Problem 4

Determine all pairs

(h,s)

of positive integers with the following property: if one draws

h

horizontal lines and another

s

lines satisfying (i) they are not horizontal, (ii) no two are parallel, and (iii) no three of the

h+s

lines are concurrent, then the number of regions formed is

1992

Step 2 of 5: Add the horizontal lines

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a_{h,s}=a_{0,s}+h(s+1)=\frac{s^2+s+2}{2}+h(s+1)

Detailed analysis

Each horizontal line intersects all $s$ non-horizontal lines and no other horizontal line, so when it is added it is cut into $s+1$ pieces. Therefore each horizontal line adds $s+1$ regions, giving the displayed formula.