Steps on The Coordinate Plane

Jacob Kirmayer

Coordinate Plane Diagram

Coordinate Plane Diagram

Rules

  1. A n-step, where n is a positive integer, is a line segment of length \(n\) and slope \(0\) or \(undefined\), or a line segment of length \(n\sqrt{2}\) and slope of \(1\) or \(-1\) . The endpoints of a n-step must be on a lattice point. For example, you may construct a 2-step segment with length of \(2\sqrt{2}\) and endpoints on \((2,2)\) and \((4,4)\).

  2. Your \(nth\) construction must be an n-step. For example, your 1st construction must be a 1-step. Your 2nd construction must be a 2-step. Your 14th construction must be a 14-step.

  3. For every \(n\geq 2\), a n-step must share an endpoint with a (n-1)-step.

  4. Two n-steps may only intersect at their endpoints.

  5. Every lattice point may contain at most two n-steps.

Tasks

  1. On the graph, construct a figure that passes through all lattice points \((a,b)\) where \(-3 \leq a \leq 3\) and \(-3 \leq b \leq 3\).

  2. On the graph, construct a figure that passes through all lattice points \((a,b)\) where \(-4 \leq a \leq 3\) and \(-4 \leq b \leq 3\).

  3. Construct a figure that passes through all lattice points \((a,b)\) where \(0 \leq a \leq x\) and \(0 \leq b \leq y.\)

    1. What is the minimum number of lattice points (including the endpoints of line segments) a figure passes through, in terms of the given coordinates?

    2. How many ways are there to construct such a figure?