Concrete mathematics : a foundation for computer science

260 SPECIAL NUMBERS
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(The center of gravity of k objects, having respective weights WI, . . . , wk
and having reSpeCtiVe Centers Of gravity at pOSitiOnS ~1, . . . pk, is at position
(WPl +. . + WkPk)/bl + . + wk).) We can rewrite this recurrence in two
equivalent forms
k&+1 = k + dl + . . . + dkp1 + dk , k 3 0;
(k-l)dk = k-l +dl +...+dk-1, k> 1.
Subtracting these equations tells us that
kdk+l -(k-l)dk = 1 +dk, k> 1;
hence dk+l = dk + l/k. The second card will be offset half a unit past the
third, which is a third of a unit past the fourth, and so on. The general
formula
&+I = Hk (6.56)
follows by induction, and if we set k = n we get dn+l = H, as the total
overhang when n cards are stacked as described.
Could we achieve greater overhang by holding back, not pushing each
card to an extreme position but storing up “potential gravitational energy”
for a later advance? No; any well-balanced card placement has
&+I 6 (l+dl)+(l-td~)+...+(l+dk)
k
,
1