The poem to the right tells how a clever innkeeper (Hilbert by name?) with nine empty rooms had no difficulty in providing separate lodgings for each of 10 travelers.  Does his plan actually work? What's the explanation?

Ten weary, footsore travelers,             All in a woeful plight, Sought shelter at a wayside inn             One dark and stormy night. "Nine rooms, no more," the landlord said,             "Have I to offer you. To each of eight a single bed,             But the ninth must serve for two." A din arose. The troubled host             Could only scratch his head, For of those tired men no two             Would occupy one bed. The puzzled host was soon at ease __             He was a clever man __ And so to please his guests devised             This most ingenious plan. In room marked A two men were placed             The third was lodged in B, The fourth to C was then assigned,             The fifth retired to D. In E the sixth he tucked away,             In F the seventh man, The eighth and ninth in G and H,             And then to A he ran, Wherein the host, as I have said,             Had laid two travelers by; Then taking one __ the tenth and last __             He lodged him safe in I. Nine single rooms __ a room for each __             Were made to serve for ten; And this it is that puzzles me             And many wiser men.