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### Russian Nim

```
Date: 02/15/99 at 14:30:24
From: Kevin Banville
Subject: Russian Nim

My math teacher told me about a game called Russian Nim. We like to
play the "20" game. It is as follows:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

You may pick a different game each time (the 15 game, the 10
game, etc.)

You play by crossing out a number and all its factors. The person who
crosses out the last number wins. Somebody beats me all the time and I
was wondering if you could offer me some strategies to win.
```

```
Date: 02/18/99 at 02:32:43
From: Doctor Schwa
Subject: Re: Russian Nim

Think which numbers get crossed out and when?

1 gets crossed out all the time. Any number from 11-20 only gets
crossed out when you hit it, because it can't be a factor of any (even
a bigger) number. That means the game lasts at least 10 turns. The key,
therefore, is to play to use up the right amount of the smaller numbers
(1-10).

There are still a lot of possibilities. Let's start with some simpler
games, like the 4 game.

1 2 3 4
If I go first, crossing out 4, you cross out 3 and win.
If I cross out 3, you take 4 and win.
If I take 2, that leaves 3 and 4; you get one, I get the other, and
I win!

Okay, now let us try the 6 game.

1 2 3 4 5 6
I go first, I take 6, you take 4 or 5, I get the other, I win!

1 2 3 4 5 6 7 8
I go first; if I take 8, that leaves 3 5 6 7.
If you take 6, I take 5 or 7, you get the other, you win.
If I take 4 instead, you take 3, that leaves 5 6 7 8; each time I take
one, you take one, so you win.
If I take 2 instead, that leaves the 3 6 pair, the 4 8 pair, 5, and 7.
Aha! Now I can win. Whatever you do to the 36 pair, I do the same to
the 48 pair, and if you take 5, I take 7. So I always get the last
number.

So far, whoever goes first should always win by doing the right thing.
The hard part is seeing any kind of pattern for what the "right thing"
is.

I don't think my analysis is good enough yet to handle the 20 game,
but maybe you can see a few nice strategies to help you out here, the
main one being to "pair up" things.

- Doctor Schwa, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Puzzles
Middle School Puzzles

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