Variation to Nim game: Alice and Bob have drawn table w/ 11 columns x 10 rows and...

[eduardo.Juan]

Alice and Bob have drawn a table with 11 columns x 10 rows and take turns to mark 1 to 3 squares at a time. Alice plays first and in each move she can mark squares only from one column, while Bob can mark squares from different columns, but at most one from each column. Each square can only be marked once. The player who marks the last square is the winner. If both players play perfectly, is there a winning strategy for any of the two? If yes, describe it!

 

[Deleted member 4993]

Alice and Bob have drawn a table with 11 columns x 10 rows and take turns to mark 1 to 3 squares at a time. Alice plays first and in each move she can mark squares only from one column, while Bob can mark squares from different columns, but at most one from each column. Each square can only be marked once. The player who marks the last square is the winner. If both players play perfectly, is there a winning strategy for any of the two? If yes, describe it!

What are your thoughts?

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[mmm4444bot]

Is this an exercise from an online class or competition?

If neither, from where does this question come? :cool:

 

[eduardo.Juan]

Is this an exercise from an online class or competition?

If neither, from where does this question come? :cool:

No it is not (I am a grown-up, not a student). Just a challenge between friends - and now we are all looking for a solution!
No idea where the original question came from.

 

[eduardo.Juan]

some ideas...

[FONT=&quot]If I am Bob, in order to win, I must leave Alice with 1+1+1 (in 3 different columns) or 1+1 squares. If I left Alice with 2+1+1, then Bob, playing perfectly, would mark 1 of the single squares, knowing that Alice would then mark either the other single or the single plus one of the 2 left, so Bob would mark the last 1 or 2 from the same column.
[/FONT]
[FONT=&quot]We will now test the cases 2+2+1 or 3+1+1.
[/FONT]
[FONT=&quot]2+2+1 : If Alice marks 2, then Bob will mark 1 from the remaining 2 and he will win.[/FONT]
[FONT=&quot]If Alice marks 1 of the 2, then Bob will be in the case 2+1+1 and he would mark 1 of the 2, so he would win.
[/FONT]
[FONT=&quot]If Alice marks a single 1, then Bob will mark 1 of 2 and will win.
[/FONT]
[FONT=&quot]3+1+1: We will examine four different cases, like above.
I do not see any clear pattern, though.[/FONT]

 

[mmm4444bot]

Just a challenge between friends - and now we are all looking for a solution!

Okay; thanks, for that. I had asked because another user posted the same text here, a few days ago. They posted from Greece; you posted from Norway.

Two things were mentioned.

1) Try working with a simpler grid, to start. Perhaps, four rows with five columns. Play the game for awhile, to see what strategies develop.

2) Try working backwards from a solution.

 

[mmm4444bot]

If I am Bob, in order to win, I must leave Alice with 1+1+1 (in 3 different columns) or 1+1 squares. If I left Alice with 2+1+1, then Bob, playing perfectly, would mark 1 of the single squares, knowing that Alice would then mark either the other single or the single plus one of the 2 left, so Bob would mark the last 1 or 2 from the same column.

We will now test the cases 2+2+1 or 3+1+1.

2+2+1 : If Alice marks 2, then Bob will mark 1 from the remaining 2 and he will win.
If Alice marks 1 of the 2, then Bob will be in the case 2+1+1 and he would mark 1 of the 2, so he would win.

If Alice marks a single 1, then Bob will mark 1 of 2 and will win.

3+1+1: We will examine four different cases, like above.
I do not see any clear pattern, though.

I have not played it, but I had thought of Bob striving to leave one unmarked spot in as many columns as possible, to hamstring Alice.

Seems like you're off to a good start, with analyzing the game.

If it's possible, a strategy for Alice would be to fill in as many columns as possible, to hamstring Bob.

I would need to play the game, awhile, to see what happens. Have fun! :cool:

 

[eduardo.Juan]

Thanks, by the way, I am posting from USA, not Norway I was just using a mobile hotspot.