3. Formulations

What is the best way to phrase this problem?

Or even if not the best way, one that gives us a hint about what is happening?

The simplest, in some way, thing to do is come up with a sequence of cards that is definitely unwinnable. This would provide an example and would positively answer the question. However, that is all the information we can get from such an example. Ideally, a rule or some deeper understanding of why things happen the way they happen would be determined. As I have not yet managed to construct an unwinnable example, this is a philosophical barrier I haven’t had to breach yet.

The first thing I’m trying to do is solve some simpler, more straightforward questions, hoping they will give me some hints as to whether the harder question is solvable, or possible paths I can follow in solving it. This is a common technique in proving a conjecture, and different simpler questions have different powers.

I will thus rephrase the question as a statement $Q$ whose truth value I will seek. This gives rise to the following statement:

Definition.

$Q$ : There exists a configuration $\mathcal{C}$ of the deck that makes it impossible to win.

Consider $S$, some other statement. Then, if the truth value of $S$ is in any way related to the truth value of $Q$, only one of the following can hold:

1. $S\Rightarrow Q$ and $Q\nRightarrow S$
2. $\overline{S}\Rightarrow Q$ and $Q\nRightarrow \overline{S}$
3. $Q\Rightarrow S$ and $S\nRightarrow Q$
4. $Q\Rightarrow \overline{S}$ and $\overline{S}\nRightarrow Q$
5. $S\iff Q$

where as per standard logic language, a bar over a statement denotes the negation of that statement. For example, $\overline{Q}$ denotes the statement that there does not exist a configuration $\mathcal{C}$ of the deck that makes it impossible to win, and is read ‘not $Q$’ or ’$Q$ bar’.

In the first case, $S$ is said to be a stronger statement than $Q$. The strength of the statements 2-4 follows analogously, whereas in the last case $S$ and $Q$ are said to be equivalent.

The ideal situation is to construct an equivalent or stronger statement, from which $Q$ (or $\overline{Q}$) would follow. However, it is likely that such a statement will also be harder to show. It is easier to construct a simpler, weaker, statement, in hope that it will eventually lead to the right direction. In the case that the truth value cannot be determined, however, the method used could still provide useful insights or extrapolatable methods in determining the stronger statement.

This is exactly the stage at which I found thinking about this question a nice activity. When I brought it up to Albert we had radically different approaches - a fascinating evening that was. When one forms a simpler question, they do so in hope it will lead them to some direction. We thus found that our initial guesses were quickly discarded by either of us, and in this space I will only present ideas that were not dismissed within 10 minutes.

Forming the first simpler question §

With this abstract preamble out of the way, we use this logic to form some simpler statements to gain a better understanding of the ‘universe of $Q$‘.

We know that play is turn-based, at each of which points the hands of the player(s) and at least one of the piles are updated. The game with more than one players involves some combinatorial complexity. On the other hand, the players collectively ‘see’ more cards in the more-than-1 player version, which could be a great advantage.
The lack of combinatorial complexity for the 1-player game implies that it is possibly easier to start by investigating that, as opposed to the case of multiple players, which can be addressed once the 1-player question has been answered, as well as the relationship between the scene of the game for different numbers of players.

We thus begin by investigating the 1-player version of the game.

A first easy observation is that the inclusion of 10-tricks makes the game more likely to be winnable. Thus, a configuration $\mathcal{C}$ is winnable without 10-tricks implies that $\mathcal{C}$ is winnable with 10-tricks

Then, consider a different ruleset ${\mathcal{R}}^{\prime }$: no cards are drawn to begin with, and a player must play the top card of the deck as drawn. Therefore, if a configuration $\mathcal{C}$ is winnable under this ruleset, $\mathcal{C}$ is winnable with the original ruleset, as the player now has the choice.

This gives rise to the first alternative statement:

Definition.

$S_1$: There exists a configuration of the cards that is unwinnable with the alternative ruleset ${\mathcal{R}}^{\prime }$ and without 10-tricks.

This is an easier preliminary question, and we have that $\overline{S_1}\Rightarrow \overline{Q}$, that is, if there doesn’t exist such a configuration, then there doesn’t exist a configuration that is unwinnable for the original game, and we have proved the falsehood of $Q$. This is the equivalent of scenario 3 in our breakdown of logic cases.