Wednesday, May 18

# Can you solve it? Oxford University Admission Questions | Math

Do you have what it takes to study philosophy at Oxford? Today’s three puzzles are “epistemic logic puzzles,” that is, puzzles related to reasoning about knowledge. But I know you know, I know you know

All three puzzles have been established in recent years during Oxford University admissions interviews for joint degrees in philosophy. In each case, there is an initial question. Almost all candidates will answer correctly, and I hope you will too. I have also included a sample of the follow-up questions. Only the best candidates will do everything right. Best of luck!

1. Stephanie’s surprise.

Stephanie has invited her friends Rowan and Colleen to her house. They are all perfectly logical. He tells them that he has hidden a surprise under one of the blue squares.

Stephanie has privately told Rowan the surprise row number and Colleen the surprise column letter, and everyone knows it. The following conversation occurs.

Rowan: I don’t know where the surprise is, but I also know that Colleen doesn’t know.

Girl: Yes, in fact, at first I did not know the location of the surprise. But now I know where it is.

Rowan: In that case, now I also know where it should be.

Question. Where is the surprise?

Follow-up: Suppose that before any conversation takes place, someone bumps into B1, which opens and reveals that it is empty. a) Could the conversation have proceeded as before? b) Were any of them surprised to see it empty? c) How could the conversation have changed, if you both already knew? d) How can adding information, that B1 is empty, makes Rowan’s statement false? (This is the most interesting aspect of the puzzle, as it seems paradoxical that adding information can reduce knowledge.)

2. Tile party.

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At a party for our perfectly logical philosophy friends Sheila and Colin, a surprise has been hidden under one of these colorful tiles:

Each friend is privately told a piece of information about where the surprise is. 􏰀

It is well known to all that this and no other information is given.

Host: Do any of you know where the surprise is?

. . . A long, awkward silence. . .

Host: Do you know now?

. . . Most awkward silence. . .

Sheila, Colin: (simultaneously): Now I know where it is!

Question. Where is the surprise?

Follow-up: a) Did any of them expect the first silence? b) What effect did this silence have on your knowledge? How did they learn anything from it? c) Did Colin know that Sheila knew that Colin did not initially know where the surprise was? d) Did any of them wait for the second silence?

3. Alice boxes.

Alice has invited her friends Caroline and Susan to her house and has placed several boxes on the table in front of them. The women are all perfectly logical.

• little red box

• medium red box

• big black box

• little blue box

• big blue box

Alice tells her friends that she has placed a gift in one of the boxes, and privately she has told Caroline the color of the box and Susan the size of the box, and they both know it. The following conversation occurs.

Caroline: I don’t know which box is in the gift, and I also know that Susan doesn’t.

Susan: I knew before you spoke that you did not know which box contains the gift.

Caroline: Ah, now that you say that, it suddenly occurs to me which box the gift should contain.

Question. What box does the gift contain?

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Follow-up: After the conversation, does Susan also know which box is in the gift? If so, who found out first, Caroline or Susan?

All the riddles come from the Oxford interview question drawer of mathematical philosopher Joel David Hamkins. Hamkins is Professor of Logic and Sir Peter Strawson Chair of Philosophy at University College, Oxford. He says the university likes candidate students to work through logical reasoning, as this gives them insight into how they approach thinking about a new topic. “We also got to see a bit of his personality, his tenacity, and his ability to discuss something rationally without knowing everything about it, including his ability to accept helpful suggestions from others. So the interview not only tests whether they can solve the puzzle on their own in isolation, but we can see how the whole process of their attempted solution unfolds as it happens, and that is what is valuable to the admissions evaluation “.

But don’t get too smart – these questions won’t be used next year!

I’ll be back with the solutions at 5pm UK. PLEASE NO SPOILERS

To find more information about the work of Professor Joel David Hamkins here is your personal website.

I am the author of several puzzle books, the most recent Puzzle book for language lovers. I also give school talks on math and puzzles (restrictions allow). If your school is interested, get in touch.

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