Snake Game Combinatorics
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I am doing an algorithmic analysis of the video game snake. It would be helpful to know the total amount of game positions in a $10times10$ board or an approximation of it.
There is another question on this site that uses a $5times5$ board and a snake of length $3$ and can solve this problem, but what I have realized is that since the snake is not allowed to overlap itself, it gets much more complicated to compute its possibilities after length $4$, and I need to know all the way up to length $100$ on a $10times10$ board.
Does anyone know how to do this?
Thanks
combinatorics number-theory
asked Dec 13 '18 at 22:38
TrigatenTrigaten
112
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add a comment |
$begingroup$
I am doing an algorithmic analysis of the video game snake. It would be helpful to know the total amount of game positions in a $10times10$ board or an approximation of it.
There is another question on this site that uses a $5times5$ board and a snake of length $3$ and can solve this problem, but what I have realized is that since the snake is not allowed to overlap itself, it gets much more complicated to compute its possibilities after length $4$, and I need to know all the way up to length $100$ on a $10times10$ board.
Does anyone know how to do this?
Thanks
combinatorics number-theory
asked Dec 13 '18 at 22:38
TrigatenTrigaten
112
$endgroup$
add a comment |
$begingroup$
I am doing an algorithmic analysis of the video game snake. It would be helpful to know the total amount of game positions in a $10times10$ board or an approximation of it.
There is another question on this site that uses a $5times5$ board and a snake of length $3$ and can solve this problem, but what I have realized is that since the snake is not allowed to overlap itself, it gets much more complicated to compute its possibilities after length $4$, and I need to know all the way up to length $100$ on a $10times10$ board.
Does anyone know how to do this?
Thanks
combinatorics number-theory
asked Dec 13 '18 at 22:38
TrigatenTrigaten
112
$endgroup$
I am doing an algorithmic analysis of the video game snake. It would be helpful to know the total amount of game positions in a $10times10$ board or an approximation of it.
There is another question on this site that uses a $5times5$ board and a snake of length $3$ and can solve this problem, but what I have realized is that since the snake is not allowed to overlap itself, it gets much more complicated to compute its possibilities after length $4$, and I need to know all the way up to length $100$ on a $10times10$ board.
Does anyone know how to do this?
Thanks
combinatorics number-theory
combinatorics number-theory
asked Dec 13 '18 at 22:38
TrigatenTrigaten
112
asked Dec 13 '18 at 22:38
TrigatenTrigaten
112
edited Dec 13 '18 at 22:54
Trigaten
asked Dec 13 '18 at 22:38
TrigatenTrigaten
112
asked Dec 13 '18 at 22:38
TrigatenTrigaten
112
asked Dec 13 '18 at 22:38
TrigatenTrigaten
112
112
add a comment |
add a comment |
1 Answer
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I do not think anyone is going to have an exact answer, but here is an upper bound.
There are $100$ positions for the snake's head, at most $4$ possibilities for the body segment next to the head, and at most $3$ for each subsequent body part. Therefore, there are at most $400cdot 3^{k-2}$ possibilities for a snake of length $k$. Summing from $k=2$ to $100$ gives $$#text{ snake positions}le 400(3^{99}-1)/2.$$I suppose you should multiply that by $100$ to account for the position of the food piece.
answered Dec 13 '18 at 22:57
Mike EarnestMike Earnest
23.9k12051
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$begingroup$
I do not think anyone is going to have an exact answer, but here is an upper bound.
There are $100$ positions for the snake's head, at most $4$ possibilities for the body segment next to the head, and at most $3$ for each subsequent body part. Therefore, there are at most $400cdot 3^{k-2}$ possibilities for a snake of length $k$. Summing from $k=2$ to $100$ gives $$#text{ snake positions}le 400(3^{99}-1)/2.$$I suppose you should multiply that by $100$ to account for the position of the food piece.
answered Dec 13 '18 at 22:57
Mike EarnestMike Earnest
23.9k12051
$endgroup$
add a comment |
$begingroup$
I do not think anyone is going to have an exact answer, but here is an upper bound.
There are $100$ positions for the snake's head, at most $4$ possibilities for the body segment next to the head, and at most $3$ for each subsequent body part. Therefore, there are at most $400cdot 3^{k-2}$ possibilities for a snake of length $k$. Summing from $k=2$ to $100$ gives $$#text{ snake positions}le 400(3^{99}-1)/2.$$I suppose you should multiply that by $100$ to account for the position of the food piece.
answered Dec 13 '18 at 22:57
Mike EarnestMike Earnest
23.9k12051
$endgroup$
add a comment |
$begingroup$
I do not think anyone is going to have an exact answer, but here is an upper bound.
There are $100$ positions for the snake's head, at most $4$ possibilities for the body segment next to the head, and at most $3$ for each subsequent body part. Therefore, there are at most $400cdot 3^{k-2}$ possibilities for a snake of length $k$. Summing from $k=2$ to $100$ gives $$#text{ snake positions}le 400(3^{99}-1)/2.$$I suppose you should multiply that by $100$ to account for the position of the food piece.
answered Dec 13 '18 at 22:57
Mike EarnestMike Earnest
23.9k12051
$endgroup$
I do not think anyone is going to have an exact answer, but here is an upper bound.
There are $100$ positions for the snake's head, at most $4$ possibilities for the body segment next to the head, and at most $3$ for each subsequent body part. Therefore, there are at most $400cdot 3^{k-2}$ possibilities for a snake of length $k$. Summing from $k=2$ to $100$ gives $$#text{ snake positions}le 400(3^{99}-1)/2.$$I suppose you should multiply that by $100$ to account for the position of the food piece.
answered Dec 13 '18 at 22:57
Mike EarnestMike Earnest
23.9k12051
answered Dec 13 '18 at 22:57
Mike EarnestMike Earnest
23.9k12051
answered Dec 13 '18 at 22:57
Mike EarnestMike Earnest
23.9k12051
answered Dec 13 '18 at 22:57
Mike EarnestMike Earnest
23.9k12051
23.9k12051
add a comment |
add a comment |
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