This was not a single book but a series, Tom Corbett -- Space Cadet. There were 8 books in the series, but there is a Mercury Ball game in the first book, Stand by for Mars.
The Tom Corbett series was loosely based on Robert A. Heinlein's great YA book, Space Cadet, by way of a TV series. For a brief time in the mid-fifties, Tom Corbett was hugely popular: TV, radio, books, comic books, a comic strip, view master reels, lunch boxes, coloring books, punch out books, cut-out costumes on the backs of serial boxes, Halloween costumes...the list goes on and on. I had most of these, still have some.
Willy Ley, a scientist, was the science advisor for the books (in addition to writing a science column for Galaxy magazine). The Heinlein juveniles, and later the YA books written by Asimov under the name Paul French, and the Winston juveniles edited by Lester del Rey, replaced the earlier view of space as a nebulous realm where clouds, planets, moons, ringed planets, asteroids, flying saucers, angels, and what-not all wandered randomly in an airy blackness, with the more realistic view of planets in orbit around suns. The first movie to have this view was also by Heinlein, Destination Moon. Many popular magazines -- Life, Look, The Saturday Evening Post -- educated the American public about the actual nature of the universe.
The Tom Corbett books can still be read today, but are not nearly at the level of Heinlein, Asimov, and Clarke.