2. Number Triples An Investigation

3. Number Triples A number triple consists of three whole numbers in a definite order. For example (4, 2, 1) is a triple and (1, 4, 2) is a different triple. The numbers in a triple do not have to be different but cannot include zeros. Thus (2, 2, 5) is a triple but (2, 0, 7) is not.

4. The sum of a triple is found by adding the three numbers together. Investigate how many different triples there are with a given sum. Try to find a formula in terms of s, to predict the number of different triples, N, with a sum of s. How many triples are there with a sum of 22 and of 50? What is the least sum to have over 400 triples?

5. Solution This is obtained using a difference table as shown in the next few slides. From the difference table a formula can be found for ‘N’ the number of triples with a particular sum ‘s’.

6. Sum Triples Number of triples 3 1 1 1 1 4 1 1 2,1 2 1,2 1 1 3 5 1 1 3,1 3 1,3 1 1,1 2 2 2 1 2,2 2 1 6 6 1 1 4,1 4 1,4 1 1,1 2 3 1 3 2,2 1 3,3 1 2,2 3 1, 321, 2 2 2 10 7 1 1 5,1 5 1,5 1 1,1 2 4 2 1 4,2 4 1,1 4 2,4 1 2 4 2 1,1 3 3,3 1 3,3 3 1 2 2 3,2 3 2,3 2 2 15

7. s3456 7 N13610 15 ∆ 2 3 4 5 ∆²111 If ‘s’ is the sum and ‘N’ the number of triples then forming a difference table gives From the difference table, the formula for N in terms of s is a quadratic. N(s) = a s² + b s + c where a, b and c are constants. a × 3² + b × 3 + c = 1.... (1) a × 4² + b × 4 + c = 3.... (2) a × 5² + b × 5 + c = 6.... (3) These solve to give a=0.5, b = -1.5 and c = 1 Hence we have the formula N=0.5s 2 -1.5s + 1

8. N(s) = ½ s²  s + 1= ½(s²  3s + 2) which may be written N(s) = ½(s  1)(s  2) and this is true for s = 1 and s = 2 So we have N(22) = ½(22  1)(22  2) = ½ × 21 × 20 = 210 N(50) = ½(50  1)(50  2) = ½ × 49 × 48 = 1176 N(29) = 378, N(30) = 406, N(31) = 435 So 30 is the smallest sum to have over 400 triples.