Question

A gardner throws 18 seeds onto an equilateral triangle shaped plot of side 1 meter. Then at least 2 seed are within distance of 25cm Statement is true or false?

Answer 1

Yes

Explanation:

Split the large triangle into `16` equilateral triangles with sides of length `25cm`...

Each seed is inside or on the boundary of at least one of the smaller triangles.

So if `17` seeds are distributed between the small triangles then at least one triangle has at least two seeds within it or on its boundary.

The distance between two seeds in or on the same triangle is at most `25cm`. If equal to `25cm` then they must lie at two corners.

At least one of those corners is shared with an adjacent small triangle. So if any seeds associate with that adjacent triangle are at maximal distance `25cm` then they must also lie at the corners of that triangle.

Continuing, we find that if we try to ensure the maximum distance of `25cm` between all of the seeds while keeping them in their small triangles, then they all lie on corners.

There are only `15` corners at which seeds can be placed, so the condition must break and at least `2` seeds are closer than `25cm`.