Three boys — Denis, Troy, and Memo — all work at the Super Duper Market after school each day. They either stock shelves or they bag the customers’ groceries at the cash register. Each day Mr. Chive, the store manager, decides which boy will do what by these rules:
- Either Denis or Memo, but not both, will stock shelves;
- If Denis stocks shelves, then Troy will bag groceries;
- Troy and Memo will not both bag groceries.
Which of the three boys could have stocked shelves yesterday AND could be bagging groceries today?
Let’s look at the boys one at a time. We’ll start with Denis.
Suppose Denis stocked shelves yesterday. Then, from (1), Memo did not, so Memo bagged groceries. Also, from (2), Troy bagged. But, from (3), Troy and Memo can’t both bag. So Denis couldn’t have stocked shelves yesterday — or any day, for that matter! He’ll always be bagging groceries.
So much for Denis. Then, from (1), Memo must always stock shelves.
So, is Troy the one who can do either? From (2), if Denis stocks, then Troy will bag. But nothing is said about when Denis doesn’t stock (which is always); Troy can bag whether Denis stocks or not. So far so good. Now, can Troy stock shelves? No problem: Troy and Memo can’t both bag groceries, but nothing says they can’t both stock shelves.
So Denis will always bag groceries, Memo will always stock shelves, and Troy can do either, presumably depending on where Mr. Chive needs him more. Troy’s the one who could have stocked shelves yesterday and could be bagging groceries today.
Mr. Chive, by the way, could seriously simplify his rules.
(And we note that this scenario could not be taking place in Europe, where customers bag their own groceries in re-usable bags they bring to the market themselves, as some folks in the United States have also begun to do.)