Page 288 - HOW TO PROVE IT: A Structured Approach, Second Edition

P1: Oyk/
0521861241c06 CB996/Velleman October 20, 2005 1:8 0 521 86124 1 Char Count= 0

274 Mathematical Induction

Figure 6

put one tile in the middle, as in Figure 8. The area remaining to be covered
now consists of four 2 × 2 subgrids with one square removed from each. Each
of these can be covered with one tile, thus completing the upper right subgrid
of Figure 7.

Figure 7

Figure 8
The remaining three quarters of Figure 7 are completed by a similar proce-
dure. The final solution is shown in Figure 9.
The method we used in solving this problem is an example of a recursive
procedure.Wesolvedtheproblemforan8 × 8gridbysplittingitintofour4 × 4
grid problems. To solve each of these, we split it into four 2 × 2 problems, each