Maximum N digit number we can make with certain number of match sticks

Question

The picture shows how many sticks you need to draw each of the digits. given a number that consists of N digits. I want to move some sticks to maximize the number. I am not allowed to change the number of digits. It had to have N digits always.I cant remove any matchstick either.

Example:

given 512
answer:977
given 079
answer:997

My solution was to count the number of matchsticks we can use. Then we start placing 9 till we run out of it. Then when we encounter that we have problem like one match stick remaining we backtrack and make a different choice for the numbers. I am not sure whether this captures all the cases. Is there any flaw in my logic. If so what is a better way?

StackOverflow is not for maths homework. It's for programmers to help each other out. — tambre, Sep 28 '16 at 18:45
this is a programming problem actually :) N can be as big 100000 — Dragon Surfer, Sep 29 '16 at 02:26
Because things can be big doesn't make it programming. Now, where's your code? What language do you want help with? What's the problem with your code? Etc. — tambre, Sep 29 '16 at 16:16

score 1 · Answer 1 · answered Oct 02 '16 at 09:00

First you can see that the actual value of the initial number doesn't matter to us since we can move the sticks freely. We just have to keep the number of sticks in the initial number (s) and the number of digits (n). So first we calculate the number of sticks in the initial number.

Because the number of digits should remain the same we can assign the digit that need the least amount of sticks to every required digits. Digit 1 requires only two sticks so we initialize the number with n 1 s. This way we are sure that we have fulfilled the number of digits requirement.

After initializing the number with 1 s we should remove 2*n sticks from s.

s -= 2*n

Because each 1 requires two sticks and we have used n ones.

Now to maximize the number we have to try and use 9 s as much as possible. Each 9 requires 6 sticks and we have already used 2 sticks for each digit so to make a 9 digit we need 4 other sticks.

There is two exception that we should be careful about:

We shouldn't have any extra sticks. So in each step we should calculate if we use all the other digits with the number that takes the most sticks, do any sticks remain or not? 8 takes the most digits, and it need 7 sticks. So in order to make an 8 digit we need 5 extra sticks. If in any step the remaining sticks s is equal to remaining digits times 5, we should use 8 for the rest of the digits.
If there are less sticks remaining than 4 ( amount of sticks needed to make a 1 to 9 ), we should take the best decision with respect to the remaining digits and remaining sticks.

Below is a python implementation:

def solve ( s, # number of sticks
            n ): #number of digits
    s -= 2*n # use n 1 digits

    result = ["1"]*n

    for i in range(n):
        if s == 0: # no sticks remained
            break
        if s == 5*(n-i): # we should use 8 from now on or else there will be extra sticks
            result[i] = "8"
            s -= 5
        elif s < 4: # not enough sticks to make a 9
            if i == n-1: # this is the last digit make the most of if
                if s == 3: # maximum digit is 5 with 5 sticks 
                    result[i] = "5"
                    s -= 3
                elif s == 2: # maximum digit is 4 with 4 sticks 
                    result[i] = "4"
                    s -= 2
                elif s == 1: # maximum digit is 7 with 3 sticks 
                    result[i] = "7"
                    s -= 1
            else:
                result[i] = "7"
                s -= 1 # maximum digit is 7 with less than 6 sticks
        else:
            result[i] = "9"
            s -= 4
    print "".join(result)


needed_stick = {"0":6, "1":2, "2":5, "3":5, "4":4, "5":5, "6":6, "7":3, "8":7, "9":6}
initial_number = raw_input("Initial number: ")
s = sum( needed_stick[c] for c in initial_number )
n = len( initial_number )
solve(s,n)

Output:

Initial number: 512
977

Initial number: 079
997

Initial number: 88888888888888
88888888888888

Initial number: 8881
9995

Maximum N digit number we can make with certain number of match sticks

1 Answers1