Week 9
Dedicated to Colm O’ Dunlain, from whom many of these questions are stolen
November 15, 2015
1. A crate contains 27 apples, arranged in a cube. A worm sits in the
centre apple. Passing only betweentouching apples, can the worm visit
each apple exactly once?
2. What if there is a worm in the centre of a 2n + 1 × 2n + 1 cube, n > 0?
3. For what m × n × p rectangular arrangements of apples can a worm
in the centre apple of the arrangement visit each apple exactly once?
Defining the centre apple must also be specified, cause it gets unclear
4. A census-taker arrives at a womans door and enquires as to the ages
x, y, z of her three sons Woman: ”xyz = 72 and x + y + z is my door
number.” Census-dude: ”That tells me nothing!” Woman: ”My eldest
son is out.” Census-dude: ”Cheers! That’s all I needed.”
What are the ages of the three children?
5. In a tall building, there are 55 unlabelled electrical wires, running from
top to bottom. An electrician needs to label them correctly at both
ends. How many journeys up and down the stairs does he need to
make? He starts at the ground floor and ends there too.
6. Can we, by appropriate choice of signs, make the expression
±1 ± 2 ± 3 ± · · · ± (4m + 1)
equal to any odd positive integer less than or equal to (2n + 1)(4n + 1)
7. On a remote island there are two tribes, the Atans and the Betans. One
tells the truth always, the other never. Three islanders are standing
on the beach when somebody comes close to shore in a boat. She calls
out to one of them, asking if he is an Atan or a Betan; the answer is
lost in the noise of the surf. Another says: He says he is an Atan. The
third says No, thats not true. He is a Betan and so am I. Which tribe
lies and which tells the truth? What tribes do they belong to?
8. For which positive integers n is n! + 5 a perfect cube?
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9. What is the last digit of 31001 71002 131003 ?
10. What is the first digit of (1 +
1/1000000 1000000
?
)
11. A total of 119 students live in a building with 120 rooms. A room
is overpopulated if there are at least 15 people in there. If a room
is overpopulated, the students living there fight and leave it at the
end of the day, each going to a diiferent room in the building to live.
Someday,will there be peace? That is, no overpopulated rooms?
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