finalized

5 files changed, 21 insertions, 683 deletions

diff --git a/challenge-123/cheok-yin-fung/blog.txt b/challenge-123/cheok-yin-fung/blog.txt
new file mode 100644
index 0000000000..10c25304a6
--- /dev/null
+++ b/challenge-123/cheok-yin-fung/blog.txt
@@ -0,0 +1 @@
+https://e7-87-83.github.io/coding/challenge_123.html
diff --git a/challenge-123/cheok-yin-fung/perl/ch-2x.pl b/challenge-123/cheok-yin-fung/perl/ch-2-cube-hypercube.pl
index 968afdfad9..a70526e819 100644
--- a/challenge-123/cheok-yin-fung/perl/ch-2x.pl
+++ b/challenge-123/cheok-yin-fung/perl/ch-2-cube-hypercube.pl
@@ -16,7 +16,7 @@ use Algorithm::Combinatorics qw(permutations);  #use for hypercube
 my $k = $ARGV[0] || 3;
 my $D = $ARGV[1] || $k;
 
-die "Usage: ch-2x.pl [2, 3 or 4] (optional)[dimension of space] " 
+die "Usage: ch-2-xxxxxxxxxx.pl [2, 3 or 4] (optional)[dimension of space] " 
     if $k > 4 or $k <= 1;
 die "How can I put a $k-polytope into $D-dim space? \n" if $k > $D;
 
@@ -51,30 +51,15 @@ sub is_cube {
     my $NW = vec_sum($N, $W);
     my $WU = vec_sum($W, $U);
     my $UN = vec_sum($U, $N);
+    my $iter_face = permutations([$NW, $WU, $UN]);
     my $bool = undef;
-    if (vec_same_f($NW, $v{$ind[3]})) {
-      if   (  vec_same_f($WU, $v{$ind[4]}) 
-          &&  vec_same_f($UN, $v{$ind[5]}) ) { $bool = 1;
-      } elsif ( vec_same_f($WU, $v{$ind[5]}) 
-           &&   vec_same_f($UN, $v{$ind[4]}) ) { $bool = 1;
-      }
-    } 
-    if (!$bool && vec_same_f($NW, $v{$ind[4]})) {
-      if   (  vec_same_f($WU, $v{$ind[3]}) 
-          &&  vec_same_f($UN, $v{$ind[5]}) ) { $bool = 1;
-      } elsif ( vec_same_f($WU, $v{$ind[5]}) 
-           &&   vec_same_f($UN, $v{$ind[3]}) ) { $bool = 1;
-      } 
-    }
-    if (!$bool && vec_same_f($NW, $v{$ind[5]})) {
-      if   (  vec_same_f($WU, $v{$ind[4]}) 
-          &&  vec_same_f($UN, $v{$ind[3]}) ) { $bool = 1;
-      } elsif ( vec_same_f($WU, $v{$ind[3]}) 
-           &&   vec_same_f($UN, $v{$ind[4]}) ) { $bool = 1;
-      } else {
-        return 0;
-      }
+    while (!$bool && (my $p = $iter_face->next)) {
+        $bool = 
+            vec_same_f($v{$ind[3]}, $p->[0]) &&
+            vec_same_f($v{$ind[4]}, $p->[1]) &&
+            vec_same_f($v{$ind[5]}, $p->[2]) ; 
     }
+
     return 0 if !$bool;
 
     my $NWU = vec_sum( $N, $WU);
@@ -185,7 +170,7 @@ sub vec_sum {
 sub vec_same {
     my $first = $_[0];
     my $second = $_[1];
-    warn "Not the same dimension in vec_same_f \n" if $first->$#* != $second->$#*;
+    warn "Not the same dimension in vec_same\n" if $first->$#* != $second->$#*;
     for my $s (0..$first->$#*) {
         return undef if $first->[$s] != $second->[$s];
     }
@@ -195,7 +180,7 @@ sub vec_same {
 sub vec_same_f {
     my $first = $_[0];
     my $second = $_[1];
-    warn "Not the same dimension in vec_same_f \n" if $first->$#* != $second->$#*;
+    warn "Not the same dimension in vec_same_f\n" if $first->$#* != $second->$#*;
     for my $s (0..$first->$#*) {
         return undef if sprintf("%f",$first->[$s]) != sprintf("%f",$second->[$s]);
     }
@@ -271,7 +256,7 @@ ok is_square(
               == 1, 
              "arctan(1/5) by atan2() of a much smaller size"
             ."(multiple by 0.0009), caught by the equalities"
-            ."(_\"not ok\" is normal_)";  
+            ."(_\"not ok\" is not rare_)";  
 
 ok is_square( [1, 2] , [4,3], [3,1], [2,4] ) == 1, "Knight's square";
 ok is_square( [1, 1] , [-1, 1], [1,-1], [-1,-1] ) == 1, "centre at origin";
diff --git a/challenge-123/cheok-yin-fung/perl/ch-2.pl b/challenge-123/cheok-yin-fung/perl/ch-2.pl
index 7421f86fc7..c39cba05e1 100644
--- a/challenge-123/cheok-yin-fung/perl/ch-2.pl
+++ b/challenge-123/cheok-yin-fung/perl/ch-2.pl
@@ -12,6 +12,9 @@ use Test::More tests => 13;
 
 my $D = $ARGV[0] || 2;
 
+say "Input coordinates of 4 points in $D dimensional space:";
+say "a point per line.";
+
 my $pt0 = [split " ", <STDIN>];
 my $pt1 = [split " ", <STDIN>];
 my $pt2 = [split " ", <STDIN>];
@@ -42,7 +45,10 @@ sub is_square {
 #       times the edge length would not be a necessary check 
 #       if we preserve the dot product test, because in
 #       Euclidean space, if two vectors are orthogonal and in equal length,
-#       we can apply the Pythagorean theorem.
+#       we can apply the Pythagorean theorem for the norm of the vector sum.
+#       For DEMOSTRATION PURPOSE, 
+#       the test of orthogonality (the dot product test) is commented out by brace,
+#       but the dot product test will be used both for cube and hypercube.
          ) { 
         return 1;
     } 
@@ -69,27 +75,6 @@ sub norm {
     return $sum;
 }
 
-sub vec_sum {
-    my $first = $_[0];
-    my $second = $_[1];
-    my $ans = [];
-    warn "Not the same dimension in vec_sum \n" if $first->$#* != $second->$#*;
-    for my $s (0..$first->$#*) {
-        push $ans->@*, $first->[$s] + $second->[$s];
-    }
-    return $ans;
-}
-
-sub vec_same {
-    my $first = $_[0];
-    my $second = $_[1];
-    warn "Not the same dimension in vec_same \n" if $first->$#* != $second->$#*;
-    for my $s (0..$first->$#*) {
-        return 0 if $first->[$s] != $second->[$s];
-    }
-    return 1;
-}
-
 sub vec_subtract {
     my $first = $_[0];
     my $second = $_[1];
@@ -106,6 +91,8 @@ ok is_square( [10,20], [20,20], [20, 10], [10, 10] ) == 1, "Example 1";
 ok is_square( [12,24], [16,10], [20, 12], [18, 16] ) == 0, "Example 2";
 ok is_square( [1, 2] , [4,3], [3,1], [2,4] ) == 1, "Knight's square";
 ok is_square( [1, 1] , [-1, 1], [ 1,-1], [-1,-1] ) == 1, "centre at origin";
+
+# =========== test cases with irrational numbers ==============
 ok is_square( [1, sqrt(3)/2, -1/2], [1, -sqrt(3)/2, 1/2], 
               [-1, sqrt(3)/2, -1/2], [-1, -sqrt(3)/2, 1/2] ) == 1, 
              "centre at origin, inclined";
diff --git a/challenge-123/cheok-yin-fung/perl/ch-2a.pl b/challenge-123/cheok-yin-fung/perl/ch-2a.pl
deleted file mode 100644
index e257c351dd..0000000000
--- a/challenge-123/cheok-yin-fung/perl/ch-2a.pl
+++ /dev/null
@@ -1,312 +0,0 @@
-#!/usr/bin/perl
-# The Weekly Challenge 123
-# Task 2 extension: Square/Cube/Hypercube Points
-# Usage: $ ch-2a.pl (optional) $k (optional)$D 
-# $k: 2 or 3 or 4, stands for square or cube or hypercube, default is 3
-# $D: 2 or above, cannot be smaller than $k, default is same as $k
-
-# Note: check out $ diff ch-2a.pl ch-2ax.pl
-# ALSO: ch-2x.pl is the best implementation.
-
-use strict;
-use warnings;
-use v5.10.0;
-use Test::More tests => 14; 
-
-use Algorithm::Combinatorics qw(permutations);  #use for hypercube
-
-
-my $k = $ARGV[0] || 3;
-my $D = $ARGV[1] || $k;
-
-die "Usage: ch-2a.pl [2, 3 or 4] (optional)[dimension of space] " 
-    if $k > 4 or $k <= 1;
-die "How can I put a $k-polytope into $D-dim space? \n" if $k > $D;
-
-
-sub is_square {
-    my ($p0,$p1,$p2,$p3) = @_;
-    my $v0 = vec_subtract($p0, $p1);
-    my $v1 = vec_subtract($p0, $p2);
-    my $v2 = vec_subtract($p0, $p3);
-    return 0 unless (vec_prod_f($v1, $v2) == 0) xor
-                    (vec_prod_f($v0, $v2) == 0) xor
-                    (vec_prod_f($v0, $v1) == 0);
-#========== BEGIN: diff of ch-2a.pl and ch-2ax.pl =========
-    return 0 unless vec_same($v0, vec_sum($v1, $v2) ) xor
-                    vec_same($v1, vec_sum($v2, $v0) ) xor
-                    vec_same($v2, vec_sum($v0, $v1) );
-#=========== END: diff of ch-2a.pl and ch-2ax.pl ==========
-# COMMENT:
-#  this test is mathematically NOT necessary, 
-#  and if you check it against ch-2ax.pl,
-#  you will find this section is the source of failed tests!
-    my @n_vector = map { norm_f($_) } ($v0, $v1, $v2);
-    @n_vector = sort {$a<=>$b} @n_vector;
-    if ( $n_vector[0] == $n_vector[1] ) { 
-        return 1;
-    } 
-    else {
-        return 0;
-    }
-}
-
-sub is_cube {
-    my @p = @_;
-    my %v;
-    $v{$_} = vec_subtract($p[0], $p[$_]) for (1..7);
-    my @ind = sort { norm_f($v{$a}) <=> norm_f($v{$b}) } keys %v;
-    my ($N, $W, $U) = ($v{$ind[0]} , $v{$ind[1]} , $v{$ind[2]}) ;
-    return 0 unless norm_f($N) == norm_f($W) && norm_f($N) == norm_f($U);
-    return 0 unless vec_prod_f($N,$W) == 0 && vec_prod_f($W,$U) == 0 
-                        && vec_prod_f($U,$N) == 0;
-    my $NW = vec_sum($N, $W);
-    my $WU = vec_sum($W, $U);
-    my $UN = vec_sum($U, $N);
-    my $bool = undef;
-    if (vec_same($NW, $v{$ind[3]})) {
-      if   (  vec_same($WU, $v{$ind[4]}) 
-          &&  vec_same($UN, $v{$ind[5]}) ) { $bool = 1;
-      } elsif ( vec_same($WU, $v{$ind[5]}) 
-           &&   vec_same($UN, $v{$ind[4]}) ) { $bool = 1;
-      }
-    } 
-    if (!$bool && vec_same($NW, $v{$ind[4]})) {
-      if   (  vec_same($WU, $v{$ind[3]}) 
-          &&  vec_same($UN, $v{$ind[5]}) ) { $bool = 1;
-      } elsif ( vec_same($WU, $v{$ind[5]}) 
-           &&   vec_same($UN, $v{$ind[3]}) ) { $bool = 1;
-      } 
-    }
-    if (!$bool && vec_same($NW, $v{$ind[5]})) {
-      if   (  vec_same($WU, $v{$ind[4]}) 
-          &&  vec_same($UN, $v{$ind[3]}) ) { $bool = 1;
-      } elsif ( vec_same($WU, $v{$ind[3]}) 
-           &&   vec_same($UN, $v{$ind[4]}) ) { $bool = 1;
-      } else {
-        return 0;
-      }
-    }
-    return 0 if !$bool;
-
-    my $NWU = vec_sum( $N, $WU);
-    if ( vec_same( $v{$ind[6]} , $NWU ) ) {
-        return 1;
-    }
-    else {
-        return 0;
-    }
-}
-
-sub is_hypercube {
-    my @p = @_;
-    my %v;
-    $v{$_} = vec_subtract($p[0], $p[$_]) for (1..15);
-    my @ind = sort { norm_f($v{$a}) <=> norm_f($v{$b}) } keys %v;
-    my ($N, $W, $U, $A) = ( $v{$ind[0]}, $v{$ind[1]} , 
-                            $v{$ind[2]}, $v{$ind[3]} );
-    return 0 unless 
-        norm_f($N) == norm_f($W) && norm_f($W) == norm_f($U)
-                             && norm_f($U) == norm_f($A);
-    return 0 unless 
-        vec_prod_f($N, $W) == 0 &&
-        vec_prod_f($N, $U) == 0 &&
-        vec_prod_f($N, $A) == 0 &&
-        vec_prod_f($A, $W) == 0 &&
-        vec_prod_f($A, $U) == 0 &&
-        vec_prod_f($W, $U) == 0 ;
-
-    my $AU = vec_sum($A, $U);
-    my $AW = vec_sum($A, $W);
-    my $AN = vec_sum($A, $N);
-    my $NW = vec_sum($N, $W);
-    my $WU = vec_sum($W, $U);
-    my $UN = vec_sum($U, $N);
-    my $bool_face = undef;
-    my $iter_face = permutations([$AU, $UN, $NW, $WU, $AW, $AN]);
-    while (!$bool_face && (my $p = $iter_face->next)) {
-        $bool_face = 
-            vec_same($v{$ind[4]}, $p->[0]) &&
-            vec_same($v{$ind[5]}, $p->[1]) &&
-            vec_same($v{$ind[6]}, $p->[2]) &&
-            vec_same($v{$ind[7]}, $p->[3]) &&
-            vec_same($v{$ind[8]}, $p->[4]) &&
-            vec_same($v{$ind[9]}, $p->[5]) ;
-    }
-    return 0 if !$bool_face;
-
-    my $UNW = vec_sum($UN, $W);
-    my $ANW = vec_sum($NW, $A);
-    my $AWU = vec_sum($WU, $A);
-    my $AUN = vec_sum($UN, $A);
-    my $bool_cube = undef;
-    my $iter_cube = permutations([$UNW, $ANW, $AWU, $AUN]);
-    while (!$bool_cube && (my $p = $iter_cube->next)) {
-        $bool_cube = 
-            vec_same($v{$ind[10]}, $p->[0]) &&
-            vec_same($v{$ind[11]}, $p->[1]) &&
-            vec_same($v{$ind[12]}, $p->[2]) &&
-            vec_same($v{$ind[13]}, $p->[3]);
-    }
-    return 0 if !$bool_cube;
-
-    my $AUNW = vec_sum($AU,$NW);
-    if ( vec_same($v{$ind[14]}, $AUNW) ) {
-        return 1;
-    }
-    else {
-        return 0;
-    }
-}
-
-sub vec_prod {
-    my $first = $_[0];
-    my $second = $_[1];
-    warn "Not the same dimension in vec_prod \n" if $first->$#* != $second->$#*;
-    my $sum = 0;
-    $sum+= ($first->[$_]*$second->[$_]) for (0..$first->$#*);
-    return $sum;
-}
-
-sub vec_prod_f {
-    return sprintf("%f", vec_prod($_[0], $_[1]));
-}
-
-sub norm {
-    my $p = $_[0];
-    my $sum = 0;
-    $sum+= ($p->[$_])**2 for (0..$p->$#*);
-    return $sum;
-}
-
-sub norm_f {
-    return sprintf("%f", norm($_[0]));
-}
-
-sub vec_sum {
-    my $first = $_[0];
-    my $second = $_[1];
-    my $ans = [];
-    warn "Not the same dimension in vec_sum \n" if $first->$#* != $second->$#*;
-    for my $s (0..$first->$#*) {
-        push $ans->@*, $first->[$s] + $second->[$s];
-    }
-    return $ans;
-}
-
-sub vec_same {
-    my $first = $_[0];
-    my $second = $_[1];
-    warn "Not the same dimension in vec_same \n" if $first->$#* != $second->$#*;
-    for my $s (0..$first->$#*) {
-        return undef if $first->[$s] != $second->[$s];
-    }
-    return 1;
-}
-
-sub vec_subtract {
-    my $first = $_[0];
-    my $second = $_[1];
-    my $ans = [];
-    warn "Not the same dimension in vec_subtract\n" if $first->$#* != $second->$#*;
-    for my $s (0..$first->$#*) {
-        push $ans->@*, $second->[$s] - $first->[$s];
-    }
-    return $ans;
-}
-
-
-
-# 9 tests
-ok is_square( [1,0], [0,1], [-1,0],[0,-1]) == 1, "on x-axis and y-axis";
-
-ok is_square( [5/sqrt(26), 1/sqrt(26)], 
-              [-1/sqrt(26), 5/sqrt(26)],
-              [-5/sqrt(26), -1/sqrt(26)], 
-              [1/sqrt(26), -5/sqrt(26)]) == 1,
-             "inclined by arctan(1/5), centre at origin"; 
-
-ok is_square( 
-              [cos(atan2(1,5)), sin(atan2(1,5))], 
-              [-sin(atan2(1,5)), cos(atan2(1,5))],
-              [-cos(atan2(1,5)), -sin(atan2(1,5))],
-              [sin(atan2(1,5)), -cos(atan2(1,5))]
-            )  
-              == 1, "arctan(1/5) by atan2(), caught by the equalities";  
-
-ok is_square( 
-              [2.7*cos(atan2(1,5)), 2.7*sin(atan2(1,5))], 
-              [-2.7*sin(atan2(1,5)), 2.7*cos(atan2(1,5))],
-              [-2.7*cos(atan2(1,5)), -2.7*sin(atan2(1,5))],
-              [2.7*sin(atan2(1,5)), -2.7*cos(atan2(1,5))]
-            )  
-              == 1, 
-             "arctan(1/5) by atan2() of larger size (multipled by 2.7)," 
-            ."caught by the equalities";  
-
-ok is_square( 
-              [2.8*cos(atan2(1,5)), 2.8*sin(atan2(1,5))], 
-              [-2.8*sin(atan2(1,5)), 2.8*cos(atan2(1,5))],
-              [-2.8*cos(atan2(1,5)), -2.8*sin(atan2(1,5))],
-              [2.8*sin(atan2(1,5)), -2.8*cos(atan2(1,5))]
-            )  
-              == 1, 
-             "arctan(1/5) by atan2() of larger size (multipled by 2.8),"
-            ."caught by the equalities";  
-
-ok is_square( 
-              [4.0*cos(atan2(1,5)), 4.0*sin(atan2(1,5))], 
-              [-4.0*sin(atan2(1,5)), 4.0*cos(atan2(1,5))],
-              [-4.0*cos(atan2(1,5)), -4.0*sin(atan2(1,5))],
-              [4.0*sin(atan2(1,5)), -4.0*cos(atan2(1,5))]
-            )  
-              == 1, 
-             "arctan(1/5) by atan2() of larger size (multipled by 4.0),"
-            ." caught by the equalities";  
-
-ok is_square( 
-              [0.0009*cos(atan2(1,5)), 0.0009*sin(atan2(1,5))], 
-              [-0.0009*sin(atan2(1,5)), 0.0009*cos(atan2(1,5))],
-              [-0.0009*cos(atan2(1,5)), -0.0009*sin(atan2(1,5))],
-              [0.0009*sin(atan2(1,5)), -0.0009*cos(atan2(1,5))]
-            )  
-              == 1, 
-             "arctan(1/5) by atan2() of a much smaller size"
-            ."(multiple by 0.0009), caught by the equalities"
-            ."(_\"not ok\" is normal_)";  
-
-ok is_square( [1, 2] , [4,3], [3,1], [2,4] ) == 1, "Knight's square";
-ok is_square( [1, 1] , [-1, 1], [1,-1], [-1,-1] ) == 1, "centre at origin";
-
-# 4 tests
-ok is_cube( [1, 1, 1], [1, 1, 0], [1, 0, 1], [1, 0, 0], 
-            [0, 1, 1], [0, 1, 0], [0, 0, 1], [0, 0, 0] ) == 1, 
-            "standard 2**3";
-ok is_cube([ -2, -2, -2], [ -2, -2,  2], [ -2,  2, -2], [ -2,  2,  2], 
-           [  2, -2, -2], [  2, -2,  2], [  2,  2, -2], [  2,  2,  2]) == 1, 
-            "centre at origin";
-ok is_cube( 
-            [-2, -2*sqrt(3), 3-2*sqrt(3)]   ,  
-            [-2, 4-2*sqrt(3), 3+2*sqrt(3)],    
-            [-2, 2*sqrt(3), -2*sqrt(3)-1 ],  
-            [-2, 4+2*sqrt(3), -1+2*sqrt(3) ], 
-            [ 6, -2*sqrt(3), 3-2*sqrt(3)],
-            [ 6, 4-2*sqrt(3), 3+2*sqrt(3)],
-            [ 6, 2*sqrt(3), -2*sqrt(3)-1],
-            [ 6, 4+2*sqrt(3), -1+2*sqrt(3)]
-           ) 
-              == 1, "a rotated cube centred at (2,2,1)";
-ok is_cube([  2,  2,  2], [  2,  3,  2], [  2,  2,  3], [  2,  4,  2], 
-           [  3,  3,  2], [  2,  2,  1], [  2,  3,  2], [  2,  7,  3]) == 0, 
-            "this is not a cube";
-
-# 1 test
-ok is_hypercube(
-                [0,0,0,0],[0,0,0,1],[0,0,1,0],
-                [0,0,1,1],[0,1,0,0],[0,1,0,1],
-                [0,1,1,0],[0,1,1,1],[1,0,0,0],
-                [1,0,0,1],[1,0,1,0],[1,0,1,1],
-                [1,1,0,0],[1,1,0,1],[1,1,1,0],
-                [1,1,1,1]
-               ) == 1, "hypercube";
-done_testing();
diff --git a/challenge-123/cheok-yin-fung/perl/ch-2ax.pl b/challenge-123/cheok-yin-fung/perl/ch-2ax.pl
deleted file mode 100644
index 5aab9c492c..0000000000
--- a/challenge-123/cheok-yin-fung/perl/ch-2ax.pl
+++ /dev/null
@@ -1,323 +0,0 @@
-#!/usr/bin/perl
-# The Weekly Challenge 123
-# Task 2 extension: Square/Cube/Hypercube Points
-# Usage: ch-2ax.pl (optional) $k (optional)$D 
-# $k: 2 or 3 or 4, stands for square or cube or hypercube, default is 3
-# $D: 2 or above, cannot be smaller than $k, default is same as $k
-
-# Note: check out $ diff ch-2a.pl ch-2ax.pl
-# ALSO: ch-2x.pl is the best implementation.
-use strict;
-use warnings;
-use v5.10.0;
-use Test::More tests => 14; 
-
-use Algorithm::Combinatorics qw(permutations);  #use for hypercube
-
-
-my $k = $ARGV[0] || 3;
-my $D = $ARGV[1] || $k;
-
-die "Usage: ch-2ax.pl [2, 3 or 4] (optional)[dimension of space] " 
-    if $k > 4 or $k <= 1;
-die "How can I put a $k-polytope into $D-dim space? \n" if $k > $D;
-
-
-sub is_square {
-    my ($p0,$p1,$p2,$p3) = @_;
-    my $v0 = vec_subtract($p0, $p1);
-    my $v1 = vec_subtract($p0, $p2);
-    my $v2 = vec_subtract($p0, $p3);
-    return 0 unless (vec_prod_f($v1, $v2) == 0) xor
-                    (vec_prod_f($v0, $v2) == 0) xor
-                    (vec_prod_f($v0, $v1) == 0);
-    my @n_vector = map { norm_f($_) } ($v0, $v1, $v2);
-    @n_vector = sort {$a<=>$b} @n_vector;
-    if ( $n_vector[0] == $n_vector[1] ) { 
-        return 1;
-    } 
-    else {
-        return 0;
-    }
-}
-
-sub is_cube {
-    my @p = @_;
-    my %v;
-    $v{$_} = vec_subtract($p[0], $p[$_]) for (1..7);
-    my @ind = sort { norm_f($v{$a}) <=> norm_f($v{$b}) } keys %v;
-    my ($N, $W, $U) = ($v{$ind[0]} , $v{$ind[1]} , $v{$ind[2]}) ;
-    return 0 unless norm_f($N) == norm_f($W) && norm_f($N) == norm_f($U);
-    return 0 unless vec_prod_f($N,$W) == 0 && vec_prod_f($W,$U) == 0 
-                        && vec_prod_f($U,$N) == 0;
-    my $NW = vec_sum($N, $W);
-    my $WU = vec_sum($W, $U);
-    my $UN = vec_sum($U, $N);
-    my $bool = undef;
-    if (vec_same($NW, $v{$ind[3]})) {
-      if   (  vec_same($WU, $v{$ind[4]}) 
-          &&  vec_same($UN, $v{$ind[5]}) ) { $bool = 1;
-      } elsif ( vec_same($WU, $v{$ind[5]}) 
-           &&   vec_same($UN, $v{$ind[4]}) ) { $bool = 1;
-      }
-    } 
-    if (!$bool && vec_same($NW, $v{$ind[4]})) {
-      if   (  vec_same($WU, $v{$ind[3]}) 
-          &&  vec_same($UN, $v{$ind[5]}) ) { $bool = 1;
-      } elsif ( vec_same($WU, $v{$ind[5]}) 
-           &&   vec_same($UN, $v{$ind[3]}) ) { $bool = 1;
-      }
-    }
-    if (!$bool && vec_same($NW, $v{$ind[5]})) {
-      if   (  vec_same($WU, $v{$ind[4]}) 
-          &&  vec_same($UN, $v{$ind[3]}) ) { $bool = 1;
-      } elsif ( vec_same($WU, $v{$ind[3]}) 
-           &&   vec_same($UN, $v{$ind[4]}) ) { $bool = 1;
-      } else {
-        return 0;
-      }
-    }
-    return 0 if !$bool;
-
-    my $NWU = vec_sum( $N, $WU);
-    if ( vec_same( $v{$ind[6]} , $NWU ) ) {
-=pod
-        return 0 unless
-            2*norm_f($N) == norm_f($NW) &&
-            norm_f($NW) == norm_f($WU) &&
-            norm_f($WU) == norm_f($UN) &&
-            3*norm_f($N) == norm_f($NWU);
-=cut
-        return 1;
-    }
-    else {
-        return 0;
-    }
-}
-
-sub is_hypercube {
-    my @p = @_;
-    my %v;
-    $v{$_} = vec_subtract($p[0], $p[$_]) for (1..15);
-    my @ind = sort { norm_f($v{$a}) <=> norm_f($v{$b}) } keys %v;
-    my ($N, $W, $U, $A) = ( $v{$ind[0]}, $v{$ind[1]} , 
-                            $v{$ind[2]}, $v{$ind[3]} );
-    return 0 unless 
-        norm_f($N) == norm_f($W) && norm_f($W) == norm_f($U)
-                             && norm_f($U) == norm_f($A);
-    return 0 unless 
-        vec_prod_f($N, $W) == 0 &&
-        vec_prod_f($N, $U) == 0 &&
-        vec_prod_f($N, $A) == 0 &&
-        vec_prod_f($A, $W) == 0 &&
-        vec_prod_f($A, $U) == 0 &&
-        vec_prod_f($W, $U) == 0 ;
-
-    my $AU = vec_sum($A, $U);
-    my $AW = vec_sum($A, $W);
-    my $AN = vec_sum($A, $N);
-    my $NW = vec_sum($N, $W);
-    my $WU = vec_sum($W, $U);
-    my $UN = vec_sum($U, $N);
-    my $bool_face = undef;
-    my $iter_face = permutations([$AU, $UN, $NW, $WU, $AW, $AN]);
-    while (!$bool_face && (my $p = $iter_face->next)) {
-        $bool_face = 
-            vec_same($v{$ind[4]}, $p->[0]) &&
-            vec_same($v{$ind[5]}, $p->[1]) &&
-            vec_same($v{$ind[6]}, $p->[2]) &&
-            vec_same($v{$ind[7]}, $p->[3]) &&
-            vec_same($v{$ind[8]}, $p->[4]) &&
-            vec_same($v{$ind[9]}, $p->[5]) ;
-    }
-    return 0 if !$bool_face;
-
-    my $UNW = vec_sum($UN, $W);
-    my $ANW = vec_sum($NW, $A);
-    my $AWU = vec_sum($WU, $A);
-    my $AUN = vec_sum($UN, $A);
-    my $bool_cube = undef;
-    my $iter_cube = permutations([$UNW, $ANW, $AWU, $AUN]);
-    while (!$bool_cube && (my $p = $iter_cube->next)) {
-        $bool_cube = 
-            vec_same($v{$ind[10]}, $p->[0]) &&
-            vec_same($v{$ind[11]}, $p->[1]) &&
-            vec_same($v{$ind[12]}, $p->[2]) &&
-            vec_same($v{$ind[13]}, $p->[3]);
-    }
-    return 0 if !$bool_cube;
-
-    my $AUNW = vec_sum($AU,$NW);
-    if ( vec_same($v{$ind[14]}, $AUNW) ) {
-=pod
-        return 0 unless 
-            2*norm_f($N) == norm_f($NW) &&
-            norm_f($NW) == norm_f($AU) &&
-            norm_f($NW) == norm_f($UN) &&
-            norm_f($NW) == norm_f($WU) &&
-            norm_f($NW) == norm_f($AW) &&
-            norm_f($NW) == norm_f($AN) &&
-            3*norm_f($N) == norm_f($UNW) &&
-            3*norm_f($N) == norm_f($ANW) &&
-            3*norm_f($N) == norm_f($AWU) &&
-            3*norm_f($N) == norm_f($AUN) &&
-            4*norm_f($N) == norm_f($AUNW);
-=cut
-        return 1;
-    }
-    else {
-        return 0;
-    }
-}
-
-sub vec_prod {
-    my $first = $_[0];
-    my $second = $_[1];
-    warn "Not the same dimension in vec_prod \n" if $first->$#* != $second->$#*;
-    my $sum = 0;
-    $sum+= ($first->[$_]*$second->[$_]) for (0..$first->$#*);
-    return $sum;
-}
-
-sub vec_prod_f {
-    return sprintf("%f", vec_prod($_[0], $_[1]));
-}
-
-sub norm {
-    my $p = $_[0];
-    my $sum = 0;
-    $sum+= ($p->[$_])**2 for (0..$p->$#*);
-    return $sum;
-}
-
-sub norm_f {
-    return sprintf("%f", norm($_[0]));
-}
-
-sub vec_sum {
-    my $first = $_[0];
-    my $second = $_[1];
-    my $ans = [];
-    warn "Not the same dimension in vec_sum \n" if $first->$#* != $second->$#*;
-    for my $s (0..$first->$#*) {
-        push $ans->@*, $first->[$s] + $second->[$s];
-    }
-    return $ans;
-}
-
-sub vec_same {
-    my $first = $_[0];
-    my $second = $_[1];
-    warn "Not the same dimension in vec_same \n" if $first->$#* != $second->$#*;
-    for my $s (0..$first->$#*) {
-        return undef if $first->[$s] != $second->[$s];
-    }
-    return 1;
-}
-
-sub vec_subtract {
-    my $first = $_[0];
-    my $second = $_[1];
-    my $ans = [];
-    warn "Not the same dimension in vec_subtract\n" if $first->$#* != $second->$#*;
-    for my $s (0..$first->$#*) {
-        push $ans->@*, $second->[$s] - $first->[$s];
-    }
-    return $ans;
-}
-
-
-
-# 9 tests
-ok is_square( [1,0], [0,1], [-1,0],[0,-1]) == 1, "on x-axis and y-axis";
-
-ok is_square( [5/sqrt(26), 1/sqrt(26)], 
-              [-1/sqrt(26), 5/sqrt(26)],
-              [-5/sqrt(26), -1/sqrt(26)], 
-              [1/sqrt(26), -5/sqrt(26)]) == 1,
-             "inclined by arctan(1/5), centre at origin"; 
-
-ok is_square( 
-              [cos(atan2(1,5)), sin(atan2(1,5))], 
-              [-sin(atan2(1,5)), cos(atan2(1,5))],
-              [-cos(atan2(1,5)), -sin(atan2(1,5))],
-              [sin(atan2(1,5)), -cos(atan2(1,5))]
-            )  
-              == 1, "arctan(1/5) by atan2(), caught by the equalities";  
-
-ok is_square( 
-              [2.7*cos(atan2(1,5)), 2.7*sin(atan2(1,5))], 
-              [-2.7*sin(atan2(1,5)), 2.7*cos(atan2(1,5))],
-              [-2.7*cos(atan2(1,5)), -2.7*sin(atan2(1,5))],
-              [2.7*sin(atan2(1,5)), -2.7*cos(atan2(1,5))]
-            )  
-              == 1, 
-             "arctan(1/5) by atan2() of larger size (multipled by 2.7)," 
-            ."caught by the equalities";  
-
-ok is_square( 
-              [2.8*cos(atan2(1,5)), 2.8*sin(atan2(1,5))], 
-              [-2.8*sin(atan2(1,5)), 2.8*cos(atan2(1,5))],
-              [-2.8*cos(atan2(1,5)), -2.8*sin(atan2(1,5))],
-              [2.8*sin(atan2(1,5)), -2.8*cos(atan2(1,5))]
-            )  
-              == 1, 
-             "arctan(1/5) by atan2() of larger size (multipled by 2.8),"
-            ."caught by the equalities";  
-
-ok is_square( 
-              [4.0*cos(atan2(1,5)), 4.0*sin(atan2(1,5))], 
-              [-4.0*sin(atan2(1,5)), 4.0*cos(atan2(1,5))],
-              [-4.0*cos(atan2(1,5)), -4.0*sin(atan2(1,5))],
-              [4.0*sin(atan2(1,5)), -4.0*cos(atan2(1,5))]
-            )  
-              == 1, 
-             "arctan(1/5) by atan2() of larger size (multipled by 4.0),"
-            ." caught by the equalities";  
-
-ok is_square( 
-              [0.0009*cos(atan2(1,5)), 0.0009*sin(atan2(1,5))], 
-              [-0.0009*sin(atan2(1,5)), 0.0009*cos(atan2(1,5))],
-              [-0.0009*cos(atan2(1,5)), -0.0009*sin(atan2(1,5))],
-              [0.0009*sin(atan2(1,5)), -0.0009*cos(atan2(1,5))]
-            )  
-              == 1, 
-             "arctan(1/5) by atan2() of a much smaller size"
-            ."(multiple by 0.0009), caught by the equalities"
-            ."(_\"not ok\" is normal_)";  
-
-ok is_square( [1, 2] , [4,3], [3,1], [2,4] ) == 1, "Knight's square";
-ok is_square( [1, 1] , [-1, 1], [1,-1], [-1,-1] ) == 1, "centre at origin";
-
-# 4 tests
-ok is_cube( [1, 1, 1], [1, 1, 0], [1, 0, 1], [1, 0, 0], 
-            [0, 1, 1], [0, 1, 0], [0, 0, 1], [0, 0, 0] ) == 1, 
-            "standard 2**3";
-ok is_cube([ -2, -2, -2], [ -2, -2,  2], [ -2,  2, -2], [ -2,  2,  2], 
-           [  2, -2, -2], [  2, -2,  2], [  2,  2, -2], [  2,  2,  2]) == 1, 
-            "centre at origin";
-ok is_cube( 
-            [-2, -2*sqrt(3), 3-2*sqrt(3)]   ,  
-            [-2, 4-2*sqrt(3), 3+2*sqrt(3)],    
-            [-2, 2*sqrt(3), -2*sqrt(3)-1 ],  
-            [-2, 4+2*sqrt(3), -1+2*sqrt(3) ], 
-            [ 6, -2*sqrt(3), 3-2*sqrt(3)],
-            [ 6, 4-2*sqrt(3), 3+2*sqrt(3)],
-            [ 6, 2*sqrt(3), -2*sqrt(3)-1],
-            [ 6, 4+2*sqrt(3), -1+2*sqrt(3)]
-           ) 
-              == 1, "a rotated cube centred at (2,2,1)";
-ok is_cube([  2,  2,  2], [  2,  3,  2], [  2,  2,  3], [  2,  4,  2], 
-           [  3,  3,  2], [  2,  2,  1], [  2,  3,  2], [  2,  7,  3]) == 0, 
-            "this is not a cube";
-
-# 1 test
-ok is_hypercube(
-                [0,0,0,0],[0,0,0,1],[0,0,1,0],
-                [0,0,1,1],[0,1,0,0],[0,1,0,1],
-                [0,1,1,0],[0,1,1,1],[1,0,0,0],
-                [1,0,0,1],[1,0,1,0],[1,0,1,1],
-                [1,1,0,0],[1,1,0,1],[1,1,1,0],
-                [1,1,1,1]
-               ) == 1, "hypercube";
-done_testing();