Quest for Double Dobbles


Let’s start searching for Double Dobble!

In previous post Double Dobble - constraints we touched upon the relation between the number $k$ of symbols on a card (well… in a block), the number $N$ of cards and the number $N$ of symbols over all cards.

In this post we will take a look at implementing some logic to actually find valid arrangements. Let’s face it: the lack of code in Double Dobble is annoying. To me, at least.

A few caveats

It’s important at this point to remember that I made the assumption that both the number of cards and the number of symbols are the same (i.e. $N$), but I have no proof to offer that this MUST be the case for Double Dobble too.

Another thing to observe is that we will take the same approach explained in Double Dobble: find out a displacement pattern in the circular arrangement of the blocks/cards that allows us to meet our goal.

While this can work - and indeed it leads to successful arrangements - I have no proof that a lack of proper arrangements of this type is sufficient to rule out that a certain number $k$ of symbols on a card has no valid solution. We can only say that this approach does not produce a result for that $k$.

Enough! Let’s look for some Double Dobbles!

Each arrangement in the circle can be seen as choosing $k$ places out of the $N$ available. This leads us to $N \choose k$ possible arrangements.

We can further chop this number down by observing that we should always include the first position in the arrangement we check. Any valid combination can be translated back to one including the first position, so why bother looking? This means that we are left with ${N - 1} \choose {k - 1}$ combinations.

To get the valid combinations we will just… count, making sure that for each position we only consider indexes greater than the selections in the previous slots.

Suppose we have an arrangement, represented by the indexes in the circular arrangement of $N$ symbols, in a range from $0$ up to $N - 1$:

\[A = (0, i_1, i_2, ...)\]

We can define the opposite of this arrangement the one where each index is mapped onto its opposite in the circle with respect to index $0$, that is:

\[A' = (0, N - i_1, N - i_2, ...)\]

The index $0$ is its self-opposite, just like counting in the group of remainders modulo $N$.

An arrangement $A$ is acceptable if, and only if, its opposite $A’$ is acceptable too - so in theory we might avoid checking arrangement that are opposite of ones we already checked.

Enough! Show me the code!

In coding our quest we will not consider the insight about skipping opposite arrangement, for sake of simplicity.

As we don’t know in anticipation $k$, we will turn to our old friend NestedLoops in Algorithm::Loop, building an iterator for our candidate arrangements:

my $it = NestedLoops(
      map {
         my $end = $N - 1 - ($k - 1) + $_;
         sub { [($_ + 1) .. $end] },
      } (1 .. $k - 1)

Now we can generate the arrangements and check until we find one that provides us with a Double Dobble:

while (my @seq = $it->()) {
   next unless check_double_dobble(@seq);
   say "(@seq)";

sub check_double_dobble (@seq) {
   my $k = @seq;
   my $N = 1 + $k * ($k - 1) / 2;
   my $max_delta = $N % 2 ? ($N - 1) / 2 : $N / 2;
   my (%one, %two);
   for my $i (0 .. $#seq - 1) {
      for my $j ($i + 1 .. $#seq) {
         my $delta = $seq[$j] - $seq[$i];
         $delta = $N - $delta if $delta > $max_delta;
         if ($one{$delta})    { $two{$delta} = delete $one{$delta} }
         elsif ($two{$delta}) { return }
         else                 { $one{$delta} = 1 }
   return if scalar(keys %one) == 1 && !$one{$max_delta};
   return 1;

The check_double_dobble function is not very… scientific, in that it bails out immediately if we hit a delta three times (it would not be a valid Double Dobble), but still allows for one of the deltas to appear exactly once (in particular, the maximum possible delta).

I currently have no demonstration of why this is true… but it seems to work 🙄

It is interesting that the code above yields the sequence of indexes $(0 1 2 4 7)$ for an input $k = 5$, which is different from the one in Double Dobble, although still working of course:

0 1 2 4
  1 1 3
    2 2 5
      4 3
4 5     7

This makes me really curious of what algorithm they used.

It’s also interesting that there are no solutions for $k \in {6, 7, 8}$, but there is one for $k = 4$ (and $N = 7$) that might have escaped the scrutiny in Double Dobble:

0 1 2
  1 1 3
    2 2
3     4

It leads to a very simple game with 7 cards only… but it’s still better than the more trivial case for $k = 3$.

Well… at least now I know how the author of Double Dobble might have been approached the problem… but didn’t!

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