Use a chalk snap line to divide the floor into four quadrants, with two lines crossing from the center of one side to the center of the side across from it and intersecting in the middle. Lay a square at the intersection as you lay the lines to ensure they are exactly 90 degrees square.
Spread vinyl tile adhesive onto the middle of the floor with a notched trowel, covering about four square feet in one of the four corners formed by intersected lines.
Press a linoleum tile onto the floor at the intersection, so it's bordered by both lines. Press the tile fully down into the adhesive.
Set the second linoleum tile of the opposite color against the first, making sure the edges are butted tightly to each other.
Repeat, laying alternately colored tiles abutting each other in a checkerboard grid, using the lines to keep it all straight. Spread more adhesive as needed, working your way toward the walls.
Using a vinyl tile cutter, cut tiles to fit around the perimeter of the room after all the full tiles are laid. Install the cut tiles with the cut sides facing the walls. Let them all set for 12 hours.