A math riddle involving sums of digits

Question

When I was in highschool I became obsessed with this strange combination I discovered. I don't really know a lot about math but I've recently re-discovered it and believe I have found an answer, but no real way to prove it. I'd like someone with a fair bit more knowledge than me to try at it.

You have 4 single digit number, 4 double digit numbers, and 2 triple digit numbers, demonstrated with "space filling" variables like so; X + X + X + X + XX + XX + XX + XX + XXX + XXX = SUM there are 18 X's above. You are permitted to use each digit from 1 to 9 exactly twice each, no more no less. No zeroes are allowed. They can be in any order, for example: 1 + 2 + 3 + 4 +56 +78 +91 +23 +456 +789 = 1503 Using this setup, how many different possible combinations of digits are there so that the SUM is equal to 1350? And can you prove there are no other possible solutions?

I believe there are 36 possible permutations that sum to 1350. If someone can mathematically prove that or disprove it by discovering additional solutions that I have not found I would be greatly appreciative.

Answer 1

Here are the first 37 elements of a list of 456 that are obtained by imposing the constraint that ones, tens, and hundreds should appear in non-decreasing order from left to right. (As noted by Abraham Zhang in a comment, permuting digits of equal weight does not change the sum.) Many more combinations are possible if that constraint is removed or weakened.

Which elements (if any) of this list contradict your rules? If some do, why?

1 : 2 + 2 + 3 + 3 + 14 + 46 + 57 + 57 + 168 + 998 = 1350
2 : 1 + 1 + 2 + 3 + 43 + 54 + 65 + 86 + 297 + 798 = 1350
3 : 1 + 2 + 3 + 4 + 14 + 25 + 65 + 68 + 379 + 789 = 1350
4 : 2 + 2 + 3 + 4 + 14 + 15 + 65 + 67 + 389 + 789 = 1350
5 : 1 + 1 + 2 + 2 + 36 + 36 + 47 + 47 + 589 + 589 = 1350
6 : 1 + 1 + 2 + 2 + 36 + 36 + 47 + 48 + 578 + 599 = 1350
7 : 1 + 1 + 2 + 3 + 24 + 37 + 47 + 68 + 568 + 599 = 1350
8 : 1 + 1 + 2 + 3 + 24 + 36 + 47 + 68 + 579 + 589 = 1350
9 : 1 + 1 + 2 + 2 + 34 + 37 + 47 + 68 + 569 + 589 = 1350
10 : 1 + 1 + 2 + 4 + 25 + 35 + 47 + 68 + 368 + 799 = 1350
11 : 1 + 2 + 2 + 3 + 15 + 45 + 47 + 68 + 368 + 799 = 1350
12 : 1 + 1 + 2 + 3 + 25 + 45 + 47 + 68 + 369 + 789 = 1350
13 : 1 + 1 + 2 + 3 + 25 + 46 + 47 + 58 + 368 + 799 = 1350
14 : 1 + 2 + 2 + 4 + 15 + 35 + 46 + 68 + 378 + 799 = 1350
15 : 1 + 1 + 2 + 2 + 43 + 54 + 65 + 87 + 397 + 698 = 1350
16 : 1 + 1 + 2 + 2 + 33 + 64 + 65 + 87 + 497 + 598 = 1350
17 : 1 + 1 + 2 + 2 + 33 + 54 + 76 + 86 + 497 + 598 = 1350
18 : 1 + 1 + 2 + 2 + 34 + 54 + 75 + 86 + 397 + 698 = 1350
19 : 1 + 2 + 2 + 3 + 53 + 54 + 64 + 76 + 197 + 898 = 1350
20 : 1 + 2 + 4 + 4 + 25 + 35 + 36 + 67 + 178 + 998 = 1350
21 : 2 + 2 + 3 + 3 + 14 + 74 + 75 + 85 + 196 + 896 = 1350
22 : 1 + 3 + 3 + 4 + 14 + 26 + 26 + 77 + 598 + 598 = 1350
23 : 2 + 2 + 3 + 4 + 14 + 35 + 56 + 67 + 178 + 989 = 1350
24 : 1 + 2 + 3 + 4 + 24 + 35 + 57 + 67 + 168 + 989 = 1350
25 : 2 + 2 + 3 + 3 + 14 + 45 + 57 + 67 + 168 + 989 = 1350
26 : 1 + 1 + 2 + 2 + 53 + 64 + 74 + 75 + 389 + 689 = 1350
27 : 1 + 2 + 2 + 3 + 53 + 64 + 64 + 75 + 187 + 899 = 1350
28 : 1 + 1 + 2 + 3 + 53 + 64 + 64 + 75 + 288 + 799 = 1350
29 : 2 + 2 + 3 + 3 + 14 + 14 + 67 + 67 + 589 + 589 = 1350
30 : 1 + 1 + 3 + 4 + 24 + 35 + 57 + 67 + 269 + 889 = 1350
31 : 1 + 1 + 3 + 3 + 25 + 45 + 47 + 67 + 269 + 889 = 1350
32 : 1 + 1 + 3 + 4 + 25 + 35 + 46 + 67 + 279 + 889 = 1350
33 : 1 + 1 + 2 + 4 + 35 + 35 + 47 + 67 + 269 + 889 = 1350
34 : 1 + 3 + 3 + 4 + 25 + 25 + 46 + 76 + 178 + 989 = 1350
35 : 1 + 1 + 2 + 2 + 34 + 64 + 75 + 85 + 387 + 699 = 1350
36 : 1 + 1 + 3 + 3 + 24 + 26 + 47 + 67 + 589 + 589 = 1350
37 : 1 + 1 + 3 + 3 + 26 + 26 + 47 + 47 + 598 + 598 = 1350

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EDIT: With the "original problem" described in a comment, in which

$$(a + bc + def + gh + i) + (i + hg + fed + cb + a) = 1350 \enspace,$$

the following are the first 37 of 480 solutions obtained by imposing the constraint that $a<i$, $c<h$, and $d<f$:

1 : 1 + 23 + 456 + 78 + 9 + 9 + 87 + 654 + 32 + 1 = 1350
2 : 6 + 14 + 397 + 25 + 8 + 8 + 52 + 793 + 41 + 6 = 1350
3 : 6 + 24 + 397 + 15 + 8 + 8 + 51 + 793 + 42 + 6 = 1350
4 : 7 + 64 + 318 + 25 + 9 + 9 + 52 + 813 + 46 + 7 = 1350
5 : 7 + 54 + 318 + 26 + 9 + 9 + 62 + 813 + 45 + 7 = 1350
6 : 7 + 24 + 318 + 65 + 9 + 9 + 56 + 813 + 42 + 7 = 1350
7 : 1 + 73 + 258 + 64 + 9 + 9 + 46 + 852 + 37 + 1 = 1350
8 : 2 + 84 + 367 + 15 + 9 + 9 + 51 + 763 + 48 + 2 = 1350
9 : 2 + 14 + 367 + 85 + 9 + 9 + 58 + 763 + 41 + 2 = 1350
10 : 8 + 63 + 427 + 15 + 9 + 9 + 51 + 724 + 36 + 8 = 1350
11 : 8 + 53 + 427 + 16 + 9 + 9 + 61 + 724 + 35 + 8 = 1350
12 : 8 + 35 + 427 + 16 + 9 + 9 + 61 + 724 + 53 + 8 = 1350
13 : 8 + 15 + 427 + 36 + 9 + 9 + 63 + 724 + 51 + 8 = 1350
14 : 8 + 61 + 427 + 53 + 9 + 9 + 35 + 724 + 16 + 8 = 1350
15 : 8 + 61 + 427 + 35 + 9 + 9 + 53 + 724 + 16 + 8 = 1350
16 : 8 + 51 + 427 + 63 + 9 + 9 + 36 + 724 + 15 + 8 = 1350
17 : 8 + 13 + 427 + 65 + 9 + 9 + 56 + 724 + 31 + 8 = 1350
18 : 8 + 31 + 427 + 65 + 9 + 9 + 56 + 724 + 13 + 8 = 1350
19 : 8 + 31 + 427 + 56 + 9 + 9 + 65 + 724 + 13 + 8 = 1350
20 : 8 + 51 + 427 + 36 + 9 + 9 + 63 + 724 + 15 + 8 = 1350
21 : 8 + 13 + 427 + 56 + 9 + 9 + 65 + 724 + 31 + 8 = 1350
22 : 6 + 12 + 397 + 54 + 8 + 8 + 45 + 793 + 21 + 6 = 1350
23 : 6 + 52 + 397 + 14 + 8 + 8 + 41 + 793 + 25 + 6 = 1350
24 : 6 + 42 + 397 + 15 + 8 + 8 + 51 + 793 + 24 + 6 = 1350
25 : 4 + 82 + 357 + 19 + 6 + 6 + 91 + 753 + 28 + 4 = 1350
26 : 4 + 92 + 357 + 18 + 6 + 6 + 81 + 753 + 29 + 4 = 1350
27 : 4 + 28 + 357 + 19 + 6 + 6 + 91 + 753 + 82 + 4 = 1350
28 : 4 + 25 + 387 + 16 + 9 + 9 + 61 + 783 + 52 + 4 = 1350
29 : 8 + 43 + 526 + 17 + 9 + 9 + 71 + 625 + 34 + 8 = 1350
30 : 8 + 34 + 526 + 17 + 9 + 9 + 71 + 625 + 43 + 8 = 1350
31 : 8 + 73 + 526 + 14 + 9 + 9 + 41 + 625 + 37 + 8 = 1350
32 : 7 + 42 + 516 + 38 + 9 + 9 + 83 + 615 + 24 + 7 = 1350
33 : 7 + 32 + 516 + 48 + 9 + 9 + 84 + 615 + 23 + 7 = 1350
34 : 7 + 23 + 516 + 48 + 9 + 9 + 84 + 615 + 32 + 7 = 1350
35 : 8 + 13 + 526 + 47 + 9 + 9 + 74 + 625 + 31 + 8 = 1350
36 : 8 + 14 + 526 + 37 + 9 + 9 + 73 + 625 + 41 + 8 = 1350
37 : 8 + 31 + 526 + 47 + 9 + 9 + 74 + 625 + 13 + 8 = 1350