Two Dice = Nine Cards

Alice has two standard dice with labels 1 thru 6. When she rolls them and adds their labels, she gets a distribution over integers in [2, 12]. Bob has nine cards, each labeled with some real number. When Bob chooses two cards (without replacement) and adds their labels, he gets exactly the same distribution over integers in [2, 12] as Alice gets by rolling her dice. What are the labels on Bob's nine cards?

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Three Boxes and a Ruby

Alice places three identical boxes on a table. She has concealed a precious ruby in one of them. The other two boxes are empty. Bob is allowed to pick one of the boxes. Among the two boxes remaining on the table, at least one is empty. Alice must then remove one empty box from the table. Finally, Bob is allowed to open either the box he picked, or the box lying on the table. If he opens the box with the ruby, he gets a kiss from Alice (which he values more than the ruby, of course). What should Bob do?

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Absent-Minded Professor

N women stand in a queue to take seats in an auditorium. Seating is pre-assigned. However, the first woman is an absent-minded professor who chooses any of the N seats at random. Subsequent women in the queue behave as follows: if the seat assigned to her is available, she takes it. Otherwise, she chooses an unoccupied seat at random. What are the chances that the last woman in the queue shall get the seat assigned to her?

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Color of My Probabilistic Hat

Three wizards are seated at a circular room. A magician shall make hats appear on their heads, one hat per wizard. Hats are either black or white, chosen uniformly at random. A wizard cannot see his own hat. At the sound of a bell, all wizards react simultaneously. A wizard reacts by either announcing a color or keeping quiet. If at least one wizard makes an announcement and if all the announcements are correct, the wizards have collectively won the game! Wizards are allowed to confer beforehand to devise a strategy. On average, can they win more than half the times the game is played?

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Kirkman's Schoolgirl Problem

Nine schoolgirls are to be arranged in three rows and three columns on four different days so that any pair of schoolgirls is in the same row on exactly one of the four days.

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Three Boxes with Two Balls Each

The first box has two red balls. The second box has two green balls. The third box has one red and one green ball. Boxes are labeled but all labels are wrong! You are allowed to open one box, pick one of its balls at random, see its color and put it back into the box (you do not get to know the color of the other ball). How many such operations are necessary to correctly label the boxes?

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Divide 100 Marbles into Two Piles

How would you divide 50 black and 50 white marbles into two piles, not necessarily of same size, so that the probability of picking a white marble as follows is maximized: we first pick one of the piles uniformly at random, then we pick a marble in that pile uniformly at random?

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Three Heavy and Three Light Balls

Three out of six lookalike balls are heavy. The other three are light. How many weighings on a beam balance are necessary to identify the heavy balls?

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Tossing with One-Third Probability

At a restaurant, how can Jack choose one out of three desserts with equal probability with the help of a coin? What if the coin is biased and the bias is unknown?

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Coins in a Row

30 coins of arbitrary denominations are laid out in a row. Simran and Tavleen alternately pick one of the two coins at the ends of the row so as to pick up as much money as possible. If Simran makes the first move, could Tavleen ever collect more money than Simran, if Simran makes the optimal choices?

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