Lazy caterer's sequence, Timus Online Judge: 1209 - 1, 10, 100, 1000...

1209-lzy.cpp

#include<iostream>
#include<cstdio>
#include<cmath>

using namespace std;

int main(){

	long n;
	cin >> n;

	while(n--){
		long double p, test;
		cin >> p;	
		
		test = sqrt(8*p - 7);
		if (test - floor(test) == 0)
			cout << 1 << ' ';
		else
			cout << 0 << ' ';
	}
	
}


// This actually is disguised as a problem which uses the Lazy caterer's sequence -- 
// central polygonal numbers, describes the maximum number of pieces of a disk 
// (a pancake or pizza is usually used to describe the situation) that can be made 
// with a given number of straight cuts. https://en.wikipedia.org/wiki/Lazy_caterer%27s_sequence


// 1 + n(n+1)/2 = input (p)
// solving if sqrt(8p-7) is perfect it's true, else false.

// AC in 1 try.

Why - 7?

Solve for $n$ in $n^2+n + 2(1- p) = 0$ using quadratic formula ( $-b \pm \sqrt{b^2-4ac} / 2a$ )

Got it. Thanks.