The Crimson Rambler — John Russell Fearn

When Warner Darnworth is shot through the back of his head in his study while seated at his desk, Chief Inspector Gossage is called in to investigate. But the study door is securely locked on the inside with the only key, the closed window is tamper-proof, no weapon can be found, the movement of everyone in the house has been accounted for, and there are no traces of footprints in the garden outside. In other words, it's an impossible crime. Gossage finds plenty of family skeletons and motives that might have caused Darnworth's murder. But who actually killed the man--and more to the point, how? A classic locked-room mystery.