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 nondistinct versions of the same
triangle by quoting the sides in nonincreasing 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
