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2/5 ≤ m/n ≤ 3/5 (Posted on 2009-10-20) Difficulty: 3 of 5
Five distinct n-digit binary numbers are such that for any two numbers chosen from them the digits will coincide in precisely m places. There is no place with the common digit for all the five numbers. At least one of the binary numbers contains leading zero.

Prove that 2/5 ≤ m/n ≤ 3/5.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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re(2): Eureka! (spoiler) | Comment 6 of 7 |
(In reply to re: Eureka! (spoiler) by Jer)

I know the problem didn't ask for values, only proof, but I was bored and decided to make a program just to see how many values there are for different cases.

There are a total of 32 solutions for n=5, m=3 with the last one being:

01111
10111
11011
11101
11110

No solutions exist for n=6

There are a total of 26,880 solutions for n=7, m=3 with the last one being:

0111111
1010110
1011001
1100101
1101010

There are a total of 430,080 solutions for n=8, m=4 with the last one being:

01111111
10111100
11011010
11100110
11110001

That's as far as the program has gotten so far, as Python isn't as fast as I would like.
  Posted by Justin on 2009-12-07 04:33:40

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