Using optimal strategy,
the game never ends.
Let's start by assuming red has a winning strategy and ignore any moves we can prove aren't winning. For example,
any position where blue can obtain the mirrored position can not be winning for red.
I'll refer to the tokens closest to the opposing home row as the inner tokens and the other two as the outer tokens. To begin with, if red starts by moving the inner token,
blue can capture it and have the same position.
|v|v|
| |v|
|>| |
>|v| |
>| | |
>| | |
Then in the case where red moves the outer token first, if blue moves the inner token
they lose.
| |v|
| |v|
>|v| |
>|v| |
|>| |
| |>|
If blue the moves the outer token they lose immediately.
| | |
|v|v|
>|v|v|
| | |
| |>|
| |>|
But if they move the inner token they still lose.
| |v|
| |v|
>| | |
|>| |
|v|>|
|v|>|
| |v|
| |v|
|>| |
|>| |
| |>|
| | |
That leaves the case where both players move the outer token first. If red then moves the inner token
blue can once again capture it and have the same position, so this is also not winning.
|v| |
|>|v|
|>| |
| | |
>|v|v|
|>| |
If instead red moves the outer token again
blue is forced to capture or they lose immediately like before.
|v| |
>| |v|
| |>|
| | |
|v|v|
>|v|v|
| | |
| |>|
| |>|
Then the question is, should they move again after they capture?
The answer is yes. If they don't then this happens:
|v| |
|v| |
>| | |
>| | |
>| |v|
|>|v|
and now blue is in zugzwang. They are forced to move their inner token before red does.
| | |
|v| |
>|v| |
| |>|
|>|v|
|>|v|
After this they can no longer force a capture and are more moves away from winning than red, so this is a loss.
If they do move again, which token should they move?
They can't move their outer token off the board. Despite being closer to a victory they are now at a disadvantage because they can only move one token.
|v| |
|v| |
>| | |
|>| |
>| | |
>| | |
Moving again and staying put both lead to defeat.
| | |
|v| |
>|v| |
| |>|
>| | |
>| | |
| | |
| | |
|v|>|
|v| |
>| | |
>| | |
| | |
|v| |
| | |
| | |
>|v| |
| |>|
If they move again:
| | |
|v| |
>| | |
>| | |
>|v| |
| |>|
| | |
|v| |
>|v| |
| |>|
| |>|
| |>|
So what about the other option?
| | |
>|v| |
>| |v|
Red can't move the outer token here or it will just serve as a springboard for blue's inner token.
| | |
| | |
>|v| |
>| | |
|>|v|
>| |v|
It's the same if red captures first.
|v| |
| | |
|>| |
>| | |
|>|v|
>| |v|
Moving the same token again after capturing leads to this position:
|v| |
| |>|
>| |v|
This is actually a mirrored version of a losing position we've seen earlier, so capturing and not moving again is the only move left.
|v| |
|>| |
>| |v|
If blue doesn't capture they lose.
|v| |
|v| |
|>| |
| |>|
>| | |
>| | |
| | |
| | |
|v|>|
|v| |
>| | |
>| | |
| | |
|v| |
| | |
| | |
>|v| |
| |>|
Moving a token off the board after capturing also doesn't work out.
| | |
|v| |
>|v| |
| |>|
>| | |
>| | |
If they move the inner token instead we get another winning position for red we've seen before.
| | |
|v| |
>| | |
>| | |
>|v|v|
|>|v|
That leaves us with capturing and not moving again.
| | |
>|v| |
>| |v|
But this is the same position we started with, so these positions will alternate forever.
So in the end
Red has no way to force a win but they can avoid losing by making the game continue forever. Some of the positions where the other player can get the mirrored position might still be losing, but this doesn't matter so long as red has at least one way to force the game to go on forever.