Chess Triangle
Description
Consider a bishop, a knight and a rook on an n × m chessboard. They are said to form a triangle if each piece attacks exactly one other piece and is attacked by exactly one piece. Calculate the number of ways to choose positions of the pieces to form a triangle.
Note that the bishop attacks pieces sharing the common diagonal with it; the rook attacks in horizontal and vertical directions; and, finally, the knight attacks squares which are two squares horizontally and one square vertically, or two squares vertically and one square horizontally away from its position.
Example
For n = 2 and m = 3, the output should be
chessTriangle(n, m) = 8.
Input/Output
- [execution time limit] 4 seconds (js)
-
[input] integer n
Guaranteed constraints:
1 ≤ n ≤ 40. -
[input] integer m
Guaranteed constraints:
1 ≤ m ≤ 40,
3 ≤ n · m. - [output] integer
[JavaScript (ES6)] Syntax Tips
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// Prints help message to the console
// Returns a string
function helloWorld(name) {
console.log("This prints to the console when you Run Tests");
return "Hello, " + name;
}
Solution
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function chessTriangle(n, m) {
return (
times(n, m, 2, 3) +
times(n, m, 3, 3) +
times(n, m, 2, 4) +
times(n, m, 3, 4)
);
}
function ways(n, m, x, y) {
if (n < x || m < y) return 0;
return (n - x + 1) * (m - y + 1);
}
function times(n, m, x, y) {
return (ways(n, m, x, y) + ways(m, n, x, y)) * 8;
}