There are 40 ways to make sums of three distinct positive integers total 25. (1+2+22 is such a sum, but 1+12+12 and 1+2+3+19 are not.)

How many different ways can three distinct positive integers sum to 1000?

The first combination, I say, is (1, 2, 997).  Now, keeping the first digit, the smallest digit, the same, we can get (1, 3, 996); (1, 4, 995) and so on.  Eventually, we will get a solution of (1, x, x+1) before going any further would give us repeating values.  Thus, we need to find when 1 + x + x + 1 = 1000 or when 2x = 998.  x = 499.  Thus, we have (1, 2, 997) through (1, 499, 500)  This is a total of 498 solutions with 1 as the first digit.  For all even values for the first number (let's call that A), x = (999-A)/2.  The possible number of combinations from (A, A + 1, 999 - 2A) to (A, (999-A)/2, (101-A)/2) is (999 - 3A)/2

Now, we set the first digit equal to 2, thus creating (2, 3, 995), (2, 4, 994), etc.  When 2 + x + x + 1 = 1000, that means 2x = 997, or x = 498.5.  However, since these are only integers we're talking about, x must be equal to 498, thus creating (2, 498, 500).  From (2, 3, 995) to (2, 498, 500) is 496 combinations.  For all even values of A, x = (998 - A)/2.  The number of combinations is (998 - 3A)/2. 

Now, it would be ideal if there were some solution (A, A+1, A+2).  Thus, we figure out the closest situation where 3A + 3 = 1000.  Thus, 3A = 997, the greatest A can be is 332.  This creates the solution (332, 333, 335) And that is the solution with the maximum values for each number.  Now, write out a few numbers.  First of all, if A = 1, you can have 498 combinations.  A = 2, C = 496.  A = 3, C = 495.  A = 4, C = 493.  A = A + 2, C = C - 3.  Now, when A = 331, 1 less than the maximum 332, there are 3 possible combinations.  When A = 1, there are 498, which is 3 * 166.  Thus, the amount of total combinations if A is odd is 3 * (166 * 167 / 2) or 41583.  For the even values of A, it's that minus 2 for each of the 166 even values of A, or 332, coincidentally, thus making a grand total of 82834 combinations of distinct numbers that add up to 1000.

Posted by John on 2004-07-24 06:15:21