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## Quiz of the Month (January 2002)

### Hector C. Parr

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#### SOLUTION TO LAST MONTH'S QUIZ

```     1.  6
2.  20
3.  126
```
Notes
 Each junction must be approached either from the West or the South. So the number of routes to any particular junction is the sum of the number to the junction immediately to its West and the number to the junction immediately to its South. For example, on the diagram, 10 = 6 + 4. Using this rule, the numbers can be built up as shown, starting at the bottom left.
(Mathematicians will notice the sequence of binomial coefficients, known as Pascal's Triangle. Computer programmers will realise that any required value can be obtained from a simple recursive function, along these lines:
```     FUNCTION ROUTES(X,Y)
IF X=0 or Y=0 then ROUTES(X,Y) = 1
ELSE ROUTES(X,Y) = ROUTES(X-1,Y) + ROUTES(X,Y-1)
END FUNCTION )```

#### THIS MONTH'S QUIZ

1. A certain board-game uses a number of circular counters, all of the same diameter, with their front faces divided into three equal sectors. Each sector is coloured either red, green or blue, with no two adjacent sectors the same colour. How many distinguishable counters can there be?

2. Another game uses similar counters, with the same colours available, but divided into four equal sectors. How many distinguishable counters can there be?

3. Yet another game has its counters divided into five equal sectors, with the same available colours. How many distinguishable counters can there be in this game?

4. (Bonus question. Not part of the quiz, but could earn you a 'Certificate of Distinction'.) How many distinguishable counters can there be with six equal sectors?

***

(c) Hector C. Parr (2002)

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