Explicit Formula

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Time limit3sMemory limit256 MB

Summary

Given 10 binary inputs, evaluate a fixed XOR-of-ORs boolean formula (equivalently just count pairs/triplets with at least one 1 and check parity) and output the result.

Level: Easy2 of 10
Topics: Bit manipulation, Implementation, Brute force
Solved: No attempts yet

Problem

Consider 10 Boolean variables x1, x2, x3, x4, x5, x6, x7, x8, x9, x10. Among these ten variables, look at every pair and every triplet of distinct variables. (There are 45 pairs and 120 triplets.) Count how many of these pairs and triplets contain at least one variable equal to 1. Define f(x1, ..., x10) = 1 if that count is odd, and f(x1, ..., x10) = 0 if that count is even.

The following explicit formula computes f(x1, ..., x10) correctly:

f(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = (x1 ∨ x2) ⊕ (x1 ∨ x3) ⊕ (x1 ∨ x4) ⊕ (x1 ∨ x5) ⊕ (x1 ∨ x6) ⊕ (x1 ∨ x7) ⊕ (x1 ∨ x8) ⊕ (x1 ∨ x9) ⊕ (x1 ∨ x10) ⊕ (x2 ∨ x3) ⊕ (x2 ∨ x4) ⊕ (x2 ∨ x5) ⊕ (x2 ∨ x6) ⊕ (x2 ∨ x7) ⊕ (x2 ∨ x8) ⊕ (x2 ∨ x9) ⊕ (x2 ∨ x10) ⊕ (x3 ∨ x4) ⊕ (x3 ∨ x5) ⊕ (x3 ∨ x6) ⊕ (x3 ∨ x7) ⊕ (x3 ∨ x8) ⊕ (x3 ∨ x9) ⊕ (x3 ∨ x10) ⊕ (x4 ∨ x5) ⊕ (x4 ∨ x6) ⊕ (x4 ∨ x7) ⊕ (x4 ∨ x8) ⊕ (x4 ∨ x9) ⊕ (x4 ∨ x10) ⊕ (x5 ∨ x6) ⊕ (x5 ∨ x7) ⊕ (x5 ∨ x8) ⊕ (x5 ∨ x9) ⊕ (x5 ∨ x10) ⊕ (x6 ∨ x7) ⊕ (x6 ∨ x8) ⊕ (x6 ∨ x9) ⊕ (x6 ∨ x10) ⊕ (x7 ∨ x8) ⊕ (x7 ∨ x9) ⊕ (x7 ∨ x10) ⊕ (x8 ∨ x9) ⊕ (x8 ∨ x10) ⊕ (x9 ∨ x10) ⊕ (x1 ∨ x2 ∨ x3) ⊕ (x1 ∨ x2 ∨ x4) ⊕ (x1 ∨ x2 ∨ x5) ⊕ (x1 ∨ x2 ∨ x6) ⊕ (x1 ∨ x2 ∨ x7) ⊕ (x1 ∨ x2 ∨ x8) ⊕ (x1 ∨ x2 ∨ x9) ⊕ (x1 ∨ x2 ∨ x10) ⊕ (x1 ∨ x3 ∨ x4) ⊕ (x1 ∨ x3 ∨ x5) ⊕ (x1 ∨ x3 ∨ x6) ⊕ (x1 ∨ x3 ∨ x7) ⊕ (x1 ∨ x3 ∨ x8) ⊕ (x1 ∨ x3 ∨ x9) ⊕ (x1 ∨ x3 ∨ x10) ⊕ (x1 ∨ x4 ∨ x5) ⊕ (x1 ∨ x4 ∨ x6) ⊕ (x1 ∨ x4 ∨ x7) ⊕ (x1 ∨ x4 ∨ x8) ⊕ (x1 ∨ x4 ∨ x9) ⊕ (x1 ∨ x4 ∨ x10) ⊕ (x1 ∨ x5 ∨ x6) ⊕ (x1 ∨ x5 ∨ x7) ⊕ (x1 ∨ x5 ∨ x8) ⊕ (x1 ∨ x5 ∨ x9) ⊕ (x1 ∨ x5 ∨ x10) ⊕ (x1 ∨ x6 ∨ x7) ⊕ (x1 ∨ x6 ∨ x8) ⊕ (x1 ∨ x6 ∨ x9) ⊕ (x1 ∨ x6 ∨ x10) ⊕ (x1 ∨ x7 ∨ x8) ⊕ (x1 ∨ x7 ∨ x9) ⊕ (x1 ∨ x7 ∨ x10) ⊕ (x1 ∨ x8 ∨ x9) ⊕ (x1 ∨ x8 ∨ x10) ⊕ (x1 ∨ x9 ∨ x10) ⊕ (x2 ∨ x3 ∨ x4) ⊕ (x2 ∨ x3 ∨ x5) ⊕ (x2 ∨ x3 ∨ x6) ⊕ (x2 ∨ x3 ∨ x7) ⊕ (x2 ∨ x3 ∨ x8) ⊕ (x2 ∨ x3 ∨ x9) ⊕ (x2 ∨ x3 ∨ x10) ⊕ (x2 ∨ x4 ∨ x5) ⊕ (x2 ∨ x4 ∨ x6) ⊕ (x2 ∨ x4 ∨ x7) ⊕ (x2 ∨ x4 ∨ x8) ⊕ (x2 ∨ x4 ∨ x9) ⊕ (x2 ∨ x4 ∨ x10) ⊕ (x2 ∨ x5 ∨ x6) ⊕ (x2 ∨ x5 ∨ x7) ⊕ (x2 ∨ x5 ∨ x8) ⊕ (x2 ∨ x5 ∨ x9) ⊕ (x2 ∨ x5 ∨ x10) ⊕ (x2 ∨ x6 ∨ x7) ⊕ (x2 ∨ x6 ∨ x8) ⊕ (x2 ∨ x6 ∨ x9) ⊕ (x2 ∨ x6 ∨ x10) ⊕ (x2 ∨ x7 ∨ x8) ⊕ (x2 ∨ x7 ∨ x9) ⊕ (x2 ∨ x7 ∨ x10) ⊕ (x2 ∨ x8 ∨ x9) ⊕ (x2 ∨ x8 ∨ x10) ⊕ (x2 ∨ x9 ∨ x10) ⊕ (x3 ∨ x4 ∨ x5) ⊕ (x3 ∨ x4 ∨ x6) ⊕ (x3 ∨ x4 ∨ x7) ⊕ (x3 ∨ x4 ∨ x8) ⊕ (x3 ∨ x4 ∨ x9) ⊕ (x3 ∨ x4 ∨ x10) ⊕ (x3 ∨ x5 ∨ x6) ⊕ (x3 ∨ x5 ∨ x7) ⊕ (x3 ∨ x5 ∨ x8) ⊕ (x3 ∨ x5 ∨ x9) ⊕ (x3 ∨ x5 ∨ x10) ⊕ (x3 ∨ x6 ∨ x7) ⊕ (x3 ∨ x6 ∨ x8) ⊕ (x3 ∨ x6 ∨ x9) ⊕ (x3 ∨ x6 ∨ x10) ⊕ (x3 ∨ x7 ∨ x8) ⊕ (x3 ∨ x7 ∨ x9) ⊕ (x3 ∨ x7 ∨ x10) ⊕ (x3 ∨ x8 ∨ x9) ⊕ (x3 ∨ x8 ∨ x10) ⊕ (x3 ∨ x9 ∨ x10) ⊕ (x4 ∨ x5 ∨ x6) ⊕ (x4 ∨ x5 ∨ x7) ⊕ (x4 ∨ x5 ∨ x8) ⊕ (x4 ∨ x5 ∨ x9) ⊕ (x4 ∨ x5 ∨ x10) ⊕ (x4 ∨ x6 ∨ x7) ⊕ (x4 ∨ x6 ∨ x8) ⊕ (x4 ∨ x6 ∨ x9) ⊕ (x4 ∨ x6 ∨ x10) ⊕ (x4 ∨ x7 ∨ x8) ⊕ (x4 ∨ x7 ∨ x9) ⊕ (x4 ∨ x7 ∨ x10) ⊕ (x4 ∨ x8 ∨ x9) ⊕ (x4 ∨ x8 ∨ x10) ⊕ (x4 ∨ x9 ∨ x10) ⊕ (x5 ∨ x6 ∨ x7) ⊕ (x5 ∨ x6 ∨ x8) ⊕ (x5 ∨ x6 ∨ x9) ⊕ (x5 ∨ x6 ∨ x10) ⊕ (x5 ∨ x7 ∨ x8) ⊕ (x5 ∨ x7 ∨ x9) ⊕ (x5 ∨ x7 ∨ x10) ⊕ (x5 ∨ x8 ∨ x9) ⊕ (x5 ∨ x8 ∨ x10) ⊕ (x5 ∨ x9 ∨ x10) ⊕ (x6 ∨ x7 ∨ x8) ⊕ (x6 ∨ x7 ∨ x9) ⊕ (x6 ∨ x7 ∨ x10) ⊕ (x6 ∨ x8 ∨ x9) ⊕ (x6 ∨ x8 ∨ x10) ⊕ (x6 ∨ x9 ∨ x10) ⊕ (x7 ∨ x8 ∨ x9) ⊕ (x7 ∨ x8 ∨ x10) ⊕ (x7 ∨ x9 ∨ x10) ⊕ (x8 ∨ x9 ∨ x10)

Here ∨ denotes logical OR, and ⊕ denotes exclusive OR (XOR).

Given the values of x1, x2, ..., x10, compute the value of f(x1, ..., x10).

Input

The input contains 10 numbers x1, x2, x3, x4, x5, x6, x7, x8, x9, x10. Each of them is either 0 or 1.

Output

Output a single value — f(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10).

Examples8

Example 1
Input
```
1 0 0 1 0 0 1 0 0 1
```
Expected output
```
0
```
Example 2
Input
```
0 0 0 0 0 0 0 0 0 0
```
Expected output
```
0
```
Example 3
Input
```
1 1 1 1 1 1 1 1 1 1
```
Expected output
```
1
```
Example 4
Input
```
1 0 0 0 0 0 0 0 0 0
```
Expected output
```
1
```
Example 5
Input
```
1 1 1 1 1 0 1 1 1 1
```
Expected output
```
1
```
Example 6
Input
```
0 0 1 1 1 1 1 1 1 1
```
Expected output
```
0
```
Example 7
Input
```
0 0 0 1 1 1 1 1 1 1
```
Expected output
```
1
```
Example 8
Input
```
1 0 1 0 1 0 1 0 1 0
```
Expected output
```
1
```