Do you have what it takes to study philosophy at Oxford? Today’s three puzzles are ‘epistemic logic puzzles’, that is, puzzles concerned with reasoning about knowledge. But I know you know I know you know that.
All three puzzles have been set in recent years during Oxford university admissions interviews for joint philosophy degrees. In each case, there is an initial question. Almost all candidates will answer this correctly, and I hope you will too. I’ve also included a sample of the follow-up questions. Only the best candidates will get everything right. Best of luck!
1. Stephanie’s surprise.
Stephanie has invited her friends Rowan and Colleen to her home. They are all perfectly logical. She tells them that she has hidden a surprise under one of the blue squares.
Stephanie has privately told Rowan the row number of the surprise and Colleen the column letter of the surprise, and everyone is aware of this. The following conversation ensues.
Rowan: I don’t know where the surprise is, but I also know that Colleen doesn’t know.
Colleen: Yes, indeed, at first I didn’t know the location of the surprise. But now I know where it is.
Rowan: In that case, I now also know where it must be.
Question. Where is the surprise?
Follow up: Suppose that before any conversation took place, someone trips over B1, which opens, revealing it to be empty. a) Could the conversation have proceeded as before? b) Were either of them surprised to see it empty? c) How can it have changed the conversation, if they both knew it already. d) How can it be that adding information, that B1 is empty, makes Rowan’s statement become false? (This is the most interesting aspect of the puzzle, since it seems paradoxical that adding information can reduce knowledge.)
2. Tile party.
At a party for our perfectly logical philosophy friends Sheila and Colin, a surprise has been hidden under one of these coloured tiles:
Each friend is privately told a piece of information about where the surprise is.
It is commonly known by all that this and no other information is given.
Host: Do either of you know where the surprise is?
. . . Awkward long silence. . .
Host: Do you know now?
. . . More awkward silence. . .
Sheila, Colin: (simultaneously): Now I know where it is!
Question. Where is the surprise?
Follow up: a) Did either of them expect the first silence? b) What effect on their knowledge did that silence serve? How did they learn anything from it? c) Did Colin know that Sheila knew that Colin didn’t know initially where the surprise was? d) Did either of them expect the second silence?
3. Alice’s boxes.
Alice has invited her friends Caroline and Susan to her home, and she has placed several boxes on the table before them. The women are all perfectly logical.
small red box
medium red box
large black box
small blue box
large blue box
Alice tells her friends that she has placed a gift into one of the boxes, and she has privately told Caroline the colour of the box and Susan the size of the box, and they both know this. The following conversation ensues.
Caroline: I don’t know which box contains the gift, and I also know that Susan doesn’t know.
Susan: I already knew before you spoke that you didn’t know which box contains the gift.
Caroline: Ah, now that you say that, it suddenly occurs to me which box must contain the gift.
Question. Which box contains the gift?
Follow up: After the conversation, does Susan also know which box contains the gift? If so, who came to the knowledge first, Caroline or Susan?
All the puzzles come from the drawer of mathematical philosopher Joel David Hamkins’ Oxford interview questions. Hamkins is Professor of Logic and the Sir Peter Strawson Fellow in Philosophy at University College, Oxford. He says that the college likes to get student candidates to work though some logical reasoning since this gives them insight into how they approach thinking about a new topic. “We also get to see a little of their personality, their tenaciousness, and their ability to discuss something rationally without yet knowing all about it, including their ability to accept helpful suggestions from others. So the interview isn’t just testing whether they can solve the puzzle on their own in isolation, but we get to see the whole process of their solution attempt unfold as it happens, and that is what is valuable for admissions evaluation.”
But don’t get too clever – these questions will not be used next year!
I’ll be back with the solutions at 5pm UK. PLEASE NO SPOILERS
UPDATE: Read the solutions here.
To find more about the work of Professor Joel David Hamkins here is his personal website.
I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.