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:

1. …returns it to her pile of unused imple­ments, then draws another at random.
2. …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.