Claim the Last Flag

Date: 10/25/2002 at 13:02:43
From: Cathy Williams
Subject: Survivor Challenge: be the one to choose last

I saw a challenge on tv last night that had two teams facing a field 
of 21 flags. The challenge was this: the teams took turns choosing 1, 
2, or 3 flags at each turn. The team that was able to claim the last 
flag won.

I know I saw a show about this problem a few months back. There's an 
equation of sorts for how to win.

Any help?
Thanks!

Date: 10/25/2002 at 14:54:02
From: Doctor Achilles
Subject: Re: Survivor Challenge: be the one to choose last

Hi Cathy,

Thanks for writing to Dr. Math.

If there are 1, 2, or 3 flags left when it's my turn to choose, then I 
can win easily.

If there are 4 flags left when it's my turn to choose, then no matter 
what I do, my opponent will be left with 1, 2, or 3 flags, and can 
win easily.

So I want to make sure that I can leave my opponent with 4 flags on 
his/her turn.

How can I do that?  Well, if there are 5, 6, or 7 flags left on my 
turn, then I can easily leave my opponent with 4 flags.

But I have to be careful. If there are 8 flags left on my turn, then I 
have to give my opponent 5, 6, or 7 flags and s/he will be able to 
leave me with 4 on my next turn and make me lose.

So I want to make sure that I can leave my opponent with 8 flags on 
his/her turn.

How can I do that? Well, if there are 9, 10, or 11 flags left on my 
turn, then I can easilty leave my opponent with 8 flags.

But I have to be careful. If there are 12 flags left on my turn, then 
I have to give my opponent 9, 10, or 11 flags and s/he will be able 
to leave me with 8 flags on my next turn and make me lose.

So I want to make sure that I can leave my opponent with 12 flags on 
his/her turn.

How can I do that?

See the pattern?

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/