## Circular Logic

### Problem 209

Published on Friday, 19th September 2008, 06:00 pm; Solved by 1623; Difficulty rating: 60%A `k`-input *binary truth table* is a map from `k` input bits
(binary digits, 0 [false] or 1 [true]) to 1 output bit. For example, the 2-input binary truth tables for the logical AND and XOR functions are:

x |
y |
x AND y |

0 | 0 | 0 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

x |
y |
x XOR y |

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

How many 6-input binary truth tables, τ, satisfy the formula

τ(

`a`,`b`,`c`,`d`,`e`,`f`) AND τ(`b`,`c`,`d`,`e`,`f`,`a`XOR (`b`AND`c`)) = 0for all 6-bit inputs (`a`, `b`, `c`, `d`, `e`, `f`)?