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How many zeroes do we write when we write all the integers from 1 to 256 in binary?

 Jan 28, 2022
 #1
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There should be: 777 zeros in base 10.

 Jan 28, 2022
 #2
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How many zeroes do we write when we write all the integers from 1 to 256 in binary?

 

I am going to look for a pattern.

 

256 = 2^8  so there is 8 zeros which is 9 digits

 

consider the numbers between 2^1 and 2^2-1          inclusive    10,11,                                           2/2=1   zeros

consider the numbers between 2^2 and 2^3-1          inclusive    100,101,110,111,                    2^2  *2 /2=4  zeros

consider the numbers between 2^3 and 2^4-1          inclusive    4 digits                                   2^3 *3/2 = 12 zeros

       ....

consider the numbers between 2^7 and 2^8-1          inclusive    8 digits                                   2^7 *7/2   zeros

 

 

    1+4 + 2^3*3/2 +   2^4*4/2    +   2^5*5/2  +  2^6*6/2   +  2^7*7/2  + 8

=  1+4+      4*3    +      8*4       +        16*5   +     32*6    +      64*7  + 8

=  1  + 4 +   12    +      32         +       80       +      192     +       448  +  8

=    777  base 10   

 

Just as guest already determined  :)

 Jan 29, 2022

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