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June 6th, 2018, 11:58 AM   #1
Joined: Feb 2018
From: California

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Addition Nim

I am unfamiliar with the game of Nim and I am just learning that there are MANY different versions out there.

I want to learn about Addition Nim, where two players count to a number (n) and the winner says n at the end.

For example, the first person to say "20" wins. Rules are that you can add either 1, 2, 3, or 4 on your turn, but you must add something. The first person must start at 0. So it can progress like:

1st person: 0
2nd person: 1, 2, 3, or 4
1st person: 5
2nd person: 6, 7, 8, or 9
1st person: 10
2nd person: 11, 12, 13, or 14
1st person: 15
2nd person: 16, 17, 18, 19
1st person wins with 20

For this example, I see how to win. You should pick multiples of 5 so that your opponent can never quite reach 20. (Or is this not accurate? That's what I came up with.)

I want to know how to figure out strategies for other examples, too. For example:

1) if the goal is 20 and you can only add 1 or 2. How does the strategy change? I can see that you want to have 17 as your last number, but how do I guarantee that throughout the game?

2) if the goal is 21 or 22 and you an add 1 through 4, what is the strategy?

3) if the goal is 21 or 22 and you can add 1 or 2, what is the strategy?

Is there an overall strategy that applies to all Addition Nim games like these?
mjsilverfly is offline  
June 6th, 2018, 12:24 PM   #2
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Actually, I think I have something. Correct me if I'm mistaken....

does it have to do with mod math? So if let's say the goal is to say 22 and you can only count by 1, 2, 3, or 4, you need to always say a number with remainder 2 when you divide it by 5, so you'd have to say 7, 12, then 17, then win.

So you divide by 5 in this example because you can only count up to 4. Does this make sense?
mjsilverfly is offline  
June 7th, 2018, 12:27 AM   #3
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It looks like you've solved it to me. You could formalise this:
n= final target
m= maximum increment allowed (4 in the example)
r= remainder of n / (m + 1)
x= any number (will want to start at 0 or 1 and increment by 1 on your turn until the target is reached)

step targets are then (m+1)x + r

If r is 0 then the first player (who starts at 0) wins, otherwise the second player wins.
weirddave is offline  
June 7th, 2018, 01:06 PM   #4
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From: California

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Awesome! Thank you!
mjsilverfly is offline  

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