ATAR Notes: Forum
VCE Stuff => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematics => Topic started by: Andiio on August 04, 2010, 11:47:55 pm
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There are four lifts in a building. Each makes three stops, which do not have to be on consecutive floors or include the ground floor. For any two floors, there is at least one lift which stops on both of them. What is the maximum number of floors that this building can have?
(A) 4, (B) 5, (C) 6, (D) 7, (E) 12.
Please provide an explanation as well!
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I do know that 4 floors is too limited and that 5 is the answer; however I'm not really sure of the 'for any two floors, there is at least one lift which stops on both of them.' part.
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it can be argued that 6 floors is too much.
Proof:
since there are exactly 3*4=12 stops in total (where we call two stops equal iff they are the same lift and the same floor). Suppose there were 6 floors. Then by the pigeonhole principle there is some floor with at most 2 stops. Ie: at most two lifts go there, call these lifts A and B. lift A can only go to two other floors, lift B can go to only two other floors. Thus only 4 other floors can be reached from this floor, but there are 5 other floors.
a case with 5 floors can be constructed I think.