Problems
Explicit Formula
Explicit Formula
Consider \textbf{10} Boolean variables \textbf{x_1},\textbf{ x_2},\textbf{ x_3},\textbf{ x_4},\textbf{ x_5},\textbf{ x_6},\textbf{ x_7},\textbf{ x_8},\textbf{ x_9}, and \textbf{x_10}. Consider all pairs and triplets of distinct variables among these ten. (There are \textbf{45} pairs and \textbf{120} triplets.) Count the number of pairs and triplets that contain at least one variable equal to \textbf{1}. Set \textbf{f(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10)} = \textbf{1} if this number is odd and \textbf{f(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10)} = \textbf{0} if this number is even.
Here's an explicit formula that represents the function \textbf{f(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10) }correctly:
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\textbf{f(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10)} = (\textbf{x_1} V \textbf{x_2}) (\textbf{x_1} V \textbf{x_3}) (\textbf{x_1} V \textbf{x_4}) (\textbf{x_1} V \textbf{x_5}) (\textbf{x_1} V \textbf{x_6}) (\textbf{x_1} V \textbf{x_7}) (\textbf{x_1} V \textbf{x_8}) (\textbf{x_1} V \textbf{x_9}) (\textbf{x_1} V \textbf{x_10}) (\textbf{x_2} V \textbf{x_3}) (\textbf{x_2} V \textbf{x_4}) (\textbf{x_2} V \textbf{x_5}) (\textbf{x_2} V \textbf{x_6}) (\textbf{x_2} V \textbf{x_7}) (\textbf{x_2} V \textbf{x_8}) (\textbf{x_2} V \textbf{x_9}) (\textbf{x_2} V \textbf{x_10}) (\textbf{x_3} V \textbf{x_4}) (\textbf{x_3} V \textbf{x_5}) (\textbf{x_3} V \textbf{x_6}) (\textbf{x_3} V \textbf{x_7}) (\textbf{x_3} V \textbf{x_8}) (\textbf{x_3} V \textbf{x_9}) (\textbf{x_3} V \textbf{x_10}) (\textbf{x_4} V \textbf{x_5}) (\textbf{x_4} V \textbf{x_6}) (\textbf{x_4} V \textbf{x_7}) (\textbf{x_4} V \textbf{x_8}) (\textbf{x_4} V \textbf{x_9}) (\textbf{x_4} V \textbf{x_10}) (\textbf{x_5} V \textbf{x_6}) (\textbf{x_5} V \textbf{x_7}) (\textbf{x_5} V \textbf{x_8}) (\textbf{x_5} V \textbf{x_9}) (\textbf{x_5} V \textbf{x_10}) (\textbf{x_6} V \textbf{x_7}) (\textbf{x_6} V \textbf{x_8}) (\textbf{x_6} V \textbf{x_9}) (\textbf{x_6} V \textbf{x_10}) (\textbf{x_7} V \textbf{x_8}) (\textbf{x_7} V \textbf{x_9}) (\textbf{x_7} V \textbf{x_10}) (\textbf{x_8} V \textbf{x_9}) (\textbf{x_8} V \textbf{x_10}) (\textbf{x_9} V \textbf{x_10}) (\textbf{x_1} V \textbf{x_2} V \textbf{x_3}) (\textbf{x_1} V \textbf{x_2} V \textbf{x_4}) (\textbf{x_1} V \textbf{x_2} V \textbf{x_5}) (\textbf{x_1} V \textbf{x_2} V \textbf{x_6}) (\textbf{x_1} V \textbf{x_2} V \textbf{x_7}) (\textbf{x_1} V \textbf{x_2} V \textbf{x_8}) (\textbf{x_1} V \textbf{x_2} V \textbf{x_9}) (\textbf{x_1} V \textbf{x_2} V \textbf{x_10}) (\textbf{x_1} V \textbf{x_3} V \textbf{x_4}) (\textbf{x_1} V \textbf{x_3} V \textbf{x_5}) (\textbf{x_1} V \textbf{x_3} V \textbf{x_6}) (\textbf{x_1} V \textbf{x_3} V \textbf{x_7}) (\textbf{x_1} V \textbf{x_3} V \textbf{x_8}) (\textbf{x_1} V \textbf{x_3} V \textbf{x_9}) (\textbf{x_1} V \textbf{x_3} V \textbf{x_10}) (\textbf{x_1} V \textbf{x_4} V \textbf{x_5}) (\textbf{x_1} V \textbf{x_4} V \textbf{x_6}) (\textbf{x_1} V \textbf{x_4} V \textbf{x_7}) (\textbf{x_1} V \textbf{x_4} V \textbf{x_8}) (\textbf{x_1} V \textbf{x_4} V \textbf{x_9}) (\textbf{x_1} V \textbf{x_4} V \textbf{x_10}) (\textbf{x_1} V \textbf{x_5} V \textbf{x_6}) (\textbf{x_1} V \textbf{x_5} V \textbf{x_7}) (\textbf{x_1} V \textbf{x_5} V \textbf{x_8}) (\textbf{x_1} V \textbf{x_5} V \textbf{x_9}) (\textbf{x_1} V \textbf{x_5} V \textbf{x_10}) (\textbf{x_1} V \textbf{x_6} V \textbf{x_7}) (\textbf{x_1} V \textbf{x_6} V \textbf{x_8}) (\textbf{x_1} V \textbf{x_6} V \textbf{x_9}) (\textbf{x_1} V \textbf{x_6} V \textbf{x_10}) (\textbf{x_1} V \textbf{x_7} V \textbf{x_8}) (\textbf{x_1} V \textbf{x_7} V \textbf{x_9}) (\textbf{x_1} V \textbf{x_7} V \textbf{x_10}) (\textbf{x_1} V \textbf{x_8} V \textbf{x_9}) (\textbf{x_1} V \textbf{x_8} V \textbf{x_10}) (\textbf{x_1} V \textbf{x_9} V \textbf{x_10}) (\textbf{x_2} V \textbf{x_3} V \textbf{x_4}) (\textbf{x_2} V \textbf{x_3} V \textbf{x_5}) (\textbf{x_2} V \textbf{x_3} V \textbf{x_6}) (\textbf{x_2} V \textbf{x_3} V \textbf{x_7}) (\textbf{x_2} V \textbf{x_3} V \textbf{x_8}) (\textbf{x_2} V \textbf{x_3} V \textbf{x_9}) (\textbf{x_2} V \textbf{x_3} V \textbf{x_10}) (\textbf{x_2} V \textbf{x_4} V \textbf{x_5}) (\textbf{x_2} V \textbf{x_4} V \textbf{x_6}) (\textbf{x_2} V \textbf{x_4} V \textbf{x_7}) (\textbf{x_2} V \textbf{x_4} V \textbf{x_8}) (\textbf{x_2} V \textbf{x_4} V \textbf{x_9}) (\textbf{x_2} V \textbf{x_4} V \textbf{x_10}) (\textbf{x_2} V \textbf{x_5} V \textbf{x_6}) (\textbf{x_2} V \textbf{x_5} V \textbf{x_7}) (\textbf{x_2} V \textbf{x_5} V \textbf{x_8}) (\textbf{x_2} V \textbf{x_5} V \textbf{x_9}) (\textbf{x_2} V \textbf{x_5} V \textbf{x_10}) (\textbf{x_2} V \textbf{x_6} V \textbf{x_7}) (\textbf{x_2} V \textbf{x_6} V \textbf{x_8}) (\textbf{x_2} V \textbf{x_6} V \textbf{x_9}) (\textbf{x_2} V \textbf{x_6} V \textbf{x_10}) (\textbf{x_2} V \textbf{x_7} V \textbf{x_8}) (\textbf{x_2} V \textbf{x_7} V \textbf{x_9}) (\textbf{x_2} V \textbf{x_7} V \textbf{x_10}) (\textbf{x_2} V \textbf{x_8} V \textbf{x_9}) (\textbf{x_2} V \textbf{x_8} V \textbf{x_10}) (\textbf{x_2} V \textbf{x_9} V \textbf{x_10}) (\textbf{x_3} V \textbf{x_4} V \textbf{x_5}) (\textbf{x_3} V \textbf{x_4} V \textbf{x_6}) (\textbf{x_3} V \textbf{x_4} V \textbf{x_7}) (\textbf{x_3} V \textbf{x_4} V \textbf{x_8}) (\textbf{x_3} V \textbf{x_4} V \textbf{x_9}) (\textbf{x_3} V \textbf{x_4} V \textbf{x_10}) (\textbf{x_3} V \textbf{x_5} V \textbf{x_6}) (\textbf{x_3} V \textbf{x_5} V \textbf{x_7}) (\textbf{x_3} V \textbf{x_5} V \textbf{x_8}) (\textbf{x_3} V \textbf{x_5} V \textbf{x_9}) (\textbf{x_3} V \textbf{x_5} V \textbf{x_10}) (\textbf{x_3} V \textbf{x_6} V \textbf{x_7}) (\textbf{x_3} V \textbf{x_6} V \textbf{x_8}) (\textbf{x_3} V \textbf{x_6} V \textbf{x_9}) (\textbf{x_3} V \textbf{x_6} V \textbf{x_10}) (\textbf{x_3} V \textbf{x_7} V \textbf{x_8}) (\textbf{x_3} V \textbf{x_7} V \textbf{x_9}) (\textbf{x_3} V \textbf{x_7} V \textbf{x_10}) (\textbf{x_3} V \textbf{x_8} V \textbf{x_9}) (\textbf{x_3} V \textbf{x_8} V \textbf{x_10}) (\textbf{x_3} V \textbf{x_9} V \textbf{x_10}) (\textbf{x_4} V \textbf{x_5} V \textbf{x_6}) (\textbf{x_4} V \textbf{x_5} V \textbf{x_7}) (\textbf{x_\{4 \}}V \textbf{x_5} V \textbf{x_8}) (\textbf{x_4} V \textbf{x_5} V \textbf{x_9}) (\textbf{x_4} V \textbf{x_5} V \textbf{x_10}) (\textbf{x_4} V \textbf{x_6} V \textbf{x_7}) (\textbf{x_4} V \textbf{x_6} V \textbf{x_8}) (\textbf{x_4} V \textbf{x_6} V \textbf{x_9}) (\textbf{x_4} V \textbf{x_6} V \textbf{x_10}) (\textbf{x_4} V \textbf{x_7} V \textbf{x_8}) (\textbf{x_4} V \textbf{x_7} V \textbf{x_9}) (\textbf{x_4} V \textbf{x_7} V \textbf{x_10}) (\textbf{x_4} V \textbf{x_8} V \textbf{x_9}) (\textbf{x_4} V \textbf{x_8} V \textbf{x_10}) (\textbf{x_4} V \textbf{x_9} V \textbf{x_10}) (\textbf{x_5} V \textbf{x_6} V \textbf{x_7}) (\textbf{x_5} V \textbf{x_6} V \textbf{x_8}) (\textbf{x_5} V \textbf{x_6} V \textbf{x_9}) (\textbf{x_5} V \textbf{x_6} V \textbf{x_10}) (\textbf{x_5} V \textbf{x_7} V \textbf{x_8}) (\textbf{x_5} V \textbf{x_7} V \textbf{x_9}) (\textbf{x_5} V \textbf{x_7} V \textbf{x_10}) (\textbf{x_5} V \textbf{x_8} V \textbf{x_9}) (\textbf{x_5} V \textbf{x_8} V \textbf{x_10}) (\textbf{x_5} V \textbf{x_9} V \textbf{x_10}) (\textbf{x_6} V \textbf{x_7} V \textbf{x_8}) (\textbf{x_6} V \textbf{x_7} V \textbf{x_9}) (\textbf{x_6} V \textbf{x_7} V \textbf{x_10}) (\textbf{x_6} V \textbf{x_8} V \textbf{x_9}) (\textbf{x_6} V \textbf{x_8} V \textbf{x_10}) (\textbf{x_6} V \textbf{x_9} V \textbf{x_10}) (\textbf{x_7} V \textbf{x_8} V \textbf{x_9}) (\textbf{x_7} V \textbf{x_8} V \textbf{x_10}) (\textbf{x_7} V \textbf{x_9} V \textbf{x_10}) (\textbf{x_8} V \textbf{x_9} V \textbf{x_10})
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In this formula V stands for logical \textbf{or}, and stands for \textit{exclusive or} (\textbf{xor}). Remember that in C++ and Java these two binary operators are denoted as "\textbf{||}" and "\textbf{^}".
Given the values of \textbf{x_1},\textbf{ x_2},\textbf{ x_3},\textbf{ x_4},\textbf{ x_5},\textbf{ x_6},\textbf{ x_7},\textbf{ x_8},\textbf{ x_9},\textbf{ x_10}, calculate the value of \textbf{f(x_1, x_2, ..., x_10)}.
\InputFile
The input file contains \textbf{10} numbers \textbf{x_1},\textbf{ x_2},\textbf{ x_3},\textbf{ x_4},\textbf{ x_5},\textbf{ x_6},\textbf{ x_7},\textbf{ x_8},\textbf{ x_9}, and \textbf{x_10}. Each of them is either \textbf{0} or \textbf{1}.
\OutputFile
Output a single value - \textbf{f(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10)}.
Input example #1
1 0 0 1 0 0 1 0 0 1
Output example #1
0