The Binary Box

Probability Level 5

Consider an integer $n$ in base 10, whose binary form can be expressed as a string $S$ . Let the minimum value of $n$ be $N$ , such that the sub-strings of $S$ include binary numbers from $1$ to $10000_2$ . What is the sum of digits of $N$ ?

Details and Assumptions

For example, the integer $9$ , whose binary form can be expressed as a string $1001$ includes sub-strings of $0,1,10,1001$ , but not $101$ .

The answer is 28.

1 solution

Prabhakar Bhargava
Aug 31, 2014

the no. is -169840 ,sum of its digit is 28

from where did u get the no. -169840???

Ahmed Sayed - 6 years, 9 months ago

In binary, 169840 is $101001011101110000.$ This does not contain the substring 1111.

@Christopher Boo , can you post your intended value of $N$ ?

Jon Haussmann - 6 years, 8 months ago

39664, which in binary, 1001101011110000.

Christopher Boo - 6 years, 8 months ago

So, how did you get that value? I want to know.

Joeie Christian Santana - 6 years, 8 months ago

def SB(x):
    li=[]
    k=format(x,'b')
    for i in range(len(k)):
        for j in range(i+1,len(k)+1):
            li.append(int(k[i:j]))
        li=sorted(list(set(li)))    
    if li[1:17]==[1,10,11,100,101,110,111,1000,1001,1010,1011,1100,1101,1110,1111,10000]:
        return True
    else:
        return False

According to my program, if a number follows your property, it prints True . It printed True for the smallest value of $33964$ , what was the way you used?

Aditya Raut - 6 years, 2 months ago

The Binary Box

The answer is 28.

1 solution

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