## Circular Logic

### Problem 209

Published on Friday, 19th September 2008, 06:00 pm; Solved by 1429

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)) = 0

for all 6-bit inputs (a, b, c, d, e, f)?