# A stroll in the park

Professor Erasmus has returned from his saturday walk in the park. He has counted the number of trees in the park and also the number of lines formed by these trees. Professor Erasmus claims that

• there are exactly 26 trees in the park,
• there are exactly 306 (distinct) lines formed by these trees,
• and that none of these lines does contain four or more trees.

(When the professor speaks of trees on a line, then he means that the centerpoints of the trees lie on a common line.)

We wonder: Has the professor once again made one of his well-known mathematical blunders, or does such a collection of 26 trees indeed exist?

• Swords at sundown? – Rand al'Thor Mar 22 '15 at 16:43

The maximum possible lines is $\sum 25=\frac{25*26}2=325$
The only way to reduce the line count is to put a point into a pre-existing line, which reduces exactly $2$ lines. Instead of $3$ lines $AB$, $BC$ and $CA$, we have a single line $ABC$. We are not allowed to add another point to the line.
Therefore, we can not have an even number of lines, as subtracting $2$s will only result in another odd number. Professor Erasmus has made a blunder.