Party Lamps

This is a explanation of this problem from USACO's training website. I have converted it to markdown. Please do not just copy code; you will not learn anything; at least type it out and understand so you can do it yourself in the future!

To brighten up the gala dinner of the IOI’98 we have a set of N (10 <= N <= 100) colored lamps numbered from 1 to N.

The lamps are connected to four buttons:

Button 1: When this button is pressed, all the lamps change their state: those that are ON are turned OFF and those that are OFF are turned ON.
Button 2: Changes the state of all the odd numbered lamps.
Button 3: Changes the state of all the even numbered lamps.
Button 4: Changes the state of the lamps whose number is of the form 3xK+1 (with K>=0), i.e., 1,4,7,…

A counter C records the total number of button presses.

When the party starts, all the lamps are ON and the counter C is set to zero.

You are given the value of counter C (0 <= C <= 10000) and the final state of some of the lamps after some operations have been executed. Write a program to determine all the possible final configurations of the N lamps that are consistent with the given information, without repetitions.

PROGRAM NAME: lamps

INPUT FORMAT

No lamp will be listed twice in the input.

Line 1:	N
Line 2:	Final value of C
Line 3:	Some lamp numbers ON in the final configuration, separated by one space and terminated by the integer -1.
Line 4:	Some lamp numbers OFF in the final configuration, separated by one space and terminated by the integer -1.

SAMPLE INPUT (file lamps.in)

In this case, there are 10 lamps and only one button has been pressed. Lamp 7 is OFF in the final configuration.

OUTPUT FORMAT

Lines with all the possible final configurations (without repetitions) of all the lamps. Each line has N characters, where the first character represents the state of lamp 1 and the last character represents the state of lamp N. A 0 (zero) stands for a lamp that is OFF, and a 1 (one) stands for a lamp that is ON. The lines must be ordered from least to largest (as binary numbers).

If there are no possible configurations, output a single line with the single word ‘IMPOSSIBLE’

SAMPLE OUTPUT (file lamps.out)

0000000000
0101010101
0110110110

In this case, there are three possible final configurations:

All lamps are OFF
Lamps 1, 3, 5, 7, 9 are OFF and lamps 2, 4, 6, 8, 10 are ON.
Lamps 1, 4, 7, 10 are OFF and lamps 2, 3, 5, 6, 8, 9 are ON.

CODE

Java

C++

Pascal

ANALYSIS

Russ Cox

There are a number of insights required for this problem.

The main insight is that no matter which switches get pressed, the light pattern will repeat every six lamps. Another insight is that the order of switch presses does not matter, and that pressing a switch twice is the same as not pressing a switch at all.

Any even number of switch presses greater than four might as well just be four switch presses, and and any odd number greater than three might as well just be three presses.

We can treat the light information as describing only the first six lamps (by treating lamp n as lamp (n mod 6)), and only try pressing each switch once or not at all.

To maintain the actual information about which lights are on, we use an integer holding six bits. The least significant bit is light 6, the next least is light 5, and so on. This has the effect that treating these bit strings as normal numbers sorts the same way that we need to print them.

Switches are represented as bit vectors that say which lights get toggled.

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <assert.h>

#define MAXLAMP	6
#define LAMPMASK	((1<<MAXLAMP)-1)

int nlamp;
int nswitch;
int ison;
int known;

int poss[1<<MAXLAMP];

int flip[4] = {
    LAMPMASK,		/* flip all lights */
    LAMPMASK & 0xAA, 	/* flip odd lights */
    LAMPMASK & 0x55,	/* flip even lights */
    LAMPMASK & ((1<<(MAXLAMP-1))|(1<<(MAXLAMP-4)))	/* lights 1, 4 */
};

/*
 * Starting with current light state ``lights'', flip exactly n switches
 * with number >= i.
 */
void
search(int lights, int i, int n)
{
    if(n == 0) {
	if((lights & known) == ison)
	    poss[lights] = 1;
	return;
    }

    for(; i<4; i++)
	search(lights ^ flip[i], i+1, n-1);
}

void
printseq(FILE *fout, int lights)
{
    int i;
    char s[100+1];

    for(i=0; i<nlamp; i++)
	s[i] = (lights & (1<<(MAXLAMP-1 - i%MAXLAMP))) ? '1' : '0';
    s[nlamp] = '\0';
    fprintf(fout, "%s\n", s);
}

void
main(void)
{
    FILE *fin, *fout;
    int a, i, impossible;

    fin = fopen("lamps.in", "r");
    fout = fopen("lamps.out", "w");
    assert(fin != NULL && fout != NULL);

    fscanf(fin, "%d %d", &nlamp, &nswitch);

    for(;;) {
	fscanf(fin, "%d", &a);
	if(a == -1)
	    break;
	a = MAXLAMP-1 - (a-1) % MAXLAMP;
	ison |= 1<<a;
	known |= 1<<a;
    }

    for(;;) {
	fscanf(fin, "%d", &a);
	if(a == -1)
	    break;
	a = MAXLAMP-1 - (a-1) % MAXLAMP;
	assert((ison & (1<<a)) == 0);
	known |= 1<<a;
    }

    if(nswitch > 4)
	if(nswitch%2 == 0)
	    nswitch = 4;
	else
	    nswitch = 3;

    for(; nswitch >= 0; nswitch -= 2)
	    search(LAMPMASK, 0, nswitch);

    impossible = 1;
    for(i=0; i<(1<<MAXLAMP); i++) {
	if(poss[i]) {
	    printseq(fout, i);
	    impossible = 0;
	}
    }
    if(impossible)
	fprintf(fout, "IMPOSSIBLE\n");

    exit(0);
}