js-rhino, 126 bytes

Result

AC

Code [DL]

c=new java.util.Scanner(java.lang.System.in);for(;c.hasNext();){l=c.nextLine();print(+!((Math.sqrt(8*parseInt(l,2)+1)-1)%2));}

stdin

00010101
00000010
10011001
00010111
01110001
01101101
10001000
00100100
01010110
01000001
00011011
00000011
10000100
00000000
10000001
00001111
01111000
01101001
00011100
10111010
11110000
00001010
01001011
00011100
00000000
10001110
01011011
00000001
10111110
11000100
11100111
01100110
10011100
00101101
01111001
11111101
00110111
11101010
01000010
10010110
01011010
11010010
11100000
11010011
00000110
11010001
10101011
00100010
01001110
10000011

stdout

1
0
1
0
0
0
1
1
0
0
0
1
0
1
0
1
1
1
1
0
0
1
0
1
1
0
1
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
0
1
0
1
0
1
0

stderr

strace