Convert number to a base where its representation has most “4”s

Question

Inspired by this. There is a number, given as either integer, string or array of digits (your choice). Find the base in which the representation of the number will have most "4"s and return that base.

Number    Result
624        5
444       10
 68       16

restrictions:

• The base returned should not be greater than the input.
• numbers less than or equal to abs(4) should not be considered valid input, so undefined returns are acceptable

Answer 1

APL (31 19)

Now tests all possible bases.

⊃⍒{+/4=K⊤⍨K⍴⍵}¨⍳K←⎕

Explanation:

• ⍳K←⎕: read user input, store in K. Make a list from 1 to K, which are the bases to try.
• {...}¨: for each of these, run the following function
• K⊤⍨K⍴⍵: encode K into that base giving a list of digits (as numbers) per base. Use K digits (a big overestimate, but it doesn't matter because the unused ones will all be zero anyway).
• 4=: see which of these are equal to 4
• +/: sum these, now we know how many fours per base
• ⊃⍒: give the indices of the list if it were sorted downwards, so the index of the biggest one is at the front. Take the first item of this list.

Answer 2

GolfScript, 30 characters

.,{[2+.2$\base{4=},,\]}%$)~p];

Works for any base - test the code online.

Comment: This solution was based on the original version of the question. It thus may return a base larger than the input, e.g. for the input 4 it correctly returns base 5 - which is no longer valid by the new rules.

Answer 3

Python 2.x, 77 chars

F=lambda x:max((sum(x/b**d%b==4for d in range(99)),b)for b in range(5,99))[1]

Works up to base 98 and numbers at most 98 digits long.

Answer 4

J, 38 characters

f=.[:(i.>./)[:+/[:|:4=(10#"0(i.37))#:]

Usage:

   p 624
5
   p 444
10
   p 68
16

Answer 5

VBA, 121

Function k(a)
For w=5 To a
Z=0:q=a:Do:c=q Mod w:Z=Z-(c=4):q=Int(q/w):Loop Until q=0
If Z>x Then x=Z:k=w
Next
End Function

usage:

• direct window: ?k(num)
• Excel formula: =k(A1)

Answer 6

GolfScript (23 chars)

~:^,2>{^\base[4]/,~}$0=

or

~:^,2>{^\base[4]/,}$-1=

or

~:^,2>{^\base[4]/,}$)\;

Note that this takes input from stdin: for a fair comparison with Howard's GolfScript version subtract one char.

---

Howard points out that the rules have changed, and it's not very logical that they now exclude 4 as a possible input when it has a valid output (any integer greater than 4). To cover that case as well requires an extra 2 characters, which can be added in all kinds of ways:

~:^)),2>{^\base[4]/,}$)\;

or

~:^,{))^\base[4]/,}$)))\;

being a couple of the obvious ones.

Answer 7

Mathematica 59

Code

Sort[{Count[IntegerDigits[n, #], 4], #} & /@ Range[5, 36]][[-1, 2]]

Let's give the above function a name.

whichBase[n_] := Sort[{Count[IntegerDigits[n, #], 4], #} & /@ Range[2, 36]][[-1, 2]]

Explanation

1. Count[IntegerDigits[n,k],4]: Count the number of fours in the base k representation of n.
2. Sort the bases from fewest to most 4s.
3. Return the base from the last item in the list, that is, the base that had the representation with the most 4's.

Some special numbers

Now let's apply whichBase to the following special numbers.

numbers= {1953124, 8062156, 26902404, 76695844, 193710244, 444444444, 
943179076, 1876283764, 3534833124, 6357245164, 10983816964, 
18325193796, 29646969124, 46672774204, 71708377284, 107789473684, 
158856009316, 229956041484, 327482302084, 459444789604, 635782877604, 
868720588636, 1173168843844, 1567178659764, 2072449425124, 
2714896551724, 3525282954756, 4539918979204, 5801435550244, 
7359635486844, 9272428079044, 11606852190676}

{5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, \ 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}

If you convert each number to the corresponding base, you will see what is special about them.

Answer 8

C - (114 characters)

In all it's golfy glory:

x,k,c,d,n;main(v){scanf("%d",&v);for(k=5;v/k;++k){x=v;c=0;while(x)c+=x%k==4,x/=k;c>=d?n=k,d=c:0;}printf("%d",n);}

And somewhat ungolfed:

x,k,c,d,n; // declare a bunch of ints, initialized to 0
main(v){   // declare one more, without using an extra comma
    scanf("%d",&v); // get the input (v)
    for(k=5;v/k;++k){ // loop over each base (k) greater than or equal to (/)
                      // our input (v)
        x=v;          // temp value (x) set to input (v)
        c=0;          // number of 4s in the current base (c) re-initialized
        while(x)       // loop over our temp until it's used up
            c+=x%k==4, // if the next digit (x%k) is 4 (==4) increment the
                       // current count (c+=)
            x/=k;      // remove the current digit
        c>=d?n=k,d=c:0; // if the number of 4s in this base (c) is greater
                       // than the current maximum number of 4s (d), then
                       // save the new best base (n), and new maximum
                       // number of 4s
    }
    printf("%d",n);   // output the result
}

Just for fun here's the output for the numbers [0,127] (these are the largest bases under the input number itself).

0, 0, 0, 0, 0, 5, 6, 7, 8, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 5, 21, 22, 23, 6, 25, 26, 27, 7, 29, 30, 31, 8, 33, 34, 35, 9, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 12, 49, 50, 51, 13, 53, 54, 55, 14, 57, 58, 59, 15, 61, 62, 63, 16, 65, 66, 67, 17, 69, 5, 71, 18, 73, 74, 75, 19, 7, 78, 79, 20, 81, 82, 83, 21, 85, 86, 87, 22, 89, 90, 91, 23, 93, 94, 5, 24, 97, 98, 99, 25, 101, 102, 103, 26, 5, 106, 107, 27, 109, 5, 111, 28, 113, 114, 5, 29, 9, 5, 5, 5, 121, 122, 123

Answer 9

C# (482 ~423 Bytes)

First attempt at a 'golfed' solution. I used basically the same algorithm as the VBA above. I could probably save some bytes inlining the conversion function, or shortening the name. Like I said this is a first attempt, so please be gentle.

With whitespace:

using System;
class Program
{
    static void Main(string[] args)
    {
        int n = int.Parse(args[0]);
        int c=0, m=0;
        string r="";
        int t = 0;
        for (int i = 5; i < 37; i++)
        {
            while (n > 0)
            {
                r = (char)((int)(n % i) + 48 + (7 * ((int)(n % i) > 9 ? 1 : 0))) + r;
                n = (int)(n / i);
            }
            t = r.Length - r.Replace("4", "").Length;
            if (t > c) { c = t; m = i; }
        }
        Console.WriteLine("Base: " + m);
    }
}

Answer 10

R - 148 137 chars

(so, far away from the rest of the competition but still)

f=function(n){s=sapply;which.max(s(lapply(strsplit(s(4:n,function(x){q=n;r="";while(q){r=paste(q%%x,r);q=q%/%x};r})," "),`==`,4),sum))+3}

Basically transform the input from base 10 to all bases from 4 to n (using modulo %% and integer division %/%) and pick the index of the first one having the most 4s.

f(624)
[1] 5
f(444)
[1] 10

Answer 11

C# with Linq 273

using System;using System.Linq;class P{static void Main(){int r,z,k=int.Parse(Console.ReadLine());if(k<=4) return;Console.WriteLine(Enumerable.Range(4, k).Select(x =>{r = 0;z = k;while (z > 0){if(z % x==4){r++;}z/=x;}return new[]{r, x};}).OrderBy(n => n[0]).Last()[1]);}}

or

using System;
using System.Linq;

class P
{
    static void Main()
    {
        int r, z, k = int.Parse(Console.ReadLine());
        if (k <= 4) return;
        Console.WriteLine(
            Enumerable.Range(4, k).Select(x =>
                {
                    r = 0;
                    z = k;
                    while (z > 0)
                    {
                        if (z % x == 4)
                        {
                            r++;
                        }
                        z /= x;
                    }
                    return new[] { r, x };
                }).OrderBy(n => n[0]).Last()[1]);

    }
}

Pretty sure the number of variables can be reduced and the if's can be converted to ?s. Oh well...

Answer 12

Python: 7 and Θ(1)

print 1

Works with any integer number.

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Given a number n you can represent it in base 1 with a sequence made of n repetitions of an arbitrarily chosen symbol. If you use 4 as a symbol you end-up with the numerical representation using the most 4s possible.