Levik has invested a lot in real estate, and now owns all the land on one side of Flooble Blvd, ending with his very own Perplexus Tower. He now has 10 clients who each want to lease some land in order to build.

Each client plans to build something with a different number of stories from 1 to 10. Levik wants to be able to see all these buildings from the top of Perplexus Tower. This is impossible if any building is immediately behind another that is two or more stories higher.

In how many different orders can Levik lease the narrow strip of land to his 10 clients? What if, after building Perplexus Tower even higher, buildings can only be obscured if the one directly in front of it is three or more stories higher? Four? More?

If 3 or more stories higher obscures (2 story limit):

For 3 buildings any order is allowed to 3! orders
To add a fourth it can go in front of 2, 3 or in the back.
Building n+1 can go in front of n, n-1 or in the back.
So for 10 buildings we have 3!*3^7 = 13122 orders

Similar reasoning yields:
3 story limit: 4!*4^6 = 98304
4 story limit: 5!*5^5 = 375000
5 story limit: 6!*6^4 = 933120
6 story limit: 7!*7^3 = 1728720
7 story limit: 8!*8^2 = 2580480
8 story limit: 9!*9^1 = 3265920
9 story limit: 10! = 3628800 (no actual limitations at all)

Posted by Jer on 2006-04-27 13:06:05