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?

If the tribe starts with X offerings, the island will last X turns.

Explanation:

If the tribe initially has X offerings, then its successive holdings are:

After turn 1: X-(2^0)
After turn 2: 2[X-2^0]-2^1 = (2^1)X-(2^1)-(2^1)
After turn 3: 2[(2^1)X-(2^1)-(2^1)]-2^2
= (2^2)X-(2^2)-(2^2)-(2^2)
After turn 4: 2[(2^2)X-(2^2)-(2^2)-(2^2)]-2^3
= (2^3)X-(2^3)-(2^3)-(2^3)-(2^3)
etc..............
After turn N: (2^[N-1])X-N(2^[N-1])
= (X-N)*(2^[N-1])

When N=X, X-N=0. The tribe will have zero.

P.S. You didn't mention what tribe this was, but I assume it is a Native American tribe. A Comanche friend of mine once related one of the
fascinating legends that are handed down in her tribe from generation to generation:

"This wagon train is heading across the desert, when all of a sudden the wagon master notices that on all sides of the valley, there are Comanche warriors! He quickly forms the wagons into the 'Hollywood' circle, to protect the families in the train. Nothing happens. Soon, drums are heard pounding out in the distance, BUM, bum, bum, bum, BUM, bum, bum, bum, BUM, bum, bum, bum.......(the famous Hollywood drumbeat from the John Ford movies). The wagon master tells the train, 'I don't like the sound of this....' From out in the distance comes the voice of the Comanche chief: 'Hey, give us a break, paleface! He's not our regular drummer!' "
Edited on February 18, 2004, 7:21 pm

Posted by Penny on 2004-02-18 19:20:14