Chapter 6.7, Problem 1PPC

Practice Problem CONCEPTUALIZE

Imagine a cobbler's business (where shoes are repaired) with trendy, V-shaped cabinets, four of which are shown here. When a pair of shoes is brought in for repair, it is kept in a shoebox in one of these cabinets. The location of each pair of shoes is recorded using a set of numbers that designate the cabinet (C), the shelf (S), and the specific box (B).

Each cabinet has a number, corresponding to the number of shelves it has. Thus, for the cabinets shown here, the value of C can be l, 2,3, or 4. Shelves within each cabinet are numbered sequentially from the bottom, starting with zero. For the smallest cabinet, with just one shelf, 0 is the only shelf designation. For the other cabinets, shelf designations can have integer values of 0 through C - 1. In addition, each individual box has a number on it. Boxes in the bottom row (row 0) all have the number 0 on them. Any box that resides directly above a box labeled 0, is also labeled 0. Boxes to the right or the left of the zero box on each shelf are numbered sequentially, starting with $+1$ (for boxes 011 the right), and starting with $-1$ (for boxes on the left). Using this numbering system, the cobbler can specify the location of a pair of shoes by designating three numbers: C, S, and B. For each of the following sets of numbers (C, S, B) determine whether or not they designate a box in one of the cabinets. For a set of numbers that does not designate a box in one of the cabinets, explain why.