So yeah, my um friend has a little game, that… yeah she plays when she sets the table for her family. And she asked me to ask you this. Because it’s been bugging me her for a while now.
There are four family members. When she puts out the silverware, she goes to the drawer and selects four each of knives, forks and spoons, and sets one knife, one fork and one spoon at each of the four place settings.
But rather than set each place all at once, she likes to play a little game. She starts at one seat, and picks a random implement and sets it down. Then she starts around the table, picking a random implement from the remaining pile and putting it on the next place setting around the table if there’s room for it. Once she’s put down an implement, she continues around and around the table—hopefully three times, right?—until she either has placed all the silverware on the four settings (which is lucky), or runs into a problem where the implement in her hand is already present at the next setting (which she thinks of as unlucky).
If she can’t complete all the circles according to her little “plan”, she just sets the rest of the silverware out as you or I might before heading off to do a little ritual involving hand-washing to fix the luck problem.
OK, so she’s eccentric, not sick. Cut her some slack. Table-setting is boring.
Given there are four seats at the table, and three kinds of implement, what proportion of the time does she end up “lucky” as opposed to “unlucky”?
What if they’re having soup, so there are four kinds of implement?
What if there are N additional guests visiting?
Which of the following kludges will most improve her expected luckiness, in each of the previous situations? Instead of failing immediately, when she can’t put down the randomly-selected implement, she:
- …returns it to her pile of unused implements, then draws another at random.
- …reverses the direction she’s walking around the table, and tries to place the current implement at the previous seat; then she carries on in that direction until she reaches an impasse and also can’t reverse again.