Digit Game
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 |
tdemselves |
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 |
| ... |
| 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 |
| ... |
| 100 |
(3+8)*9+1 |
4 |
| ... |
| 1,000 |
(7*8*9-4)*2 |
5? |
| ... |
| 2,019 |
4*7*8*9+3 |
5? |
| ... |
| 10,000 |
(4*7*9-2)*5*8 |
6? |
Solutions for 1,000
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 I don't think it's possible with only four digits.
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.
Solutions for 10,000
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.
Solutions for 100,000
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.
Other Notable Solutions
- 10 is the smallest natural number number requiring two digits for a solution: 9+1.
- 19 is the smallest natural number number requiring three digits for a solution: 2*9+1.
- 72 is the largest number requiring only two digits for a solution: 8*9.
- 73 is* the smallest natural number requiring four digits for a solution using only subtraction and multiplication: 2*5*8-7.
- 82 is* the smallest natural number requiring four digits for a solution: (2+7)*9+1.
- 146 is* the smallest natural number requiring five digits for a solution using only subtraction and multiplication: (6*9-5)*3-1.
- 298 is* the smallest natural number requiring five digits for a solution: (4+7)*9*3+1.
- 504 is the largest number requiring only three digits for a solution: 7*8*9.
- 506 is* the smallest natural number requiring six digits for a solution using only subtraction and multiplication: ((9-1)*8*4-3)*2.
- 647 is the smallest natural number requiring six digits for a solution using only addition and multiplication: (5*7*9+8)*2+1.
- 659 is* the smallest natural number requiring six digits for a solution: (7*9+3)*5*2-1.
- a number between 1,600 and 60,481 is* the smallest natural number requiring seven digits for a solution: ???.
- a number between 1,600 and 362,878 is* the smallest natural number with no solution.
- 3,024 is the largest number requiring only four digits for a solution: 6*7*8*9.
- 15,120 is the largest number requiring only five digits for a solution: 5*6*7*8*9.
- 60,480 is the largest number requiring only six digits for a solution: 4*5*6*7*8*9.
- 181,440 is the largest number requiring only seven digits for a solution: 3*4*5*6*7*8*9.
- 362,880 is the largest number requiring only eight digits for a solution: 2*3*4*5*6*7*8*9.
- 544,320 is the largest number with a solution: (1+2)*3*4*5*6*7*8*9.
Other Notable Facts
- All numbers from 1 to 45 are constructible using only addition, and none above are.
- 39 numbers below 100 are constructible using only multiplication.
- All numbers below 600 are constructible using only subtraction and multiplication.
- All numbers below 1,000 are constructible using only addition and multiplication.