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-# Perl Weekly Challenge #123
+# Perl Weekly Challenge #124
 
 You can find more information about this weeks, and previous weeks challenges at:
 
@@ -10,136 +10,207 @@ submit solutions in whichever language you feel comfortable with.
 
 You can find the solutions here on github at:
 
-https://github.com/drbaggy/perlweeklychallenge-club/tree/master/challenge-123/james-smith/perl
+https://github.com/drbaggy/perlweeklychallenge-club/tree/master/challenge-124/james-smith/perl
 
-# Task 1 - Ugly Numbers
+# Task 1 - Happy Women's Day
 
-***You are given an integer `$n >= 1`. Write a script to find the $nth element of Ugly Numbers.***
-
-**Defn:** Ugly numbers are those number whose prime factors are 2, 3 or 5. For example, the first 10 Ugly Numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12.
+***Write a script to print the Venus Symbol, international gender symbol for women. Please feel free to use any character.***
 
 ## The solution
 
-There are two ways of working out the *nth* ugly number - we either have to search all numbers starting at 1 counting ugly numbers -OR- do something more "intellegent".
+We will first look at the symbol defined in the question...
 
-The former works well for small `n`, but doesn't scale well as the ugly numbers become more sparse.
+```
+    ^^^^^
+   ^     ^
+  ^       ^
+ ^         ^
+ ^         ^
+ ^         ^
+ ^         ^
+ ^         ^
+  ^       ^
+   ^     ^
+    ^^^^^
+      ^
+      ^
+      ^
+    ^^^^^
+      ^
+      ^
+```
 
-### Method
+We note there are 3 types of rows:
 
-Any Ugly number is either a multiple of 2, 3 or 5 of another Ugly number. So to find the next ugly number we multiple all ugly numbers by 2, 3 or 5 and find the lowest value greater than the last ugly seen.
+ * Type I: a line of 5 symbols (centered)
+ * Type II: a single symbol in the middle of the row
+ * Type III: two symbols either side of the middle at a given distance.
+
+We encode these in an array -1 -> line of 5 symbols; 0 -> a single symbol at the centre; values > 0 - two points at the given distance away from the centre. The code becomes this:
 
 ```perl
-sub nth_ugly {
-  my $n = shift;
-  state @uglies = (1);
-  while(1) {
-    return $uglies[$n-1] if $n <= @uglies;
-    my $next = 0;
-    foreach my $l (5,3,2) {
-      foreach(@uglies) {
-        next if $_ * $l <= $uglies[-1];
-        $next = $_*$l if !$next || $next > $_*$l;
-        last;
-      }
-    }
-    push @uglies, $next;
-  }
-}
+my @pts = qw(-1 3 4 5 5 5 5 5 4 3 -1 0 0 0 -1 0 0);
+say $_ < 0 ? '    ^^^^^'
+  : !$_    ? '      ^'
+  : ' ' x (6-$_) . '^' . ' 'x($_*2-1) .'^'
+  foreach @pts;
 ```
 
-We cache the values internally in the function - in the `state` variable `@uglies`
+### Now for a more generic solution!
 
-### Optimization
+This symbol is just a circle and cross below. We can use trig to work out the points of the circle. To ensure we don't leave gaps we sweep the arcs away from the cardinal points (N,S,E,W) up to the ordinal points (NE,NW,SE,SW) - 8 different 45deg arcs. This way we just need to compute one point for each line and then compute the other co-ordinate using pythagorus' theorem.
 
-We can speed this up by short-cutting the inner loop. 
-  * All uglies are either a multiple of 2, 3 or 5 times another ugly (with the exception of 1).
-  * We keep track of the next ugly that is a multiple of 2, 3, 5 etc - we call these `$v2`, `$v3` and `$v5` respectively. These are `2*$uglies[$i2]`, `3*$uglies[$i3]`, `5*$uglies[$i5]`.
-  * Initially the values of `$l2`, `$l3`, `$l5`, `$v2`, `$v3`, `$v5` are `0`, `0`, `0`, `2`, `3`, `5`, and the list `@uglies` is initialized with the value `(1)`.
-  * Every time we need a new ugly, we find it as the lowest value of `$v2`, `$v3`, `$v5`. We then `push` it onto `@uglies`.
-  * Now we need to update `$v2`, `$v3` and `$v5` if they are equal to this value. We do this by incrementing the index `$i2`, `$i3` and/or `$i5` and then setting
-    `$v? = ?*$uglies[$i?]`... This will often update 2 or even all 3 of the values...
+Why do we do this? If we just did 4 arcs of 90 degrees we would find that once we passed 45 degrees we would miss out points...
 
-Additionally we speed up the code by keeping a cache of ugly values we have found, and if we are asked for one we return that value from the cache, if not as we have
-kept the state of the loop we just continue from where we left off with the values of `$l2`, `$l3`, `$l5`, `$v2`, `$v3`, `$v5` which are also held in the state
-variable.
+Our process has 4 steps.
 
-This gives us the following optimized perl code.
+ 1. Create a blank canvas
+ 2. Draw the circle (note when we compute the y value we take half off the radius - this gives a better circle as we are tracing a line through the centre of the "squares"...
+ 3. Draw the cross
+ 4. Display the canvas...
 
 ```perl
-sub nth_ugly_opt {
-  my $n = shift;
-  state $l2 = 0; state $l3 = 0; state $l5 = 0;
-  state $v2 = 2; state $v3 = 3; state $v5 = 5;
-  state @uglies = (1);
-  return $uglies[$n-1] if $n <= @uglies;
-  while( @uglies < $n ) {
-    push @uglies, my $next = $v2<$v3 && $v2<$v5 ? $v2 : $v3<$v5 ? $v3 : $v5;
-    $v2 = $uglies[++$l2]*2 if $v2 == $next;
-    $v3 = $uglies[++$l3]*3 if $v3 == $next;
-    $v5 = $uglies[++$l5]*5 if $v5 == $next;
-  }
-  return $uglies[-1];
+## Create the canvas..
+my @a = map { ' ' x ($radius*2+1) } 0..$radius*2+$cross;
+
+## Now we draw the circle...
+foreach my $x (0 .. ceil($radius*0.71)) {
+  my $y = int sqrt( ($radius-.5)**2 - $x**2 );
+  substr $a[ $radius - $x ],$radius-$y,1,'^';
+  substr $a[ $radius + $x ],$radius-$y,1,'^';
+  substr $a[ $radius - $x ],$radius+$y,1,'^';
+  substr $a[ $radius + $x ],$radius+$y,1,'^';
+  substr $a[ $radius - $y ],$radius-$x,1,'^';
+  substr $a[ $radius + $y ],$radius-$x,1,'^';
+  substr $a[ $radius - $y ],$radius+$x,1,'^';
+  substr $a[ $radius + $y ],$radius+$x,1,'^';
 }
-```
-
-Below is the performance of the methods. Note these were tested without using `state` variables, as the caching nature of
-state variables prevents benchmarking (values are obtained directly from the cache) - although if you were using this in a
-real world situation that would be an advantage! Scanning where `n` is greater than 500 takes too long to get accurate
-benchmarks.
 
-|      n |                     Ugly_n | scan /s   | simple /s     | opt /s        | opt vs sim % | sim vs scn % | opt vs scn % |
-| -----: | -------------------------: | --------: | ------------: | ------------: | -----------: | -----------: | -----------: |
-|      1 |                          1 | *938,492* | **3,005,191** |   1,799,451   |          -40 |          220 |           92 |
-|      2 |                          2 | *536,552* |     816,345   | **1,089,848** |           34 |           52 |          103 |
-|      5 |                          5 | *234,116* |     238,716   |   **455,051** |           91 |            2 |           94 |
-|     10 |                         12 |   98,061  |     *77,865*  |   **250,411** |          222 |          -21 |          155 |
-|     20 |                         36 |   32,105  |     *21,225*  |   **130,707** |          516 |          -34 |          307 |
-|     50 |                        243 |   *4,289* |       5,504   |    **43,065** |          682 |           28 |          904 |
-|    100 |                      1,536 |     *724* |       1,203   |    **24,768** |        1,959 |           66 |        3,321 |
-|    200 |                     16,200 |   *63.50* |         272   |    **12,470** |        4,485 |          328 |       19,538 |
-|    500 |                    937,500 |    *0.57* |          48   |     **4,639** |        9,565 |        8,306 |      812,334 |
-|  1,000 |                 51,200,000 | *-*       |       10.60   |     **2,503** |       23,513 | -            | -            |
-|  2,000 |              8,062,156,800 | *-*       |        2.75   |     **1,187** |       43,064 | -            | -            |
-|  5,000 |         50,837,316,566,580 | *-*       |        0.41   |       **375** |       91,812 | -            | -            |
-| 10,000 |    288,325,195,312,500,000 | *-*       |        0.08   |       **230** |      273,710 | -            | -            |
-| 13,282 | 18,432,000,000,000,000,000 | *-*       |        0.05   |       **148** |      302,757 | -            | -            |
+## And the two parts of the cross...
+substr $a[2*$radius+$_],$radius,1,'^' foreach 0..$cross;
+substr $a[2*$radius+$cross/2],$radius-$cross/2,$cross+1,'^'x($cross+1);
 
+### Finally we render the canvas...
+say $_ foreach @a;
+```
 
-# Task 2 - Square Points
+Example output...
+```
+                ^^^^^^^^^
+             ^^^         ^^^
+           ^^               ^^
+         ^^                   ^^
+        ^                       ^
+       ^                         ^
+      ^                           ^
+     ^                             ^
+    ^                               ^
+    ^                               ^
+   ^                                 ^
+   ^                                 ^
+  ^                                   ^
+  ^                                   ^
+  ^                                   ^
+ ^                                     ^
+ ^                                     ^
+ ^                                     ^
+ ^                                     ^
+ ^                                     ^
+ ^                                     ^
+ ^                                     ^
+ ^                                     ^
+ ^                                     ^
+  ^                                   ^
+  ^                                   ^
+  ^                                   ^
+   ^                                 ^
+   ^                                 ^
+    ^                               ^
+    ^                               ^
+     ^                             ^
+      ^                           ^
+       ^                         ^
+        ^                       ^
+         ^^                   ^^
+           ^^               ^^
+             ^^^         ^^^
+                ^^^^^^^^^
+                    ^
+                    ^
+                    ^
+                    ^
+                    ^
+                    ^
+              ^^^^^^^^^^^^^
+                    ^
+                    ^
+                    ^
+                    ^
+                    ^
+                    ^
+```
 
-***You are given coordinates of four points i.e. (x1, y1), (x2, y2), (x3, y3) and (x4, y4). Write a script to find out if the given four points form a square.***
+# Task 2 - Tug of War
 
-## Assumption
+***You are given a set of `$n` integers `(n1, n2, n3, ...)`. Write a script to divide the set in two subsets of `n/2` sizes each so that the difference of the sum of two subsets is the least. If `$n` is even then each subset must be of size `$n/2` each. In case `$n` is odd then one subset must be `($n-1)/2` and other must be `($n+1)/2`.***
 
-We will assume that the points are not in any particular order (The sequare may be p1->p2->p3->p4 OR p1->p3->p2->p4)
 ## Solution
 
-First we need to think how we define a square - it has 4 sides of equal length and sides at right angles. If we want to define it terms of distances between points we have 4 pairs of points that are the same distance apart and two pairs of points which are at `sqrt(2)` times this distance.
+We will use an iterative solution. We start by allocating person 1 to team 1, we then iterate down allocating each person to either team 1 or team 2. If either team gets too big we bomb out (this makes this solution more efficient than the non-iterative solution). As we go we keep a tally of the difference between the two teams weights.
 
-**Note:** There are two other combinations of points for which 4 of the distances are the same and 2 of the distances and the same. These are:
-  * an isosceles triangle with an inscribed equilateral triangle - the ratio of the two squares is `2+sqrt(3)`
-  * a kite - for which one half is an equilateral triangle and the other has height `1-sqrt(3)/2` - the ratio of the two squares is `2-sqrt(3)`.
+As we do a pre-allocation stage - we need to split the routine into two functions, the first function preps the data for interation and then handles the data at the end. The second does the interative step.
 
-There are three other combinations of points for which there are only two distances.
-  * With 5 of one length and 1 of another - we have an rhomobus consisting of two equilateral triangles (ratio of squares is 3)
-  * With 3 of one length and 3 of another - we have a equilateral triangle with the fourth point at it's centre (ratio of squares is also 3) and trapezium which is a regular pentagon with one point knocked off (with ratio of squares `2:3+sqrt(5)`
-`
-![Layout of points](sets-of-points.png?raw=true)
+At each step we need to know:
+ 1) What is the max-size of the group;
+ 2) Who is in team 1;
+ 3) Who is in team 2;
+ 4) What the difference in weight is;
+ 5) What is the smallest difference we have found;
+ 6) The weights of people left to be allocated.
 
-We therefore measure the squares of the distances between the points, and collect them together. If the list of distances is 2, and the ratio of the squares of the distances is 2 then we have a square.
-
- * The `while/foreach` loops calculate the square of the distances between points, and stores these in the hash `%dist` where the distance is the key.
- * We flip the hash so that the keys become values and values become keys. This allows us to check to see if we have one length 4 times and one length 2 times, and check the ratio of the length of the diagonal vs the length of the edges of the sides to see that it is 2.
+So to start - we allocate person 1 to group 1, and set the difference to his weight. `$best` is an object to collect the information about the best allocation (the members of the two teams and the smallest difference)...
 
 ```perl
-sub is_square {
-  my @pts = @_;
-  my %dist;
-  while(@pts>1) {
-    my $p = shift @pts;
-    $dist{ ($p->[0]-$_->[0]) ** 2 + ($p->[1]-$_->[1]) ** 2 }++ foreach @pts;
+sub match_teams {
+  my( $diff, @n ) = @_;
+  separate( 1 + int(@n/2), [$diff], [], $diff, my $best = [], @n );
+  return "Team 1: [@{$best->[0]}]; Team 2: [@{$best->[1]}]; difference $best->[2]";
+}
+
+sub separate {
+  my($maxsize,$team1,$team2,$diff,$be,@nums) = @_;
+  unless(@nums) {
+    if( !defined $be->[0] || $be->[2]>abs $diff ) {
+      $be->[0] = $team1;
+      $be->[1] = $team2;
+      $be->[2] = abs $diff;
+    }
+    return;
   }
-  my %flip = reverse %dist;
-  return exists $flip{2} && exists $flip{4} && $flip{2} == 2*$flip{4} || 0;
+  my $next = shift @nums;
+  separate( $maxsize, [@{$team1},$next], $team2, $diff+$next, $be, @nums ) if @{$team1} < $maxsize;
+  separate( $maxsize, $team1, [@{$team2},$next], $diff-$next, $be, @nums ) if @{$team2} < $maxsize;
 }
 ```
+### Notes:
+ * Notice the yoda inequality `$be->[2]>abs $diff` - it makes it clearer that you are only computing the absolute value of `$diff` not that of `$diff < $be->[2]`.
+ * `$team1` / `$team2` are refs - so when we update them we make copies `[@{$team2},$next]` rather than pushing to them.
+ * We keep the running total as it avoids the need to do the sum each time.
+
+### Timings
+
+| players | rate/time |
+| ------- | --------: |
+| 10      |   2,273/s |
+| 12      |     598/s |
+| 14      |     157/s |
+| 16      |      41/s |
+| 18      |      10/s |
+| 20      |    2.68/s |
+| 22      |    0.57/s |
+| 24      |   ~    6s |
+| 26      |   ~   23s |
+| 28      |   ~   94s |
+| 30      |   ~  365s |
+
+```