### Solitaire game

Posted:

**Tue Oct 11, 2016 2:58 pm UTC**Hi,

This a new game designed to be played on computer.

Mobius solitaire game.

You take an urn with 100 balls numbered from 1 to 100.

You start picking randomly one ball noted k(1) :

- you compute v=mu(k(1)) where mu() is the Mobius function

- if v=0 then you put back the picked ball to the urn

- if v=1 you place a new ball numbered (max(A)+1) where A is the set the remaining balls and put aside the just picked ball with v=1

- if v=-1 you choose "smartly" one ball from the urn and you replace it by the ball just picked with v=-1

You do the process until you empty completely the urn.

If you play smartly how many picks do you need in the worst case to empty the urn?

Example :

URN : 1,2,3,......,100

Pick 1 : k(1)=15 then v=mu(15)=1 as the max(A)=100 you place a new ball numbered 101 on the urn and remove from the game the ball numbered 15

Pick 2 : k(2)=4 then v=mu(4)=0 you put back the picked ball 4 to the urn

Pick 3 : k(3)=7 then v=mu(7)=-1 you need to choose one ball from the remaining balls

URN contains :1,2,3,...,14,16,17,....,100,101 (15 was removed and 101 was added)

Let us say that removing a ball with v=0 is "smart" choice (if is up to the player to choose the best strategy not me). So you remove the ball 8 (v=0)

and so on

Is there any coder?

This a new game designed to be played on computer.

Mobius solitaire game.

You take an urn with 100 balls numbered from 1 to 100.

You start picking randomly one ball noted k(1) :

- you compute v=mu(k(1)) where mu() is the Mobius function

- if v=0 then you put back the picked ball to the urn

- if v=1 you place a new ball numbered (max(A)+1) where A is the set the remaining balls and put aside the just picked ball with v=1

- if v=-1 you choose "smartly" one ball from the urn and you replace it by the ball just picked with v=-1

You do the process until you empty completely the urn.

If you play smartly how many picks do you need in the worst case to empty the urn?

Example :

URN : 1,2,3,......,100

Pick 1 : k(1)=15 then v=mu(15)=1 as the max(A)=100 you place a new ball numbered 101 on the urn and remove from the game the ball numbered 15

Pick 2 : k(2)=4 then v=mu(4)=0 you put back the picked ball 4 to the urn

Pick 3 : k(3)=7 then v=mu(7)=-1 you need to choose one ball from the remaining balls

URN contains :1,2,3,...,14,16,17,....,100,101 (15 was removed and 101 was added)

Let us say that removing a ball with v=0 is "smart" choice (if is up to the player to choose the best strategy not me). So you remove the ball 8 (v=0)

and so on

Is there any coder?