Step 1: Approach this by matching the syndrome to its tumour suppressor gene. Cowden syndrome is the prototype of the $PTEN$ hamartoma tumour syndromes, so the linked gene is in the name of the family of conditions itself.
Step 2: $PTEN$ encodes a phosphatase that normally restrains the PI3K-AKT growth signalling pathway. A germline loss-of-function mutation removes this brake, leading to widespread hamartomas and a high lifetime risk of breast and thyroid cancers.
Step 3: Hallmark clues that should trigger $PTEN$: facial trichilemmomas, oral papillomas, GI hamartomatous polyps, and macrocephaly in an autosomal dominant pattern.
Step 4: Cross off the other genes by their own syndromes - $P53$ goes with Li-Fraumeni, $Rb$ with retinoblastoma, and $Ras$ is an oncogene driving sporadic tumours rather than an inherited hamartoma syndrome.
So Cowden syndrome maps to PTEN.\[\boxed{\text{PTEN}}\]