Binary Pattern (1)
In binary puzzles , a grid must be filled with ones and zeroes. Each column and row must be a sequence following these rules:
- The numbers of zeroes and ones are equal.
- There may be no subsequence of three or more of the same number (zero or one).
For instance, is a valid sequence; is invalid because there are three zeroes in a row; is invalid because there are more ones than zeroes.
How many valid sequences of length six are possible?
Want a greater challenge? Try this one .
The answer is 14.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
The number of ways to place three ones in six positions is ( 6 3 ) = 2 0 . However, we must rule out patterns with three successive ones or zeroes. These are 0 0 0 1 1 1 ; 1 1 1 0 0 0 ; 0 1 1 1 0 0 ; 1 0 0 0 1 1 ; 0 0 1 1 1 0 ; 1 1 0 0 0 1 . Therefore there are 2 0 − 6 = 1 4 valid sequences.