# [ugrads] [Mathclub-members] Fwd: Item of interest

Anthony Ruth aruth at hawk.iit.edu
Thu Nov 17 14:31:20 CST 2011

```The question of whether it is better to attack or defend in risk I have
considered many times and I wrote a program to find the answer. Normally
the attacker can use at most 3 dice and the defender at most two. The die
are matched highest to highest and if there is a tie the defender wins. I
was always told that the defender had the advantage. The attached program,
written in C, uses the rules for determining who wins a roll and goes
through every single possible dice roll to determine the attacker's and
defender's win probability. It does this for every combination from 1
attacker dice and 1 defender dice to 4 attacker dice and 3 defender dice. I
have attached both the output it produced and the program itself. It's not
commented but if you are familiar with C, you should be able to understand
how it works. For 3 attacker dice and 2 defender dice you are better off
being the attacker with a ratio of 1.17:1

On Thu, Nov 17, 2011 at 11:41 AM, Math Club IIT <mathclub.iit at gmail.com>wrote:

> Begin forwarded message:
>
> *From:* Bill <wmolchan at hawk.iit.edu>
> *Date:* November 2, 2011 8:59:15 PM CDT
> *To:* "mathclub at iit.edu" <mathclub at iit.edu>
> *Subject:* *Item of interest*
>
> I am looking to form a club based on using mathematical analysis to
> determine winning strategies in games. So whether it is better to attack or
> defend in risk, or the payoff of a monopoly property. Other ideas might be
> blackjack and poker.
>
> The events we would like to do would be things like a free sealed magic
> tournament, and a no-buy-in poker tournament with prizes.
>
> As the mathclub, I thought that you may have interested members.
> If your interested you can join the facebook group:
> http://www.facebook.com/groups/177489822339629/
> or email me at wmolchan at hawk.iit.edu
> or text me at 732 455 1159
> I am still looking for sigs to get the club started.
> You would not have to attend say a blackjack meeting to attend the magic
> tournament. If you cared about magic and not blackjack you would only need
> to attend the magic meeting to participate in the magic tournament.
>
> If you could distribute this email to your members, I would appreciate it.
> Bill Molchan
>
>
>
> _______________________________________________
> Mathclub-members mailing list
> Mathclub-members at math.iit.edu
> http://math.iit.edu/mailman/listinfo/mathclub-members
>
>

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The results for 1 attack dice and 1 defender dice is

The attackers win probability is 0.416667 and the defenders win probability is 0.583333, the ratio is 0.714286 to 1

The results for 1 attack dice and 2 defender dice is

The attackers win probability is 0.25463 and the defenders win probability is 0.74537, the ratio is 0.341615 to 1

The results for 1 attack dice and 3 defender dice is

The attackers win probability is 0.173611 and the defenders win probability is 0.826389, the ratio is 0.210084 to 1

The results for 2 attack dice and 1 defender dice is

The attackers win probability is 0.578704 and the defenders win probability is 0.421296, the ratio is 1.37363 to 1

The results for 2 attack dice and 2 defender dice is

The attackers win probability is 0.38966 and the defenders win probability is 0.61034, the ratio is 0.638432 to 1

The results for 2 attack dice and 3 defender dice is

The attackers win probability is 0.253279 and the defenders win probability is 0.746721, the ratio is 0.339189 to 1

The results for 3 attack dice and 1 defender dice is

The attackers win probability is 0.659722 and the defenders win probability is 0.340278, the ratio is 1.93878 to 1

The results for 3 attack dice and 2 defender dice is

The attackers win probability is 0.539545 and the defenders win probability is 0.460455, the ratio is 1.17176 to 1

The results for 3 attack dice and 3 defender dice is

The attackers win probability is 0.377283 and the defenders win probability is 0.622717, the ratio is 0.605865 to 1

The results for 4 attack dice and 1 defender dice is

The attackers win probability is 0.707433 and the defenders win probability is 0.292567, the ratio is 2.41802 to 1

The results for 4 attack dice and 2 defender dice is

The attackers win probability is 0.624914 and the defenders win probability is 0.375086, the ratio is 1.66606 to 1

The results for 4 attack dice and 3 defender dice is

The attackers win probability is 0.46414 and the defenders win probability is 0.53586, the ratio is 0.866159 to 1

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```

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