Invite a child to play. To reduce the chance of rejection, select the kid least likely to have other social commitments — say, the one who eats paper, or who's always all crusty and rabbity-looking around the nostrils, or whose mother talks about him on the Internet.
Determine a mutually agreeable time. Schedule it between fights with your spouse.
Plan an activity. Perhaps the children would enjoy playing Dickensian chimney sweep. Yeah...that's it. "Playing." So adorable with those sooty faces and that deep-seated rattling cough...
Tidy up. The floor should be clean enough to eat off when you fling that handful of Cheddar Bunnies down onto the kitchen tile at snack time.
Make sure your space is child-friendly. Consider hiding the porn. Move it from the family room coffee table to the end table, perhaps discreetly under a coaster.
Identify and resolve potential safety issues. Block off any stairs you don't want the children to use, perhaps with that kitchen chair that no one ever sits in because it's too rickety. Clean rust off any sharp exposed metal. Consider replacing live ammunition in all accessible firearms with blank rounds. Contain snakes.
Welcome, friends! Greet your guests at the door with a smile and a cocktail. Then look at the mother blankly and say in a slurred voice, "Oh...did you want one of these, too?"
While your children construct a basement play fort out of spare fiberglass insulation, talk to the mother. "So! How much money does your husband make?" is a surefire icebreaker. (If she works outside the home herself, consider this friendly conversational gambit: "Wow. I could never do that to my children.")
Supervise the children adequately. Offer helpful cues to good behavior. An unambiguous "YOU PUT THAT DOWN. RIGHT. NOW." is a good way to make sure the visiting child understands the house rules. Use the other child's name frequently while giving gentle guidance so that he feels included in the fun. (If you can't remember the child's name, substitute "MISTER" or "LITTLE GIRL." Kids love affectionate nicknames.)
Provide a tasty snack. Too bad the other child didn't bring one, too. Make sure the other mother knows that if it were up to you, you'd offer her child some, "but Junior doesn't like to share."
Say a pleasant goodbye. It's a good idea to thank your guests for coming, which you may do in an intimate spitty whisper as you search their pockets for your possessions.
Follow up. Make sure the other parent knows what a good time you and your child had, and suggest another playdate. But it doesn't do to be too eager; you can simply include this in the e-mail you send innocently asking if the pack of cigarettes you found belongs to her preschooler. Oh, and do insist on using the term "playdate." It doesn't make you sound like an uptight ass at all.