Linear Algebra

422 Chapter Five. Similarityof the relation has the form f(n) = c 1 r n 1 + c 2r n 2 + · · · + c kr n k for c 1, . . . , c n ∈ R.(The case of repeated roots is similar but we won’t cover it here; see any text onDiscrete Mathematics.)Now we bring in the initial conditions. Use them to solve for c 1 , . . . , c n . Forinstance, the polynomial associated with the Fibonacci relation is −λ 2 + λ + 1,whose roots are (1± √ 5)/2 and so any solution of the Fibonacci equation has theform f(n) = c 1 ((1 + √ 5)/2) n + c 2 ((1 − √ 5)/2) n . Including the initial conditionsfor the cases n = 0 and n = 1 givesc 1 + c 2 = 0(1 + √ 5/2)c 1 + (1 − √ 5/2)c 2 = 1which yields c 1 = 1/ √ 5 and c 2 = −1/ √ 5, as we found above.We close by considering the nonhomogeneous case, where the relation hasthe form f(n + 1) = a n f(n) + a n−1 f(n − 1) + · · · + a n−k f(n − k) + b for somenonzero b. We only need a small adjustment to make the transition from thehomogeneous case. This classic example illustrates.In 1883, Edouard Lucas posed the following problem, today called the Towerof Hanoi.In the great temple at Benares, beneath the dome which marksthe center of the world, rests a brass plate in which are fixed threediamond needles, each a cubit high and as thick as the body of abee. On one of these needles, at the creation, God placed sixty fourdisks of pure gold, the largest disk resting on the brass plate, andthe others getting smaller and smaller up to the top one. This is theTower of Brahma. Day and night unceasingly the priests transferthe disks from one diamond needle to another according to the fixedand immutable laws of Bram-ah, which require that the priest onduty must not move more than one disk at a time and that he mustplace this disk on a needle so that there is no smaller disk belowit. When the sixty-four disks shall have been thus transferred fromthe needle on which at the creation God placed them to one of theother needles, tower, temple, and Brahmins alike will crumble intodusk, and with a thunderclap the world will vanish. (Translation of[De Parville] from [Ball & Coxeter].)How many disk moves will it take? Instead of tackling the sixty four disk problemright away, we will consider the problem for smaller numbers of disks, startingwith three.To begin, all three disks are on the same needle.