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View from Perplexus Tower (Posted on 2006-04-27) Difficulty: 3 of 5
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?

  Submitted by Tristan    
Rating: 4.0000 (2 votes)
Solution: (Hide)
All the possibilities can found this way:
1) Start with the 1-story building
2) For each of the other buildings from 2 to 10, insert anywhere into the list such that it does not obscure any other buildings

For step 1, there will be 1 possibility.
For step 2, we will multiply the possibilities for each building we insert into the list. The 2-story building can go in front of or behind the 1. The 3-story building can go in front of the 2 or in the very back. The 4-story building can go in front of the 3 or in the very back. Continue like this to the last building.

As you might notice, there is a pattern in the number of possibilities: there are always 2 for each building. The number of possibilities is therefore equal to this product:
1*2*2*2*2*2*2*2*2*2 = 512

For the second question, going through the same steps, we will arrive at this product:
1*2*3*3*3*3*3*3*3*3 = 13122

For the third question,
1*2*3*4*4*4*4*4*4*4 = 98304

The general solution for N buildings, where a building is obscured if and only if it is directly behind a building M or more stories higher, and where M and N are positive integers and M ≤ N:
M! * M^(N-M)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Interesting variant - numberssnark2006-05-02 23:19:23
Some Thoughtsre: Interesting Variant Solution for 2 stories blocked, stories > 2 challengeTristan2006-05-01 18:35:36
Interesting Variant Solution for 2 stories blocked, stories > 2 challengejuju2006-05-01 16:29:37
re: interesting variantCharlie2006-04-28 10:33:57
re: Interesting variant of problemTristan2006-04-27 21:56:49
Interesting variant of problemjuju2006-04-27 18:35:26
SolutionThe rest of them.Jer2006-04-27 13:06:05
SolutionFirst part, alternate expl.Jer2006-04-27 12:55:14
SolutionSolution to first parttomarken2006-04-27 10:10:21
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