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max no r of horses Billy can own standing on lattice points

Source: 2021 Dürer Math Competition Finals Day2 E+10 https://artofproblemsolving.com/community/c2749870_

January 9, 2022
combinatorics

Problem Statement

Billy owns some really energetic horses. They are hopping around on points of the plane having two integer coordinates. Each horse can make the following types of jumps. They can hop to a point obtained from their current position via a translation by vector (15,9)(15, 9), (−9,15)(-9, 15), (−15,−9)(-15,-9) or (9,−15)(9,-15). The horses are now standing on lattice points such that no two can meet by making jumps as above. What is the maximal possible number of horses Billy can own?
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