Back in 1947 Richard W. Hamming had access to a computer only on weekends. Some three decades later he recalled his frustration over its perverse behavior:

That question initiated the development of error-correcting codes. The offending computer, a mechanical relay model at Bell Telephone Laboratories, always came to an angry stop and switched to the next program whenever it detected an error. This behavior impelled Hamming, a pure mathematician with an applied bent, to devise the first error-correcting code.

We shall follow the devious trail that wends its way through a quarter-century of mathematics, starting with Hamming's work, which led almost immediately to that of M. J. E. Golay. The latter sparked, some twelve years later, giant steps in the packing of congruent spheres by John Leech, which, in turn, branched off through the work of J. H. Conway, into the field of simple groups. By tracing some of the twists, turns, switchbacks and dead ends of this path, we hope to provide a small window on the history of mathematics of the twentieth century. How historians ultimately will treat this golden age of mathematical creativity we cannot even guess.