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Unusual binary representation (Posted on 2005-06-14) Difficulty: 3 of 5
Show that every positive integer is an alternating sum of strictly increasing powers of 2.

For example, 5=2^0 -2^2 +2^3 and 8=-2^3 +2^4 are alternating sums of strictly increasing powers of 2 (8=2^3 is ok too).

10=-2^1 -2^2 +2^4 is a strictly increasing sum but not alternating.

4=2^1 -2^1 +2^2 is alternating but not strictly increasing.

(author: prof Dan Shapiro of Ohio State University)

See The Solution Submitted by McWorter    
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No Subject | Comment 1 of 9

I don't know if I can prove this, but I can show how it works with any number.  For an example I will use 105 which is

1101001 in binary

subtract 10000000 leaving -10111 (notice this changes all digits from 0 to 1 and 1 to 0 except the last)

add 100000 giving 1001 (again changing all but last)

subtract 10000 leaving -111

add 1000 giving 1

subtract 1 leaving 0

written in base 10 this is
105-128 = -23
-23 + 32 = 9
9-16 = -7
-7+8 = 1
1-1=0

so 128-32+16-8+1=105

I see the pattern.  When the number is written in binary with a leading 0, use the power of each digits whose following digit is different.

I think I can work out a proof in a separate post later.


  Posted by Jer on 2005-06-14 17:03:51
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