Möbius function
The classical Möbius function: μ(n) is an important multiplicative function in number theory and combinatorics.
There are several ways to implement a Möbius function.
A fairly straightforward method is to find the prime factors of a positive integer n, then define μ(n) based on the sum of the primitive factors. It has the values {−1, 0, 1} depending on the factorization of n:
- μ(1) is defined to be 1.
- μ(n) = 1 if n is a square-free positive integer with an even number of prime factors.
- μ(n) = −1 if n is a square-free positive integer with an odd number of prime factors.
- μ(n) = 0 if n has a squared prime factor.
- Task
- Write a routine (function, procedure, whatever) μ(n) to find the Möbius number for a positive integer n.
- Use that routine to find and display here, on this page, at least the first 99 terms in a grid layout. (Not just one long line or column of numbers.)
- See also
- Related Tasks
AWK
<lang AWK>
- syntax: GAWK -f MOBIUS_FUNCTION.AWK
- converted from Java
BEGIN {
printf("first 199 terms of the mobius sequence:\n ")
for (n=1; n<200; n++) {
printf("%3d",mobius(n))
if ((n+1) % 20 == 0) {
printf("\n")
}
}
exit(0)
} function mobius(n, i,j,mu_max) {
if (n in MU) {
return(MU[n])
}
mu_max = 1000000
for (i=0; i<mu_max; i++) { # populate array
MU[i] = 1
}
for (i=2; i<=int(sqrt(mu_max)); i++ ) {
if (MU[i] == 1) {
for (j=i; j<=mu_max; j+=i) { # for each factor found, swap + and -
MU[j] *= -i
}
for (j=i*i; j<=mu_max; j+=i*i) { # square factor = 0
MU[j] = 0
}
}
}
for (i=2; i<=mu_max; i++) {
if (MU[i] == i) {
MU[i] = 1
}
else if (MU[i] == -i) {
MU[i] = -1
}
else if (MU[i] < 0) {
MU[i] = 1
}
else if (MU[i] > 0) {
MU[i] = -1
}
}
return(MU[n])
} </lang>
- Output:
first 199 terms of the mobius sequence:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1
0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1
0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1
0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1
0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0
0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1
0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1
0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1
0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1
0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
Factor
The mobius word exists in the math.extras vocabulary. See the implementation here.
<lang factor>USING: formatting grouping io math.extras math.ranges sequences ;
"First 199 terms of the Möbius sequence:" print 199 [1,b] [ mobius ] map " " prefix 20 group [ [ "%3s" printf ] each nl ] each</lang>
- Output:
First 199 terms of the Möbius sequence:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1
0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1
0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1
0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1
0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0
0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1
0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1
0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1
0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1
0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
Go
<lang go>package main
import "fmt"
func möbius(to int) []int {
if to < 1 {
to = 1
}
mobs := make([]int, to+1) // all zero by default
primes := []int{2}
for i := 1; i <= to; i++ {
j := i
cp := 0 // counts prime factors
spf := false // true if there is a square prime factor
for _, p := range primes {
if p > j {
break
}
if j%p == 0 {
j /= p
cp++
}
if j%p == 0 {
spf = true
break
}
}
if cp == 0 && i > 2 {
cp = 1
primes = append(primes, i)
}
if !spf {
if cp%2 == 0 {
mobs[i] = 1
} else {
mobs[i] = -1
}
}
}
return mobs
}
func main() {
mobs := möbius(199)
fmt.Println("Möbius sequence - First 199 terms:")
for i := 0; i < 200; i++ {
if i == 0 {
fmt.Print(" ")
continue
}
if i%20 == 0 {
fmt.Println()
}
fmt.Printf(" % d", mobs[i])
}
}</lang>
- Output:
Möbius sequence - First 199 terms:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1
0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1
0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1
0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1
0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0
0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1
0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1
0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1
0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1
0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
Java
<lang java> public class MöbiusFunction {
public static void main(String[] args) {
System.out.printf("First 199 terms of the möbius function are as follows:%n ");
for ( int n = 1 ; n < 200 ; n++ ) {
System.out.printf("%2d ", möbiusFunction(n));
if ( (n+1) % 20 == 0 ) {
System.out.printf("%n");
}
}
}
private static int MU_MAX = 1_000_000;
private static int[] MU = null;
// Compute mobius function via sieve
private static int möbiusFunction(int n) {
if ( MU != null ) {
return MU[n];
}
// Populate array
MU = new int[MU_MAX+1];
int sqrt = (int) Math.sqrt(MU_MAX);
for ( int i = 0 ; i < MU_MAX ; i++ ) {
MU[i] = 1;
}
for ( int i = 2 ; i <= sqrt ; i++ ) {
if ( MU[i] == 1 ) {
// for each factor found, swap + and -
for ( int j = i ; j <= MU_MAX ; j += i ) {
MU[j] *= -i;
}
// square factor = 0
for ( int j = i*i ; j <= MU_MAX ; j += i*i ) {
MU[j] = 0;
}
}
}
for ( int i = 2 ; i <= MU_MAX ; i++ ) {
if ( MU[i] == i ) {
MU[i] = 1;
}
else if ( MU[i] == -i ) {
MU[i] = -1;
}
else if ( MU[i] < 0 ) {
MU[i] = 1;
}
else if ( MU[i] > 0 ) {
MU[i] = -1;
}
}
return MU[n];
}
} </lang>
- Output:
First 199 terms of the möbius function are as follows:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1
0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1
0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1
0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1
0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0
0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1
0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1
0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1
0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1
0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
Julia
<lang julia>using Primes
- modified from reinermartin's PR at https://github.com/JuliaMath/Primes.jl/pull/70/files
function moebius(n::Integer)
@assert n > 0 m(p, e) = p == 0 ? 0 : e == 1 ? -1 : 0 reduce(*, m(p, e) for (p, e) in factor(n) if p ≥ 0; init=1)
end μ(n) = moebius(n)
print("First 199 terms of the Möbius sequence:\n ") for n in 1:199
print(lpad(μ(n), 3), n % 20 == 19 ? "\n" : "")
end
</lang>
- Output:
First 199 terms of the Möbius sequence:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1
0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1
0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1
0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1
0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0
0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1
0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1
0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1
0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1
0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
Pascal
See Mertens_function#Pascal
Perl
<lang perl>use utf8; use strict; use warnings; use feature 'say'; use List::Util 'uniq';
sub prime_factors {
my ($n, $d, @factors) = (shift, 1);
while ($n > 1 and $d++) {
$n /= $d, push @factors, $d until $n % $d;
}
@factors
}
sub μ {
my @p = prime_factors(shift); @p == uniq(@p) ? 0 == @p%2 ? 1 : -1 : 0;
}
my @möebius; push @möebius, μ($_) for 1 .. (my $upto = 199);
say "Möbius sequence - First $upto terms:\n" .
(' 'x4 . sprintf "@{['%4d' x $upto]}", @möebius) =~ s/((.){80})/$1\n/gr;</lang>
- Output:
Möbius sequence - First 199 terms:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1
0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1
0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1
0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1
0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0
0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1
0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1
0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1
0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1
0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
Phix
<lang Phix>function Moebius(integer n)
if n=1 then return 1 end if
sequence f = prime_factors(n,true)
for i=2 to length(f) do
if f[i] = f[i-1] then return 0 end if
end for
return iff(and_bits(length(f),1)?-1:+1)
end function
sequence s = {" ."} for i=1 to 199 do s = append(s,sprintf("%3d",Moebius(i))) end for puts(1,join_by(s,1,20," "))</lang>
- Output:
. 1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1 0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1 0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1 0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1 0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0 0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1 0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1 0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1 0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1 0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
Raku
(formerly Perl 6)
Möbius number is not defined for n == 0. Raku arrays are indexed from 0 so store a blank value at position zero to keep n and μ(n) aligned.
<lang perl6>use Prime::Factor;
sub μ (Int \n) {
return 0 if n %% 4 or n %% 9 or n %% 25 or n %% 49 or n %% 121; my @p = prime-factors(n); +@p == +@p.unique ?? +@p %% 2 ?? 1 !! -1 !! 0
}
my @möbius = lazy flat , 1, (2..*).hyper.map: -> \n { μ(n) };
- The Task
put "Möbius sequence - First 199 terms:\n",
@möbius[^200]».fmt('%3s').batch(20).join: "\n";</lang>
- Output:
Möbius sequence - First 199 terms:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1
0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1
0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1
0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1
0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0
0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1
0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1
0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1
0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1
0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
REXX
Note that the Möbius function is also spelled Mobius and/or Moebius, and it is also known as the mu function, where mu is the Greek symbol μ.
Programming note: This REXX version supports the specifying of the low and high values to be generated,
as well as the group size for the grid (it can be specified as 1 which will show a vertical list).
A null value will be shown as a bullet (•) when showing the Möbius value of for zero (this can be changed in the 2nd line of the mobius function).
The above "feature" was added to make the grid to be aligned with other solutions.
The function to computer some prime numbers is a bit of an overkill, but I wanted to keep in general (in case of larger/higher ranges for a Möbius sequence). <lang rexx>/*REXX pgm computes & shows a value grid of the Möbius function for a range of integers.*/ parse arg LO HI grp . /*obtain optional arguments from the CL*/ if LO== | LO=="," then LO= 0 /*Not specified? Then use the default.*/ if HI== | HI=="," then HI= 199 /* " " " " " " */ if grp== | grp=="," then grp= 20 /* " " " " " " */
/* ______ */
call genP HI /*generate primes up to the √ HI */ say center(' The Möbius sequence from ' LO " ──► " HI" ", max(50, grp*3), '═') /*title*/ $= /*variable holds output grid of GRP #s.*/
do j=LO to HI; $= $ right( mobius(j), 2) /*process some numbers from LO ──► HI.*/
if words($)==grp then do; say substr($, 2); $= /*show grid if fully populated,*/
end /* and nullify it for more #s.*/
end /*j*/ /*for small grids, using wordCnt is OK.*/
if $\== then say substr($, 2) /*handle any residual numbers not shown*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ mobius: procedure expose @.; parse arg x /*obtain a integer to be tested for mu.*/
if x<1 then return '∙' /*special? Then return symbol for null.*/
#= 0 /*start with a value of zero. */
do k=1; p= @.k /*get the Kth (pre─generated) prime.*/
if p>x then leave /*prime (P) > X? Then we're done. */
if p*p>x then do; #= #+1; leave /*prime (P**2 > X? Bump # and leave.*/
end
if x//p==0 then do; #= #+1 /*X divisible by P? Bump mu number. */
x= x % p /* Divide by prime. */
if x//p==0 then return 0 /*X÷by P? Then return zero*/
end
end /*k*/ /*# (below) is almost always small, <9*/
if #//2==0 then return 1 /*Is # even? Then return postive 1 */
return -1 /* " " odd? " " negative 1. */
/*──────────────────────────────────────────────────────────────────────────────────────*/ genP: @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6= 13; nP=6 /*assign low primes; # primes.*/
do lim=nP until lim*lim>=HI /*only keep primes up to the sqrt(HI).*/
end /*lim*/
do j=@.nP+4 by 2 to HI /*only find odd primes from here on. */
parse var j -1 _;if _==5 then iterate /*Is last digit a "5"? Then not prime*/
if j// 3==0 then iterate /*is J divisible by 3? " " " */
if j// 7==0 then iterate /* " " " " 7? " " " */
if j//11==0 then iterate /* " " " " 11? " " " */
if j//13==0 then iterate /* " " " " 13? " " " */
do k=7 while k*k<=j /*divide by some generated odd primes. */
if j // @.k==0 then iterate j /*Is J divisible by P? Then not prime*/
end /*k*/ /* [↓] a prime (J) has been found. */
nP= nP+1; if nP<=HI then @.nP= j /*bump prime count; assign prime to @.*/
end /*j*/; return</lang>
- output when using the default inputs:
Output note: note the use of a bullet (•) to signify that a "null" is being shown (for the 0th entry).
══════════ The Möbius sequence from 0 ──► 199 ═══════════ ∙ 1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1 0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1 0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1 0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1 0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0 0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1 0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1 0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1 0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1 0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
Sidef
Built-in:
<lang ruby>say moebius(53) #=> -1 say moebius(54) #=> 0 say moebius(55) #=> 1</lang>
Simple implementation: <lang ruby>func μ(n) {
var f = n.factor_exp.map { .tail }
f.any { _ > 1 } ? 0 : ((-1)**f.sum)
}
with (199) { |n|
say "Values of the Möbius function for numbers in the range 1..#{n}:"
[' '] + (1..n->map(μ)) -> each_slice(20, {|*line|
say line.map { '%2s' % _ }.join(' ')
})
}</lang>
- Output:
Values of the Möbius function for numbers in the range 1..199:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1
0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1
0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1
0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1
0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0
0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1
0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1
0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1
0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1
0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1
zkl
<lang zkl>fcn mobius(n){
pf:=primeFactors(n);
sq:=pf.filter1('wrap(f){ (n % (f*f))==0 }); // False if square free
if(sq==False){ if(pf.len().isEven) 1 else -1 }
else 0
} fcn primeFactors(n){ // Return a list of prime factors of n
acc:=fcn(n,k,acc,maxD){ // k is 2,3,5,7,9,... not optimum
if(n==1 or k>maxD) acc.close();
else{
q,r:=n.divr(k); // divr-->(quotient,remainder) if(r==0) return(self.fcn(q,k,acc.write(k),q.toFloat().sqrt())); return(self.fcn(n,k+1+k.isOdd,acc,maxD)) # both are tail recursion
}
}(n,2,Sink(List),n.toFloat().sqrt());
m:=acc.reduce('*,1); // mulitply factors
if(n!=m) acc.append(n/m); // opps, missed last factor
else acc;
}</lang> <lang zkl>[1..199].apply(mobius) .pump(Console.println, T(Void.Read,19,False), fcn{ vm.arglist.pump(String,"%3d".fmt) });</lang>
- Output:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1 0 1 1 -1 0 0 1 0 0 -1 -1 -1 0 1 1 1 0 -1 1 1 0 -1 -1 -1 0 0 1 -1 0 0 0 1 0 -1 0 1 0 1 1 -1 0 -1 1 0 0 1 -1 -1 0 1 -1 -1 0 -1 1 0 0 1 -1 -1 0 0 1 -1 0 1 1 1 0 -1 0 1 0 1 1 1 0 -1 0 0 0 -1 -1 -1 0 -1 1 -1 0 -1 -1 1 0 -1 -1 1 0 0 1 1 0 0 1 1 0 0 0 -1 0 1 -1 -1 0 1 1 0 0 -1 -1 -1 0 1 1 1 0 1 1 0 0 -1 0 -1 0 0 -1 1 0 -1 1 1 0 1 0 -1 0 -1 1 -1 0 0 -1 0 0 -1 -1 0 0 1 1 -1 0 -1 -1 1 0 1 -1 1 0 0 -1 -1 0 -1 1 -1 0 -1 0 -1