# Find all solutions (A,B,C,D) (where these variables are all natural numbers) that satisfy the constraints A+B+C+D=6 and 3A+1B+2C+5D = x , 6<=x<=30 Give some ideas about how this might be...

Find all solutions (A,B,C,D) (where these variables are all natural numbers) that satisfy the constraints

A+B+C+D=6 and

3A+1B+2C+5D = x , 6<=x<=30

Give some ideas about how this might be solved in Excel

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### 1 Answer

To satisfy the first constraint, namely

A+B+C+D=6

we need four natural numbers (0 and integers) that sum to 6. The possibilities are (where the string of four numbers is in no particular order):

**0006, 0015, 0024, 0033, 0114, 0123, 1113, 1122**

(I think you can work these strings out using the *Simplex Algorithm*. In Excel you could sum all possible combinations of each of A,B,C,D being from 0-6 and use an if statement to find which A+B+C+D cells satisfy the total of 6)

For the next constraint *order *the constraint equation in terms of the size of the coefficients. That is write,

k: 5D+3A+2C+1B

For the possible string combinations above, test the ** minimal and maximal sets **only. By which I mean, say for the combination

**0006**, check that

i) 6<=1B<=30 when B=6 (A,C,D all equal to 0) and ii) 6<=5D<=30 when D=6 (A,B,C all equal to 0).

For combination **0123** for example, check that

i) 6<=3A+2C+1B<=30 when B=3,C=2,A=1 (min) and when B=1,C=2,A=3 (max) (D=0), and

ii) 6<=5D+3A+2C<=30 when D=1,A=2,C=3 (min) and when D=3,A=2,C=1 (max) (B=0)

The combinations of A,B,C,D that satisfy the constraint k can again be done in Excel using if statements on the minimal and maximal sums.

If you work through all of this you should find that all combinations of letters satisfy the possible strings adding to 6 given above, so that there are

0006 (**1 x 4**), 0015 (**2 x 6**), 0024 (**2 x 6**), 0033 (**6**), 0114 (**3 x 4**), 0123 (**3 x 4**), 1113 (**4 x 1**), 1122 (**6 x 1**) **= 4 + 12 + 12 + 6 + 12 + 12 + 4 + 6 = 68 solutions**

**Answer: there are 68 solutions that satisfy the combination of the two constraints, where each of A,B,C,D are in the range 0-6.**