Warwick Mathematics Institute – Open Day Quiz –February 12th, 2014
The quiz is for fun. Ignore any question that doesn’t grab your attention. Hints and solutions
can be found at http://go.warwick.ac.uk/mathsadmissions/opendays/quiz
A: To think about - no paper needed
1. Can this be true? Farmer A and Farmer B are competing to grow the biggest pumpkins. When
A steals a pumpkin from B and adds it to his own collection, the average weight of A’s pumpkins
goes up - and so does the average weight of B’s.
2. You are floating on an airbed in the hotel swimming pool, and drop your mobile phone. It sinks
to the bottom of the pool. Does the level of the water in the pool (i) go up (ii) go down (iii) stay
the same?
3. A sugar cube is suspended from one corner and lowered into a cup of tea, so that exactly half of
its volume is submerged. The submerged half dissolves; the rest is unaffected. What is the shape
of the bottom of the part that is left?
4. Can you write a correct equation using the digits 2, 3, 4, 5, and the symbols + and =, each
exactly once?
5. If you start at the north pole, walk a mile south (along
any line of longitude, it doesn’t matter which), then walk
a mile east and finally a mile north, you’ll end up back
where you started. Is the north pole the only point on the
Earth’s surface that has this property?
6. This question has two parts. One is mathematical. The other is . . . a true story.
Part 1 If you toss a fair coin twice, what is the probability of getting two heads?
Part 2 97 MPs were asked that question in a 2012 survey by the Royal Statistical Society.
How many of them got the right answer?
(i) 97 (ii) 72 (iii) 37 (iv) 14
How many felt “confident” when dealing with numbers? (i) 97 (ii) 72 (iii) 37 (iv) 14
7. A farmer is standing by a river. He has a boat,
and wants to transport a goat, a wolf and a cabbage
to the other side. The boat is small, and so he can
carry only one at a time. He mustn’t leave the wolf
with the goat, as the wolf will eat the goat. Similarly,
he can’t leave the goat alone with the cabbage. How
can he transport all three to the other side of the
river?
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B: To answer by experiment?
8. Imagine a bicycle inner tube, made out of very soft and flexible
rubber. By poking your finger into a small hole in the tube, you
turn it inside out. What shape does it then have?
9. Using a flat blade, you have to cut a cube of cheese into two
identical pieces. What shape can the cut have? Hint: You can begin
with the answer to Question 3, but there are other correct answers.
Draw as many as you can.
10. How can you hang a picture using two nails, in
such a way that if you remove either nail, the picture
falls down?
C: To ponder at your leisure
11. Here is a proof by induction that if you put n horses in a field, then they all have the same
colour.
Initial Step: obviously it’s true if n = 1 – any horse is the same colour as itself.
Induction Step: Suppose it’s true for n. Put n + 1 horses in your field. Take one horse out. By
the inductive hypothesis, all the remaining ones have the same colour. Bring the extra horse back
again and take out another. Again you have n horses in the field, so by the induction hypothesis
they all have the same colour. So all n + 1 horses have the same colour.
Are you convinced?
12. This is not “a puzzle”, but a puzzling phenomenon. If you list the surface areas of the countries
of the world, or the populations of its cities, or the net wealth of millionaires, you find that about
30% of the numbers in the list have 1 as first digit, about 18% have 2 as first digit, and so on down,
with only about 3% having 9 as first digit. Can you explain why this is?
This phenomenon is known as Benford’s Law. Its Wikipedia entry explains how it can be used
to detect tax fraud: fake lists do not have the right distribution of digits.
Mathematicians and Puzzles
One of the things that puts many people off mathematics is the need to get the right answer. It’s
great that there is a right answer, but it can be frustrating, or even humiliating, when we can’t find
it easily. Mathematicians and mathematics students have to wrestle with this issue, especially at
the start of our careers. It can take time to learn that we are good enough at our subject, and that
we gain nothing by feeling anxious when we can’t see the solution immediately. All mathematical
activity has a certain amount of being stuck, and we all have to learn to cope with frustration. There
are compensatory rewards! With time and practice, we move towards that happy state of confidence
and acceptance of our own level of ability, which means that we are glad when we come across an
“elementary” problem that we cannot immediately solve. “How interesting”, we say, instead of “Oh
dear, I don’t know the answer”.
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