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## Quiz of the Month (October 2006)

### Hector C. Parr

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#### SOLUTION TO LAST MONTH'S QUIZ

```1.  1/64
2.  7/256
3.  3/64```

Notes.
```  Notation: H = Head,  T = Tail,  . = either
Total number of equi-probable permutations of H and T = 210 = 1024

1.  Number of perms. of form HHHHHH....  = 16
So probability = 16/1024 = 1/64

2.  HHHHHHT...      Number of perms. = 8
THHHHHHT..      Number of perms. = 4
.THHHHHHT.      Number of perms. = 4
..THHHHHHT      Number of perms. = 4
...THHHHHH      Number of perms. = 8
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Total            = 28
So probability = 28/1024 = 7/256

3. Using a similar method, find the number of permutations with exactly

#### THIS MONTH'S QUIZ

A coin (resembling a British 20p piece) is in the shape of a "curve of constant width". Each of its seven sides is an arc of a circle whose centre is the opposite vertex. The "radius" of the coin (the distance from the centre to any vertex) is 1 cm.

1. If the coin lies between two parallel lines, as shown, find the distance between the lines.

2. How much less is the circumference of the coin than the circumference of a circular coin with radius 1 cm?

3. How much less is the area of one face of the coin than that of a circular coin with radius 1 cm?

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