# Steps on The Coordinate Plane

# Rules

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 \((4,4)\).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*.For every \(n\geq 2\), a

*n-step*must share an endpoint with a*(n-1)-step*.Two

*n-steps*may only intersect at their endpoints.Every lattice point may contain at most two

*n-steps*.

# Tasks

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

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

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

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

How many ways are there to construct such a figure?