Problem 4

There is an integer

n > 1

. There are

n^2

stations on a slope of a mountain, all at different altitudes. Each of two cable car companies,

A

and

B

, operates

k

cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The

k

cable cars of

A

have

k

different starting points and

k

different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for

B

. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer

k

for which one can guarantee that there are two stations that are linked by both companies.

Step 3 of 8: Test the candidate value

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In plain words

At the proposed threshold, each company has only n−1 chains to accommodate all n² stations, which is the counting pressure we need.

k=n^2-n+1\quad\Longrightarrow\quad c_A=c_B=n-1.

Detailed analysis

Set k=n²−n+1. By the preceding count, both A and B decompose the n² stations into n²−(n²−n+1)=n−1 chains. We now show that this forces a pair linked by both companies.