HOME

TREATISE:
. Quantum
Nodal Theory

ROMALDKIRK:
. Village
. Reservoirs

ESSAYS: COSMOLOGY
. Infinity
. Universe
. Dark Matter

ESSAYS: PHILOSOPHY
. Free Will
. Representation
. Conditionals
. Postscript

ESSAYS: ORGAN MUSIC
. Practising
. British Organs
. Hymn Playing
. Music Lovers

QUIZ QUESTIONS
. Quiz Archive

SERVICES
. Book Reviews

## Quiz of the Month (January 2005)

### Hector C. Parr

***

#### SOLUTION TO LAST MONTH'S QUIZ

```   1.  8
2.  6
3.  825```

Notes
```  1. Namely: 9,9,2     9,8,3     9,7,4     9,6,5
8,8,4     8,7,5     8,6,6     7,7,6

2. Namely:  5, 4, 3  10, 8,6  15,12,9
20,16,12  13,12,5  17,15,8

3. We can avoid repeating non-distinct versions of the same
triangle by quoting the sides in non-increasing order
of length, as above. So we use the following conditions:
a >= b,  b >= c  (to avoid repetitions)
b + c > a        (the triangle inequality)
Then we can either (a) count systematically,
or (b) devise a formula,
or (c) write a short computer program.
Here is a suggested counting method:
The number of triangles of form (20,20,?) is 20.
The number of triangles of form (20,19,?) is 18.
The number of triangles of form (20,18,?) is 16, etc.
So the number of form (20,?,?) is 20+18+16+... = 110,
Similarly the number of form (19,?,?) is 19+17+15+... = 100,
the number of form (18,?,?) is 18+16+14+... =  90,
the number of form (17,?,?) is 17+15+13+... =  81,
.... and so on.
Adding, the total number is 825.```

Alternatively, suppose friendly triangles are defined as those with sides not exceeding N, rather than not exceeding 20 (where N is any even number). Then using a similar method to above the following formula may be obtained:
Total number = N(N+2)(2N+5) / 24

#### THIS MONTH'S QUIZ

```  The unit of currency in Ruritania is the Ruro (R), and 100
Rurocents (c) equal one Ruro. Coins are minted in the
following denominations: 1c, 2c, 5c, 10c, 20c, 50c.
Ruritanian law forbids the tendering of more than four
coins of any one denomination.

1. What is the smallest number of Ruritanian coins you need
to allow exact payment for any one article which may cost
anything from 1c to 60c ?

2. In how many different ways can you make exact payment for
an article costing 25c ?

3. What is the largest sum you can have in Ruritanian coins
and yet be unable to give change for one Ruro ?```
***

© Hector C. Parr (2005)

Go to Quiz Archive