Multiple Sums
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There are 12 different ways you can add two numbers, between 1 and 25,
to total 25:
(1+24, 2+23, ..., 12+13).
No number is used more than once. How many different combinations using
3 numbers to total 25 are there? Remember, use the numbers only once for
each grouping. (i.e. 25 = 1+2+22 = 1+4+20 = 2+4+19, but not 1+12+12)
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Can you fill a table for the number of possible sums for totals
from 20 to 30, with the number of terms
in each sum from 2 to 7? Can you find a formula which would provide the
answer for the number of possible sums of N, with k terms in each sum?
Source: Mind Bending Puzzles Calendar,
Terry Stickles, January 14, 2000.
Solutions were received from Henry Bottomley and Sandy Thompson. There
are 40 ways to make sums of three elements total 25. The full table is
below:
## 2 3 4 5 6 7
-----------------------------
20 9 24 23 7 0 0
21 10 27 27 10 1 0
22 10 30 34 13 1 0
23 11 33 39 18 2 0
24 11 37 47 23 3 0
25 12 40 54 30 5 0
26 12 44 64 37 7 0
27 13 48 72 47 11 0
28 13 52 84 57 14 1
29 14 56 94 70 20 1
30 14 61 108 84 26 2
No one found a proven general formula, but Henry offered these notes:
A formula could be hard, given the subject of partitions,
but it is fairly obvious that the no partition is possible if N
Given that and the generating functions for partitions, my guess is
that the number of possibilities
is equal to the co-efficicent of x^N in the expansion of:
x^((k*(k+1)/2) / product over i=1 to k of (1-x^i)
Mail to Ken