Molly does not match a maternal first cousin or a mutual second cousin at 23andMe, but this is not enough to convince me that she is a paternal match. However, I observed that she matches my sister and me at an X-DNA location where we only share paternal X-DNA.
|some half and completely identical segments shared with my sister|
Our sex-determinining chromosomes have unique inheritance rules. Females generally have two X chromosomes, one from each parent, while males generally have one X chromosome inherited from their mothers and one Y chromosome inherited from their fathers. As paternal sisters share the same X chromosome inherited unchanged from their father (his only X chromosome), I know that the lighter shaded half-identical segment on the right of my 23rd chromosome pair is paternal X-DNA. We share very little maternal X-DNA, indicated by the relatively small darker shaded completely-identical segment on the left.
My working hypothesis is therefore that Molly is a paternal match to us. As males only inherit X-DNA from their mothers I will also assume that Molly is related to us via our paternal grandmother's line (Liverpool Irish), excluding three-quarters of our ancestral lines.