Bing's house, but with n rooms?

Question

In Chapter 0 of his Algebraic Topology, Hatcher presents Bing's house with two rooms as an example of a space that is contractible, but "not in any obvious way." He then asks, as an exercise (Chapter 0, exercise 8), for the reader to construct a similar house, but with $n$ rooms. Here's my problem. I can easily make a house with two rooms that is contractible in an obvious way. Start with a brick of clay; poke a hole in the top and hollow out a room in the top half of the brick; then do the same starting from the bottom. Clearly I can make an $n$-room house in the same way, poking holes in $n$ spots on the brick, but this is also clearly not what Hatcher is looking for. Somehow, Bing's house--which you get by poking a hole in the top and hollowing out a room in the bottom, and vice versa--is more interesting than mine, even though they're topologically equivalent. I need to understand precisely what the difference is in order to construct an $n$-room analog with the same qualities. What is it?