The Digit Game is simple: you have a pool of the digits {1, 2, ..., 9}. You can add, subtract, multiply, and divide and combination of elements from this pool in any order you wish. Note that this means a digit can only ever appear once in a solution.
| n | minimal solution(s)(?) | length |
|---|---|---|
| 0 | 3-2-1 | 3 |
| 1-9 | themselves | 1 |
| 10-18 | trivial sums & products | 2 |
| 19 | 2*9+1 | 3 |
| 20 | 4*5 | 2 |
| 21 | 3*7 | 2 |
| 22 | 3*7+1 | 3 |
| 23 | 3*8-1 | 3 |
| 24 | 3*8 | 2 |
| 25 | 3*8+1 | 3 |
| 26 | 3*9-1 | 3 |
| 27 | 3*9 | 2 |
| 28 | 4*7 | 2 |
| 29 | 4*7+1 | 3 |
| 30 | 5*6 | 2 |
| 31 | 5*6+1 | 3 |
| 32 | 4*8 | 2 |
| 33 | 4*8+1 | 3 |
| 34 | 5*7-1 | 3 |
| 35 | 5*7 | 2 |
| 36 | 4*9 | 2 |
| 37 | 4*9+1 | 3 |
| 38 | 4*9+2 | 3 |
| 39 | 5*8-1 | 3 |
| 40 | 5*8 | 2 |
| 41 | 5*8+1 | 3 |
| 42 | 6*7 | 2 |
| 43 | 6*7+1 | 3 |
| 44 | 5*9-1 | 3 |
| 45 | 5*9 | 2 |
| 46 | 5*9+1 | 3 |
| 47 | 6*8-1 | 3 |
| 48 | 6*8 | 2 |
| 49 | 6*8+1 | 3 |
| 50 | 6*8+2 | 3 |
| 51 | 6*8+3 | 3 |
| 52 | 6*8+4 | 3 | 53 | 6*8+5 | 3 |
| 54 | 6*9 | 2 |
| 55 | 6*9+1 | 3 |
| 56 | 7*8 | 2 |
| 57 | 7*8+1 | 3 |
| 58 | 7*8+2 | 3 |
| 59 | 7*8+3 | 3 |
| 60 | 7*8+4 | 3 |
| 61 | 7*8+5 | 3 |
| 62 | 7*8+6 | 3 |
| 63 | 7*9 | 2 |
| 64 | 7*9+1 | 3 |
| 65 | 7*9+2 | 3 |
| 66 | 7*9+3 | 3 |
| 67 | 7*9+4 | 3 |
| 68 | 7*9+5 | 3 |
| 69 | 7*9+6 | 3 |
| 70 | 7*5*2 | 3 |
| 71 | 8*9-1 | 3 |
| 72 | 8*9 | 2 |
| 73 | 8*9+1 | 3 |
| 74 | 8*9+2 | 3 |
| 75 | 8*9+3 | 3 |
| 76 | 8*9+4 | 3 |
| 77 | 8*9+5 | 3 |
| 78 | 8*9+6 | 3 |
| 79 | 8*9+7 | 3 |
| 80 | 2*5*8 | 3 |
| 81 | (1+8)*9 | 3 |
| 82 | (2+7)*9+1 | 4 |
| 83 | 2*6*7-1 | 4 |
| 84 | 2*6*7 | 3 |
| 85 | (9+8)*5 | 3 |
| 86 | (9+8)*5+1 | 4 |
| 87 | (9+8)*5+2 | 4 |
| 88 | (9+2)*8 | 3 |
| 89 | (9+2)*8+1 | 4 |
| 90 | 9*5*2 | 3 |
| 91 | (9+4)*7 | 3 |
| 92 | (9+4)*7+1 | 4 |
| 93 | (9+4)*7+2 | 4 |
| 94 | (9+4)*7+3 | 4 |
| 95 | 8*6*2-1 | 4 |
| 96 | 8*6*2 | 3 |
| 97 | 8*6*2+1 | 4 |
| 98 | (9+5)*7 | 3 |
| 99 | (6+5)*9 | 3 |
| 100 | (3+8)*9+1 | 4 |
| ... | ||
| 1,000 | (7*8*9-4)*2 | 5 |
| ... | ||
| 2,019 | 4*7*8*9+3 | 5 |
| 2,020 | (9*8*7+1)*4 | 5 |
| 2,021 | 4*7*8*9+5 | 5 |
| 2,022 | 4*7*8*9+6 | 5 |
| 2,023 | (9*8*4+1)*7 | 5 |
| 2,024 | (9*8*7+2)*4 | 5 |
| 2,025 | (8+7)*9*5*3 | 5 |
| 2,026 | (8+7)*9*5*3+1 | 6 |
| ... | ||
| 10,000 | (4*7*9-2)*5*8 | 6? |
The minimal solution must have more than three digits, since the largest number you can make with three is 7*8*9 = 504. 2*7*8*9 = 1,008 is a near solution, but it is not possible with only four digits, as this would require a total of three factors of 5 to appear but the only digit like this is 5 itself. I was able to find the solution shown above requiring five digits. I haven't tried every possibility (just most) so it's possible you can do it in four, but I have many, many doubts.
The minimal solution must have more than four digits, since the largest number you can make with four is 6*7*8*9 = 3,024. 4*5*7*8*9 = 10,080 is a near solution, but I don't think it's possible with only five digits. I was able to find the solution shown above requiring six digits. I haven't tried every possibility (just most) so it's possible you can do it in five, but I have many, many doubts.
The minimal solution must have more than six digits, since the largest number you can make with six is 4*5*6*7*8*9 = 60,480. Closest found solutions using seven digits: 96,768 (4*(5+3)*6*7*8*9) and 105,840 ((4+3)*5*6*7*8*9). I'm not sure this is even constructible...