mathematics. The Four-Color Problem, superficially of interest only to mapmakers, led to the creation of entire new fields of mathematical inquiry. Even Fermat’s conjecture, which many mathematicians apparently considered a theoretical dead end, has opened up inquiries into the larger problems of which it was a special case, and its successful (and widely reported) solution by Andrew Wiles in 1993 has inspired the creation of prize funds for the solution of other mathematical “mountaintops.”

Excellent insights into the current shape of mathematics for those who don’t mind following a few equations.