what was the actual scheme to catch the robber? who really figured it out? would something like this work today?

The five hundred dollar reward for the knife was publicly advertised, while the reward for the thief was privately communicated to the pawnbrokers. That way, the thief - unaware that there was a bounty on her head - would feel comfortable going into a pawnshop to sell her loot, and would promptly be captured. Tom points out that this scheme is foolproof, and that had the knife actually been stolen, the thief surely would have been uncovered by now. The only possible explanation, he postulates, is either that no such knife ever existed, or the twins still possess it themselves. He further notes that this would allow the twins to look wealthy and important by offering up the reward, without actually costing them anything. This explanation completely wins over the constable, and even causes Wilson to doubt the twins.