Open Day Quiz - Hints and Solutions
1. B steals one of the smallest of A’s collection, so A’s average goes up. A’s pumpkins are almost
all bigger than almost all of B’s, so even this smaller one of A’s raises the average for B’s collection.
2. Let W be the weight of the mobile phone, in grammes, and V its volume in cubic centimetres.
Let w be the weight of an equal volume of water. The volume of an equal weight of water (i.e. W
grammes) is (W/w) × V . Archimedes principle says that an object submerged in water experiences
an upward force (buoyancy) equal to the weight of the water it has displaced. When your mobile
phone is on the airbed with you, its weight is being supported by buoyancy; therefore its presence
on the airbed increases the amount of water displaced by (W/w) × V cubic centimetres. The fact
that when dropped it sinks to the bottom of the pool tells you that it weighs more than the water
it displaces: W > w. Once submerged, it displaces just V cubic centimetres of water. Since W > w
it follows that (W/w) × V > V , so when you drop the phone, less water is displaced, and the level
of the water in the pool goes down.
3. The bottom of the un-dissolved part of the cube is the set of points equidistant from the vertex
the cube is suspended from and the opposite vertex. You can see the shape by picking two opposite
vertices in the drawing of the cube here, and finding the set of points equidistant from them.
H
E
D
A
F
G
C
B
Consider the opposite vertices G and D. The midpoints of the edges AB, BC, CF, F E, EH and
HA are clearly equidistant from G and D. Call these points m1 , m2 , m3 , m4 , m5 and m6 . If you
draw a straight line from m1 to m2 , from m2 to m3 , and so on, and finally from m6 to m1 , you see
the shape you are looking for.
4. 4 + 5 = 32 .
5. At a certain (short) distance from the South Pole, the complete circle of latitude (running
East-West) is exactly one mile in circumference. Starting at any point on this circle and walking
one mile east (or west), you end up back where you started. Now walk one mile due north from
any point on this circle. You are now standing at a point with the same property as the North
Pole. Clearly, by this recipe we have found a whole circle of points with this property. But there
are still more to be found! Where are they?
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6. The answers are Part 1: 1/4; Part 2:(i) 37 (ii) 72. Be afraid! Be very afraid!
7. The answer is quite surprising, and fun to discover for yourself. You end up with another torus
(the mathematical name for the shape of an inner tube), but it looks very different from the one
you started with. Bicycle inner tubes are not flexible enough to do this experimentally, so make a
torus out of cloth or the kind of netting they put round bottles at duty-free shops in airports.
8. There are infinitely many possibilities. Some are shown here.
The third one can be seen as a stage on a transition between the first and the second. As we
deform one to the other we get an infinite number of different shapes. The fourth is also a stage
on a transition - between what and what? Can you find all the possibilities?
9. The shaded discs represent the two nails. Removing one of them, as in the right hand figure,
allows the picture to fall.
10. This is a famous old problem, and you can find solutions online. But first try using the kiddygame version at http://www.mathcats.com/explore/river/crossing.html. It’s remarkable how much
having a geometrical representation of the problem makes it easier. Words to live by!
11. The induction step sounds plausible, but for a certain (small) value of n it just doesn’t work.
Can you see which?
12. Take a look at the Wikipedia article on Benford’s Law. It’s also quite striking to verify it by
looking at the list of areas of the countries of the world at e.g.
https://en.wikipedia.org/wiki/List of countries and dependencies by area.
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