by nsguy1350 » Sat Apr 06, 2013 10:01 pm
"Solution 1: You can arrange the coins in 4 ways. DDDPPP , DDPDPP , PPDPDD , and DPDPDP.
The problem is somewhat unclear. Imagine it like this: You are loading your grandma's 6-shooter. You have a 3 red bullets and 3 blue bullets. Grandma is particular about how they are arranged in the gun, but she won't like kill you if she doesn't have to unload. So, how many different ways can you load the gun?
Solution 2:
I define a "circle" of coins to be an arrangement of 6 coins in the 6 fixed spots. This distinguishes between different rotations of an arrangement.
I define a "position" to be an arrangement of 6 coins in 6 unfixed spots. This does not distinguish between rotations of an arrangement.
The only time you can rotate a circle of coins less than 6 places and get the same circle is when it is periodic: PDPDPD or DPDPDP. There are 20 ways to arrange the 6 coins in a line, and thus to arrange them in the 6 fixed spots around the circle. Two of these 20 ways will be the same position: DPDPDP, PDPDPD. All the others come in groups of 6, because you have to rotate them 6 times before it looks the same again. That means there is the one plus (20-2)/3 positions, which is 4."
This is quoted from AuSmith from another thread after a google search on the problem. Apparently, it is a bit different than similar problems I did before. It's a bit tricky, but apparently it didn't trip up Mark Zhang, who got a 360 on this test in competition.
2013 District 1/Region:
NS - 319/355
MA - 340/332
CA - 294/287
SC - 344/292
CS - 212/124 (fail)