Some Puzzle Problems

  1. Martin Gardner

    Imagine a planet covered entirely with water with the exception of a single small island. On this island is a sizeable fleet of airplanes and an endless supply of fuel. These airplanes are very peculiar though: they can carry only enough fuel to fly half-way around the planet, and they are capable of exchanging fuel between themselves instantaneously midair. What is the fewest number of planes that would have to be used to allow a single plane to fly completely around the planet and back to the small island along a great circle without harming any pilots?

  2. Suppose you are on a large alien planet; looking out around you, in every the direction the surface of the planet curves gently, disappearing into the horizon. But is this a “planet” as you know it? How do you know this planet is spherical like your home planet Earth? For all you know, this planet might be toroidal — doughnut-shaped, with a big hole going through it.

    Luckily, you have with you a paintbrush and a magical paint can containing infinitely much red paint. Using these tools, how can you tell whether or not the planet you’re on is spherical or toroidal?

    Challenge

    Suppose you do discover the planet you’re on is toroidal and has a hole. Can you be sure that there is only one hole? In case there are more, can you use your paint and paintbrush to determine how many holes this planet has?

  3. Aldous and Buxley sit together at a circular table in a laundromat, bored, waiting for their clothes to dry. Fiddling with their hoard of laundry quarters, they create a game. They take turns placing quarters flat on the table. Once a quarter is placed, it cannot be moved. Quarters’ edges can touch, and they can hang over the edge of the table, but may not lay on each other: they must lay flat on the table. The last person who can place a quarter on the table is the winner. Prove that, if he plays cleverly, whoever goes first can always win.
  4. Three friends go to a restaurant and the bill comes out to $30. They each contribute $10. Later the waiter realizes that there was a mistake, and the bill should only have been $25. The waiter gives $5 to the bellboy and asks him to return it to the friends. The bellboy decides to pocket $2 for himself and gives back $1 to each of the friends. Now, each friend has paid $9, totaling $27, and the bellboy has $2, which adds up to $29. What happened to the missing dollar?

  5. BdMO 2016

    Fifty natural numbers are written in such a way so that the sum of any four consecutive numbers is \(53\). The first number is \(3\), the 19th number is eight times the 13th number, and the 28th number is five times the 37th number. What is the 44th number?

  6. Terence Tao

    On a remote mountaintop there’s a small village of one hundred people. Everyone in the village has either brown eyes or blue eyes, and devoutly follows the same religion that has a single peculiar feature: each villager is forbidden from knowing the color of their own eyes, or even from discussing the topic. So each villager sees the eye color of every other villager, but has no way of discovering their own. (They’ve forbidden reflective surfaces long ago.) If a villager does discover their eye color, their religion compels them to proceed to the village square at high noon the following day, and call out for all their fellow villagers to come witness as they pluck out their eyes.

    Of the one hundred villagers, it turns out that ten of them have blue eyes and the rest have brown eyes, although they are not aware of this fact. Each villager is highly logical and religiously devout, and they all know that each villager is highly logical and devout (and they all know that they all know that each villager is highly logical and religiously devout, and …).

    One day a blue-eyed mountaineer, tired and hungry, stumbles across the village asking for help. That evening at the communal village dinner she addresses the whole gathering to thank them for their hospitality. However, not knowing their customs, the mountaineer remarks in her address how unusual it is to see another blue-eyed person like myself in this remote region. What effect, if any, does this faux pas have on the village?

  7. MIT

    Slips of paper with the numbers from \(1\) to \(99\) are placed in a hat. Five numbers are randomly drawn out of the hat one at a time (without replacement). What is the probability that the numbers are chosen in increasing order?

  8. Putnam 2004 A1

    Basketball star Shanille O’Keal’s team statistician keeps track of the number, \(S(N)\), of successful free throws she has made in her first \(N\) attempts of the season. Early in the season, \(S(N)\) was less than 80% of \(N\), but by the end of the season, \(S(N)\) was more than 80% of \(N\). Was there necessarily a moment in between when \(S(N)\) was exactly 80% of \(N\)?

  9. Putnam 2002 B1

    Shanille O’Keal is shooting free-throws. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots?

  10. Putnam 1992 B1

    Let \(S\) be a set of \(n\) distinct real numbers. Let \(A_S\) be the set of numbers that occur as averages of two distinct elements of \(S\). For a given \(n \geq 2\), what is the smallest possible number of elements in \(A_S\)?

  11. Putnam 2017 A4

    A class with \(2N\) students took a quiz, on which the possible scores were \(0, 1, \dotsc, 10\). Each of these scores occurred at least once, and the average score was exactly \(7.4\). Show that the class can be divided into two groups of \(N\) students in such a way that the average score for each group was exactly \(7.4\).

  12. Putnam 2013 B6

    Let \(n \gt 1\) be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of \(n\) spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either
    • places a stone in an empty space, or
    • removes a stone from a nonempty space \(s\), places a stone in the nearest empty space to the left of \(s\) (if such a space exists), and places a stone in the nearest empty space to the right of \(s\) (if such a space exists).
    Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?

  13. Nick’s Mathematical Puzzles ☆☆☆☆

    Two information theoreticians, Alice and Bob, perform a trick with a shuffled deck of 52 cards. Alice asks a member of the audience to select five cards at random from the deck. The audience member passes the five cards to Alice, who examines them, and hands one back. Alice then arranges the remaining four cards in some way and places them face down in a neat pile,

    Bob, who has not witnessed these proceedings, then enters the room, looks at the four cards, and determines the missing fifth card held by the audience member.

    1. How is this trick done? Note the only communication between Alice and Bob is via the arrangement of the four cards. There is no encoded speech or hand signals or ESP, no bent or marked cards, no clue in the orientation of the pile of four cards, etc.
    2. Alice and Bob follow this performance with an even more ambition trick. They toss aside their 52-card deck and repeat the trick using a non-standard 124-card deck what has four suits of thirty-one cards each. How is this trick done?
    3. Prove that 124 is the largest possible size of a deck of cards for which this trick can work.