How to keep matplotlib grid constrained while drawing outside of the grid?

I'm trying to draw the following image in matplotlib:

I don't care about the red dot or yellow circle for now. All I'm trying to do is get the grid drawn with the semicircles. A key point is that I want the outlines to show like in the picture. At the moment, the code I have is this:

import matplotlib.pyplot as plt
    
N = 5
colors = [(111,187,255),(185,239,133)] # Blue and green
plaquettecolors = ["#b9ef85ff", "#6fbbffe5"] # Green and blue
rowcolor = colors*((N-1)//2)
fig, ax = plt.subplots()


# Build the matrix for the colours
bulkcolors = []
for y in range(N-1):
    bulkcolors.extend(rowcolor[::(-1)**y])
    greenCircle = plt.Circle(
        (y,-0.5 + (1+(-1)**(y+1))*(N-1)/2),
        0.5,
        color=plaquettecolors[0],
        fill=True,
        ec="black",
        lw = 2,
        zorder=0
        )
    blueCircle = plt.Circle(
        (-0.5 + (1+(-1)**y)*(N-1)/2,y),
        0.5,
        color=plaquettecolors[1],
        fill=True,
        ec = "black",
        lw = 2,
        zorder=0)
    ax.add_patch(greenCircle)
    ax.add_patch(blueCircle)

# Reshape things.
bulkcolors = np.array(bulkcolors)
bulkcolors = bulkcolors.reshape((N-1, N-1,-1))

#ax.axis("off")
ax.matshow(bulkcolors)
plt.xticks(range(-1,N))
plt.yticks(range(-1,N))
ax.grid(which='major', axis='both', linestyle='-', color='k', linewidth=2)
ax.set_xticks(np.arange(-0.5, N-1, 1))
ax.set_yticks(np.arange(-0.5, N-1, 1))
ax.tick_params(which="major", bottom=False, left=False)
plt.show()

The issue is that I get the following:

What I'm hoping for is a way to remove the grid lines outside of the checkerboard pattern, as well as the numbers. Any tips to get this working would be appreciated!

Solution

Hide the gridlines and draw horizontal and vertical lines instead:

ax.axis("off")
ax.matshow(bulkcolors)
plt.xlim(-1.1, N-.9)
plt.ylim(-1.1, N-.9)
for i in np.arange(-.5, N-1, 1):
    plt.hlines(i, -.5, N-1.5, lw=2, color='k')
    plt.vlines(i, -.5, N-1.5, lw=2, color='k')