KissaBoo

I'm so impatient with brainteasers, so now I'm going to ask the loyal readers of Kissaboo for help. If only there were a MathCounts coach here or something...

There are three rows of pearls:

xxx
xxxx
xxxxx

You can remove any number of pearls from any ONE row on your turn. The object is to leave the last pearl for your opponent to pick up. You move first.

Any suggestions? I'm pretty disappointed in myself for not figuring it out, but, then again, I spent like 2 minutes on it and got pissed.

Posted by: maggette at August 4, 2004 02:51 PM

I spent more than 2 minutes on this, but not much more since I am working.

Quick Points:
You’ve won if on your turn:
a. There is only one row left with more than 1 token
b. There are two rows left and one of which only has 1 token

Corollary:
You’ve lost if on your turn you have the following situation:
x x
x x

Thus to win:
Take all 5 tokens.
Your opponent will be left with
x x x
x x x x

If they take one from the 4, take one from either row.
Otherwise, you can leave them with 2 by 2 or are already in a winning situation.

I will simplify this in the future and extend for all cases.

Posted by: boo at August 4, 2004 04:41 PM

I took all 5 tokens, and then they took one from the row of 4. From this position:

xxx
xxx

I always lose.

By the way, the actual game is here:

http://www.albinoblacksheep.com/flash/pearl.php

After I asked, somebody put up a solution, but I'm not entirely clear on it:

http://yarrthepirate.com/v-web/bulletin/bb/viewtopic.php?t=765

Posted by: maggette at August 4, 2004 06:19 PM

Short answer: take 2 away from the row with 3

As an exercise, show that this is in fact the only guaranteed winning first move.

I will take a look at the links after I finish this, but I realized after I posted my previous response that I lost track of whose turn it was.

One day I will make this elegant, but here are some fast and dirty points to explain how to win the game. When I use n, it stands for the number of x's in the row.

Obvious losing positions:
x and x
x
x

Obvious winning positions:

a. n (n > 1)

b.
x
n (n > 0)

Thus losing position from a and b:
c.
x x
x x

One can therefore extrapolate that 2 even rows of two or more x’s is a losing position and 2 uneven rows is a winning position.

Therefore for three rows the following is a winning position (except for n = m = 1):
n
n
m

since when n > 1, deleting all m leads to a losing position, and when n = 1, deleting all but 1 in the m row leads to a losing position.

Hence anytime the opponent is forced to leave the above scenario, you've won.

x
xx
xxx

and

x
xxxx
xxxxx

are two arrangements that forces a winning position to be left by the opponent.

Posted by: boo at August 5, 2004 12:20 AM

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