We see that all those solutions lie on a straight line. In fact, every pair (x, y) that solves that equation will be the coördinates of a point on that line. On that line, every coördinate pair is (x, 2x + 1). y = 2x + 1. That line, therefore, is called the graph of the equation y = 2x + 1. And y = 2x + 1 is called the equation of that line. The tiger-cages of the kingdom were searched for the most savage and relentless beasts, from which the fiercest monster might be selected for the arena; and the ranks of maiden youth and beauty throughout the land were carefully surveyed by competent judges in order that the young man might have a fitting bride in case fate did not determine for him a different destiny. Some cells already contain numbers, known as "givens". The goal is to fill in the empty cells, one number in each, so that each column, row, and region contains the numbers 1 through 9 exactly once. Each number in the solution therefore occurs only once in each of three "directions", hence the "single numbers" implied by the puzzle's name.