Puzzles
P1: dice
I play a game with five differently shaped (but fair) dice:
- a tetrahedron with sides marked 1, 2, 3, 4 (the side that falls down is counted)
- an octahedron with sides 1, 2, 3, 4, 5, 6, 7, 8
- a ten-sided die with sides 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
- a dodecahedron with sides 1, 2, ..., 11, 12
- an icosahedron with sides 1, 2, ..., 19, 20
In the game, I throw all five dice, and score the sum of the lowest and the highest number achieved (in any one go, only two dice contribute, and the intermediate numbers are ignored).
Characterise the statistical distribution of this score. How much difference does having the octahedron, ten-sided die and dodecahedron make - in other words, how is the distribution different if only played with the tetrahedron and the icosahedron?
P2: currencies
Shopping in the train station on the way home from holiday, I find I have three currencies to use up. I have €10, £10 and $10 as notes. There are three shops, selling:
- bars of chocolate priced in euros at €x each
- silk ties priced in dollars at $y each
- MP3 music players priced in pounds at £z each
where x, y, and z are whole numbers, such that x + z = y.
Although the products are priced in specific currencies, for each transaction the shopkeeper will accept payment in any single currency, and give change in either the same, or, if asked, a single different currency. Standard exchange rates are used throughout the station and they are the same for purchases and change. None of the exchange rates is greater than 2.
Looking at the prices, I work out that I can afford several chocolate bars, a silk tie and an MP3 player, if I make the purchases in an appropriate order.
First, I buy the silk tie, and ask for change in pounds: it's a whole number of pounds change, £w. Only then do I have enough pounds to buy the MP3 player. I hand over all my pounds, but ask for change in dollars. It's a whole number of dollars, and coincidentally it's the same number as the previous change in pounds, $w. Finally, with all the remaining money, I purchase as many bars of chocolate as I can. This leaves me just $0.64. How many bars of chocolate did I buy?
P3: rolling
A rigid rubber hollow cylinder is 5 metres long, with radius 1 metre. One half (the red part) is made from rolling up a high density piece of mass 200 kg, the other half (the blue part) from a lower density piece of mass 100 kg. The pieces are firmly fastened together to make a perfect hollow cylinder.
The cylinder is set to roll on an idealised undulating surface. This is in the form of a sine wave of amplitude h and wavelength λ.
Despite having an off-centre centre of mass, the cylinder has good enough contact with the undulating surface that it can roll perfectly without slipping. We ignore any energy losses from rolling resistance and it doesn't noticeably deform as it rolls.
The cylinder starts from stationary at position A on top of a hill. It is aligned with the heavier half (red) uppermost with the join between the halves horizontal.
After a tiny nudge, the cylinder rolls 99.5 rotations, taking it to position C. At the moment it passes C, the blue half is uppermost. The cylinder continues rolling in the direction of D. What is its angular velocity at point C?
P4: elephants
According to a well-loved work of fiction, there is a multi-coloured elephant called Elmer. On one day a year (Elmer's day) Elmer covers himself with grey berries so that he looks like a normal elephant, and the other elephants in his group paint themselves multi-coloured so they look much like Elmer normally does.
My prior belief is that this is extremely unlikely to be true - say, a probability of one in a million. My son, however, is more open-minded, and thinks it is 10% likely to be true.
We go on holiday to a place where elephants live. On Monday, we see a grey elephant. On Tuesday, we see a multi-coloured elephant.
Discuss how our beliefs changed as a result of our experiences.
Last updated 5 July 2009
Authors: email mc at hamble dot eclipse dot co dot uk
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