SwitchFlipping.java

public class SwitchFlipping {
    public static void main (String[] args) {
        doSlowWay();
        doFastWay();
    }

    private static boolean isSquare (int i) {
        int s = (int)Math.sqrt(i);
        return (s*s) == i;
    }

    private static void doFastWay() {
        for (int i = 0; i < 100; i++) {
            if (isSquare(i)) {
                System.out.println("Switch " + i + " is on");
            }
         }
    }

    private static void doSlowWay() {
        boolean[] switches = new boolean[100];
        for (int i = 1; i < switches.length; i++) {
            for (int j = 0; j < switches.length; j += i) {
                switches[j] = !switches[j];
            }
        }
        for (int i = 0; i < switches.length; i++) {
            if (switches[i]) {
                System.out.println("Switch " + i + " is on");
            }
        }
    }
}

Flipping switches or opening locker doors.
Open all the doors/flip all the switches, then do every 2nd, 3rd, 4th, etc. Which doors are still open/switches still on at the end.
Square numbers have an odd number of factors; 9 -> 1, 3, 9
Not square number have an even number of factors; 10 -> 1, 2, 5, 10
Square numbers will consequently still be on/open at the end of the process while all other numbers will have closed/turned off.