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

Question

I am stuck in a question in which lower bound L and Upper bound U is given.
Now suppose in the decimal representation of integer X

Answer 1

If I understood the problem correctly, the following should work:

• Assume all numbers have the same number of digits (if e.g. L has less digits than U, we can just fill in the beginning with 0 s).
• Let Z = U - L.
• Now we go from the first (/highest/leftmost) digit to the last one. If we are looking at the i th digit, let L(i), U(i), Z(i) and X(i) be the corresponding digit.
  • for all leading Z(i) which are 0, we set X(i) = L(i) (we don't have a choice).
  • For the first not 0 Z(i) check: is there a 4 or a 7 in the interval [L(i), U(i)-1]? If yes let X(i) be that 4 or 7 otherwise let X(i) = U(i)-1.
  • Now fill up the rest of X with 4s and 7s such that you choose a 4 if you have assigned more 7s so far and vice versa.

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Maybe an example can help in understanding this:

Given U = 5000 and L = 4900.

Now Z = 0100.

From the algorithm we 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 already had a 4)
• X(4) = 4 (can be chosen arbitrarily)

Leading to X = 4974 with an objective of 2*1=2