Do you have the knowledge you need to study philosophy at Oxford? We bring to your attention riddles on epistemic logic – that is, related to reasoning about knowledge itself. But I know that you know that I know that you know that.

All three problems in recent years have been asked during interviews for admission to the University of Oxford in the Faculty of Philosophy. Each problem has an initial question, and almost all candidates answered it correctly. Further in the text are additional questions that only the best of the best have dealt with.

Stephanie invited her friends, Roma and Sveta, to her home. They all have impeccable logic. She told them that there was a surprise under one of the blue squares.

She told Roma the number of the row, which contains the surprise, and Sveta, the number of the column, and each of them knows about it. The following dialogue takes place:

Roma: I don’t know where the surprise is, but I know that Sveta doesn’t know about it either.

Sveta: yes, at first I really didn’t know where he was. But now I know.

Roma: well, in that case, I also know now.

Question: where is the surprise located?

Additional questions. Let’s say that before this conversation, one of the guests tripped over cell B1, and it turned out that it was empty. A) Will the described conversation make sense? B) Was any of the guests surprised that cell B1 is empty? C) How would the dialogue change if both guests knew that cell B1 was empty? D) Why, because of the information that B1 is empty, Roma’s statement becomes false? (this is the most interesting point in the puzzle, as it seems paradoxical that adding information can reduce knowledge).

At a party with very logical philosophers Sveta and Kolya, the surprise was hidden under one of these colored tiles.

Each of them was given information about the location of the surprise:

– Sveta knows the shape of the tiles;

– Kolya knows the color of the tiles.

Everyone knows that they were only told about this, and nothing else.

Party Host: Does anyone know where the surprise is?

An awkward silence.

Party Host: Does anyone know now?

An awkward silence.

Sveta and Kolya, at the same time: Now I know where the surprise is!

Question: where is the surprise?

Additional questions. A) Did any of the guests expect the first awkward silence? B) How did it affect them? What did they learn from him? C) Did Kolya know that Sveta knows that Kolya did not initially know where the surprise was? D) Did any of them expect a second silence?

Alice invited her friends Caroline and Susan to her home and put several boxes on the table in front of them. It is known that all guests have impeccable logic.

On the table are:

– small red box;

– middle red box;

– large black box;

– small blue box;

– large blue box.

Alice tells her friends that one of the boxes contains a gift and that she told Caroline the color of the box and Susan the size. The following conversation takes place:

Caroline: I don’t know what box the gift is in, and I know Susan doesn’t know it.

Susan: I knew even before your reply that you didn’t know which box the gift was in.

Caroline: Oh, well, now I know what box he is in.

Q: What box is the gift in?

Additional questions. Does Susan now know which box the gift is in? If so, which one of them guessed first – Caroline or Susan?

All the puzzles come from the writing desk of the mathematical philosopher Joel David Hamkins. Hamkins is a professor of logic at University College, Oxford. He says college professors like to give applicants reasoning problems because it shows how the student gets to thinking about new topics. “We also reveal the personality of the applicant, his persistence, the ability to reason rationally about a problem about which not everything is known, the ability to take useful advice from others. Therefore, at the interview, we not only look if the applicant can solve the problem on his own, but also observe the process of his reasoning. All this is valuable information for assessing his abilities. “

Just in case, keep in mind – next year the tasks will be different!

The surprise is in square A2.

If Roma does not know where the surprise is, then we can cross out the 3rd row (because if he were told that the surprise is in the third row, he would immediately calculate the required square). If he knows that Sveta does not know where the surprise is, we can exclude the 4th row (since Sveta would know where the surprise is only if he was in square C4). If Sveta guessed where the surprise lies, then she should know the only column for which there is only one option left. This is column A – hence the surprise is in square A2.

Additional questions. A) No. Now the second part of Roma’s first statement is wrong. B) No. They both knew there was no surprise in B1. C) They learned about each other’s knowledge. D) Roma knew that Sveta didn’t know something. Adding information led him to lose this knowledge, because then he would not have known that she did not know something. Additional information could give her new knowledge. If the surprise was in square B2, she would know about it, knowing that there is nothing in B1.

The surprise is under the red triangle.

Sveta could have known where the surprise is, even before this dialogue, only if she had been told that the surprise is under the square tiles. Kolya would know if they told him the color yellow. Since they don’t know this, we can cross out the square and yellow tiles. If Sveta were told “circle”, she would now know the correct tile – since we only have one tile left. If Kolya were told “blue”, he would also understand where the surprise is. But since they both remained silent after the second question, it can be assumed that they do not know the correct tile. From this it follows that the color should be red and the shape should be triangular.

A) Yes. Sveta initially knows that Kolya does not know the location of the surprise, since both triangles have paired figures of the same color. Kolya also knows that Sveta does not know where the surprise is, since she knows that the color is red, and both red tiles have pairs of the same shape. B) They learned that the other person now knows that they do not know the location of the prize. C) No. Although he knew that the color was red, it could be a round tile, and in that case Sveta would not know that Kolya did not know where the surprise was – she could decide that he might be under the yellow circle. D) No, none of them expected him. From Kolya’s point of view, the surprise could be under the red circle, and then Sveta would answer after the first silence – but this did not happen. From the point of view of Sveta, the surprise could be under the blue triangle, and then Kolya would have answered after the first silence, but this did not happen.

Surprise in a small blue box.

The puzzle seems more complicated due to the lack of a drawing. You can make it easier by drawing a 3×3 grid in which red, black, and blue correspond to columns, and small, medium, and large correspond to rows. Possible box options:

If Caroline does not know the box, she is not told “black”. If she knows that Susan does not know the box, she was not told “red” – otherwise there would be a chance that Susan was said “average”, and in that case Caroline could not say that she knows for sure that Susan is unknown in which box surprise. It follows that Caroline was told “blue.”

If, before Caroline spoke her line, Susan had known that Caroline did not know the correct box, we could rule out the possibility that Susan said “big.” Since then, she would not know for sure that Caroline does not know which box the surprise is in, she could not rule out the possibility that the prize is in a large box, because then Caroline would know where the surprise is. Since the box is blue and not large, there is only one option left – a small blue box.

Susan was the first to know which box the surprise was in. She found out about this as soon as Caroline spoke the first sentence, because after that Susan knows that the box is blue. And since she knows the box is small, there is only one option – a small blue box.