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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)

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