In a certain tribe, you have a certain amount of tribal offerings, at the start of year 1. However, at the start of each year (including this one), you must feed the black hole with a number of tribal offerings equal to the size of the black hole. On year 1, the black hole starts as size 1 and doubles each year that you pay the tribal offerings. (If it was 4, it's 8 now.) If you can't pay this cost, the island will explode in the middle of this year.

However, your workers are very industrious with investing, and always manage to double the number of tribal offerings that you had at the beginning of the year after paying the black hole.

For example, if you started with 4 offerings: (B = beginning of year before feeding the black hole, A = after you fed the black hole, E = end of year after your tribal offerings have doubled)

--B-A-E
1|4 3 6
2|6 4 8
3|8 4 8
4|8 0 0
5|0

Since there wasn't enough to pay 16 tribal offerings, the island lasted 5 years.

How would you find the number of turns this island would last if you started with x tribal offerings?

I know this problem is old, but I'm new to the site, so I'm still catching up. :) 

While I basically agree with the solutions already posted, I'm a little confused by the wording.  For example, the problem starts off discussing "years", but then ends asking about the number of "turns" - how exactly is a turn defined? 

In the example given, it says that "the island lasted 5 years."  This isn't exactly right, because the island had nothing to offer at the beginning of the 5th year, and so only lasted 4 years (or actually, 4 1/2 years, because the question states that "if you can't pay this cost, the island will explode in the middle of this year."

I'm not trying to be nitpicky, nor am I questioning the validity of the solutions - I just think the question is a little ambiguously worded, which is why some people came up with x years and some with x+1 years, even though the method to their solutions were the same.  Either way, it's still a good puzzle.

 

Posted by tomarken on 2006-02-28 15:29:27