algorithm to find a number in which product of number of 4 & 7 is maximum in given range -


i stuck in question in lower bound l , upper bound u given.
suppose in decimal representation of integer x digit 4 appears a times , digit 7 appears b times.
problem find x has maximum value of a*b l<=x<=u.
there efficient algorithm solve it?

if understood problem correctly, following should work:

  • assume numbers have same number of digits (if e.g. l has less digits u, can fill in beginning 0 s).
  • let z = u - l.
  • now go first (/highest/leftmost) digit last one. if looking @ i th digit, let l(i), u(i), z(i) , x(i) corresponding digit.
    • for leading z(i) 0, set x(i) = l(i) (we don't have choice).
    • for first not 0 z(i) check: there 4 or 7 in interval [l(i), u(i)-1]? if yes let x(i) 4 or 7 otherwise let x(i) = u(i)-1.
    • now fill rest of x 4s , 7s such choose 4 if have assigned more 7s far , vice versa.

maybe example can in understanding this:

given u = 5000 , l = 4900.

now z = 0100.

from algorithm set

  • x(1) = l(1) = 4 (we have no choice)
  • x(2) = u(2)-1 = 9 (the first non 0 digit in z)
  • x(3) = 7 (we had 4)
  • x(4) = 4 (can chosen arbitrarily)

leading x = 4974 objective of 2*1=2


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