We present yet another new family of masks for high-contrast imaging, as required for the planned Terrestrial Planet Finder space telescope. We call these masks "checkerboard" masks. They consist of two "bar-code" masks, one rotated 90° with respect to the other. Each bar-code mask provides contrast to the 10-5 level. The checkerboard mask then achieves a 10-10 level of contrast everywhere except along the two axes of symmetry, where the contrast remains at the 10-5 level. With these designs, we are able to reduce the inner working angle to 2λ/D for each bar code, which translates to 2λ/D along the diagonal of the associated checkerboard mask. We show that by combining a Lyot-plane checkerboard mask with an image-plane occulter we can achieve even tighter inner working angles, although as with occulting designs in general pointing error and stellar size become nontrivial issues. Checkerboard masks can be thought of as the binary-mask analog of Nisenson's apodized square aperture concept.