Variants of the game

Modes of gameplay:

There are two modes of gameplay- normal play and misère play. In so called normal play, the player who makes the last move wins. In misère play, the player who makes the last move loses. It seems the 2 modes are exactly opposite of each other but the analysis shows that the game scenarios for the two mode are not related to each other.

There exists a pattern for outcome of the game as the number of spots increases in case of normal play. But no such pattern is discovered yet in the case of misère play.

Analyzing the game scenarios for different number of initial dots is a tedious job. As we saw in the sample game in previous post, X had 3 distinct options for her 1^st move. Then again Y had 6 distinct options for his turn. Thus if we consider all the possibilities (i.e. build a game tree) then we will have a large number of possible game scenarios (number of leaf nodes in the game tree) even for initial game size of 2 (i.e. for initial number of spots=2).

[ Let’s assume X & Y are playing and X makes the first move of the game.

  Let the number of initial spots be n. ]

Scientists have researched and found some interesting fact related to the gameplay.

They have found certain patterns in outcome of the game if both the players play optimally.

[ Here optimally means X, in her turn, will make a move such that it enhances her chances of winning.
  Likewise, in Y’s turn, he will make a move which enhances his chances of winning ]

For Normal Play:

// Spots = number of initial spots

//Normal Outcome = Result for the player who starts the game (X in our case)

Spots (n)	1	2	3	4	5	6	7	8	9	10	11	12	...
Normal Outcome	Loss	Loss	Win	Win	Win	Loss	Loss	Loss	Win	Win	Win	Loss	…

The result pattern for X is – L L W W W L and this repeats after every 6^th game.

In other words, X will win if number of initial spots (n) divided by 6 leaves remainder 3, 4 or 5, when X plays optimally. (i.e. when n % 6 = 3, 4 or 5)

For Misère Play:

// Spots = number of initial spots

//Misère Outcome = Result for the player who starts the game (X in our case)

Spots (n)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	...
Misère Outcome	Win	Loss	Loss	Loss	Win	Win	Loss	Loss	Loss	Win	Win	Win	Loss	Loss	Loss	...

This mode of gameplay is more interesting because no definite pattern has been discovered yet.

Apart from the modes of gameplay there are other variants of the game:

•  Brussels Sprouts

This variant starts with a number of crosses, i.e. spots with four free ends. Each move involves joining two free ends with a curve (again not crossing any existing line) and then putting a short stroke across the line to create two new free ends. So each move removes two free ends and introduces two more. Despite this, the game is finite, and indeed the total number of moves is predetermined by the initial number of crosses: the players cannot affect the result by their play. With n initial crosses, the total number of moves possible = 5n−2.

So if n is odd, the player who starts the game wins and vice versa.

The outcome is fixed and not affected by playing style of the players.

Refer to this link for prove of number of moves in a game (= 5n-2).

Again this variant can be played in misère mode which will produce exactly opposite outcome as in normal play, i.e. – for odd n, the player who starts the game loses.

This variant can be made interesting we make the stroke on current curve optional. The number of moves will become variable in that case.

• Black and White Sprouts

This variant is similar to the one that is stated in my previous post. More analysis on that variant will be done later.

In this variant the player has an option to mark a new spot on current curve drawn by him/her.

In the variant stated in the previous post the player has the option to mark a new spot on the current curve drawn by his/her opponent, i.e. he/she has to mark a new spot on current curve before making his/her move. 

So although these two variant seem similar, this slight difference changes the game playing strategies completely.

Refer to this link for more info on black & white sprouts.

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Now let’s have a look at another variant. Let’s call this version - ‘Sprouts 99’.

I didn't like the idea of making a new dot on the current curve after every move. Here is what I do when I play with my friends – (assume X & Y are two players playing the game) If Y made a curve (say c1) in his current move then X has the power to decide if she wants to create a dot on c1. Likewise Y has the power to decide if he wants to make a dot on the current curve after X’s move. Thus adding a new spot is possible but not mandatory. Also a player cannot add a new spot to his current spot. She/he can add a new spot only on the last move (curve) made by her/his opponent.

This may seem to complicate things.
In optimal play (optimal play means X, in her turn, will make a move such that it enhances her chances of winning. Likewise, in Y’s turn, he will make a move which enhances his chances of winning) before making a move the player has to decide if he/she should mark a dot on the current curve, how the outcome will change with/without the new dot, possible outcome(s) with/without new dot after his/her move … in short- a lot of computations in one’s head.

This actually makes things more complicated but that will lead to more interesting gameplay.

In sample game on the previous post we saw that there are a large number of unique game scenarios possible even for n=2.  This number increases exponentially as we increase the number of initial spots.

This was for normal sprout.

In sprouts99 after each move the player has the option to add/not add a new spot.  So for every move that was possible in normal sprouts we now have another move possible in sprouts99 (when we remove the new spot on current curve). So if p moves are possible after a turn in normal sprouts, 2p moves will be possible in sprouts99. Thus after every turn we have twice more options with us in sprouts99.

Hence the number of unique ways to play a particular game is exponentially more than what we had for the original sprouts game.

The next post consists of a general analysis and some facts about sprouts99.