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Quiz of the Month (October 2005)

Hector C. Parr

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SOLUTION TO LAST MONTH'S QUIZ

```  1.  6
2.  8
3.  8.727 cm2```

Notes.
3. (Spherical Trigonometry teaches us a simple relationship between the area of a spherical triangle and the excess of its angle-sum over 180o. But the following argument solves the problem from first principles).

If two great circles intersect at an angle qo, it is easily seen that each of the two enclosed "lunes" has an area equal to q/360 of the total surface of the sphere. So if a represents the area of one of the green regions as a fraction of the whole surface, b the area of one red region, c the area of one yellow region, and x the area of triangle ABC (and of its image, A'B'C'), we have:
 a + x = 45/360 = 1/8 b + x = 90/360 = 1/4 c + x = 50/360 = 5/36 Also 2a + 2b + 2c + 2x = 1 Solving, x = 1/144 But total surface area = 4 * PI * 102 So Area ABC = 4 * PI * 102 / 144 = 8.7266 cm2

THIS MONTH'S QUIZ

By coincidence the symbols used by Martians to write the numbers 1 to 9 are identical with those used on earth. But because Martians have six fingers on each tentacle, their number system is based on twelve rather than ten. Thus 20M stands for twenty four and 100M for one hundred and forty four.

1. What does 999M stand for in earthling's notation?

2. How would you write "seven thousand" in Martian notation?

3. The only money available on Mars is the Marsdollar coin. A visitor from earth was paying for a kilogram of Mars rock, but forgetting that the bill was written in Mars notation, counted out one hundred fewer Marsdollars than the Martian salesman expected. What can you say about the true cost of the rock?

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