In response to a challenge from Twitter user @7puzzle I was inspired to write a small PHP script that would generate all whole numbers created by a given function. Here's the original challenge:
"Using 1 2 3 and + - x ÷ available, it IS http://mathematicallypossible.com to make ALL target numbers 1-9. Is this statement TRUE or FALSE?" - @7puzzle
and here's the output of my routine:
1 = 1 - 1 + 1
1 = 1 - 2 + 2
1 = 1 - 3 + 3
1 = 1 * 1 * 1
1 = 1 * 1 / 1
1 = 1 * 2 - 1
1 = 1 * 2 / 2
1 = 1 * 3 - 2
1 = 1 * 3 / 3
1 = 1 / 1 * 1
1 = 1 / 1 / 1
1 = 1 / 2 * 2
1 = 1 / 3 * 3
1 = 1 + 1 - 1
1 = 1 + 2 - 2
1 = 1 + 3 - 3
1 = 2 - 1 * 1
1 = 2 - 1 / 1
1 = 2 - 2 / 2
1 = 2 - 2 + 1
1 = 2 - 3 / 3
1 = 2 - 3 + 2
1 = 2 * 1 - 1
1 = 2 * 1 / 2
1 = 2 * 2 - 3
1 = 2 / 1 - 1
1 = 2 / 1 / 2
1 = 2 / 2 * 1
1 = 2 / 2 / 1
1 = 2 + 1 - 2
1 = 2 + 2 - 3
1 = 3 - 1 - 1
1 = 3 - 1 * 2
1 = 3 - 2 * 1
1 = 3 - 2 / 1
1 = 3 - 3 + 1
1 = 3 * 1 - 2
1 = 3 * 1 / 3
1 = 3 / 1 - 2
1 = 3 / 1 / 3
1 = 3 / 3 * 1
1 = 3 / 3 / 1
1 = 3 + 1 - 3
2 = 1 - 1 + 2
2 = 1 - 2 + 3
2 = 1 * 1 * 2
2 = 1 * 1 + 1
2 = 1 * 2 * 1
2 = 1 * 2 / 1
2 = 1 * 3 - 1
2 = 1 / 1 * 2
2 = 1 / 1 + 1
2 = 1 + 1 * 1
2 = 1 + 1 / 1
2 = 1 + 2 - 1
2 = 1 + 2 / 2
2 = 1 + 3 - 2
2 = 1 + 3 / 3
2 = 2 - 1 + 1
2 = 2 - 2 + 2
2 = 2 - 3 + 3
2 = 2 * 1 * 1
2 = 2 * 1 / 1
2 = 2 * 2 - 2
2 = 2 * 2 / 2
2 = 2 * 3 / 3
2 = 2 / 1 * 1
2 = 2 / 1 / 1
2 = 2 / 2 * 2
2 = 2 / 2 + 1
2 = 2 / 3 * 3
2 = 2 + 1 - 1
2 = 2 + 2 - 2
2 = 2 + 3 - 3
2 = 3 - 1 * 1
2 = 3 - 1 / 1
2 = 3 - 2 / 2
2 = 3 - 2 + 1
2 = 3 - 3 / 3
2 = 3 - 3 + 2
2 = 3 * 1 - 1
2 = 3 * 2 / 3
2 = 3 / 1 - 1
2 = 3 / 3 * 2
2 = 3 / 3 + 1
2 = 3 + 1 - 2
2 = 3 + 2 - 3
3 = 1 - 1 + 3
3 = 1 * 1 * 3
3 = 1 * 1 + 2
3 = 1 * 2 + 1
3 = 1 * 3 * 1
3 = 1 * 3 / 1
3 = 1 / 1 * 3
3 = 1 / 1 + 2
3 = 1 + 1 * 2
3 = 1 + 1 + 1
3 = 1 + 2 * 1
3 = 1 + 2 / 1
3 = 1 + 3 - 1
3 = 2 - 1 + 2
3 = 2 - 2 + 3
3 = 2 * 1 + 1
3 = 2 * 2 - 1
3 = 2 * 3 - 3
3 = 2 * 3 / 2
3 = 2 / 1 + 1
3 = 2 / 2 * 3
3 = 2 / 2 + 2
3 = 2 + 1 * 1
3 = 2 + 1 / 1
3 = 2 + 2 - 1
3 = 2 + 2 / 2
3 = 2 + 3 - 2
3 = 2 + 3 / 3
3 = 3 - 1 + 1
3 = 3 - 2 + 2
3 = 3 - 3 + 3
3 = 3 * 1 * 1
3 = 3 * 1 / 1
3 = 3 * 2 - 3
3 = 3 * 2 / 2
3 = 3 * 3 / 3
3 = 3 / 1 * 1
3 = 3 / 1 / 1
3 = 3 / 2 * 2
3 = 3 / 3 * 3
3 = 3 / 3 + 2
3 = 3 + 1 - 1
3 = 3 + 2 - 2
3 = 3 + 3 - 3
4 = 1 * 1 + 3
4 = 1 * 2 * 2
4 = 1 * 2 + 2
4 = 1 * 3 + 1
4 = 1 / 1 + 3
4 = 1 + 1 * 3
4 = 1 + 1 + 2
4 = 1 + 2 + 1
4 = 1 + 3 * 1
4 = 1 + 3 / 1
4 = 2 - 1 + 3
4 = 2 * 1 * 2
4 = 2 * 1 + 2
4 = 2 * 2 * 1
4 = 2 * 2 / 1
4 = 2 * 3 - 2
4 = 2 / 1 * 2
4 = 2 / 1 + 2
4 = 2 / 2 + 3
4 = 2 + 1 * 2
4 = 2 + 1 + 1
4 = 2 + 2 * 1
4 = 2 + 2 / 1
4 = 2 + 3 - 1
4 = 3 - 1 + 2
4 = 3 - 2 + 3
4 = 3 * 1 + 1
4 = 3 * 2 - 2
4 = 3 / 1 + 1
4 = 3 / 3 + 3
4 = 3 + 1 * 1
4 = 3 + 1 / 1
4 = 3 + 2 - 1
4 = 3 + 2 / 2
4 = 3 + 3 - 2
4 = 3 + 3 / 3
5 = 1 * 2 + 3
5 = 1 * 3 + 2
5 = 1 + 1 + 3
5 = 1 + 2 * 2
5 = 1 + 2 + 2
5 = 1 + 3 + 1
5 = 2 * 1 + 3
5 = 2 * 2 + 1
5 = 2 * 3 - 1
5 = 2 / 1 + 3
5 = 2 + 1 * 3
5 = 2 + 1 + 2
5 = 2 + 2 + 1
5 = 2 + 3 * 1
5 = 2 + 3 / 1
5 = 3 - 1 + 3
5 = 3 * 1 + 2
5 = 3 * 2 - 1
5 = 3 / 1 + 2
5 = 3 + 1 * 2
5 = 3 + 1 + 1
5 = 3 + 2 * 1
5 = 3 + 2 / 1
5 = 3 + 3 - 1
6 = 1 * 2 * 3
6 = 1 * 3 * 2
6 = 1 * 3 + 3
6 = 1 + 2 + 3
6 = 1 + 3 + 2
6 = 2 * 1 * 3
6 = 2 * 2 + 2
6 = 2 * 3 * 1
6 = 2 * 3 / 1
6 = 2 / 1 * 3
6 = 2 + 1 + 3
6 = 2 + 2 * 2
6 = 2 + 2 + 2
6 = 2 + 3 + 1
6 = 3 * 1 * 2
6 = 3 * 1 + 3
6 = 3 * 2 * 1
6 = 3 * 2 / 1
6 = 3 * 3 - 3
6 = 3 / 1 * 2
6 = 3 / 1 + 3
6 = 3 + 1 * 3
6 = 3 + 1 + 2
6 = 3 + 2 + 1
6 = 3 + 3 * 1
6 = 3 + 3 / 1
7 = 1 + 2 * 3
7 = 1 + 3 * 2
7 = 1 + 3 + 3
7 = 2 * 2 + 3
7 = 2 * 3 + 1
7 = 2 + 2 + 3
7 = 2 + 3 + 2
7 = 3 * 2 + 1
7 = 3 * 3 - 2
7 = 3 + 1 + 3
7 = 3 + 2 * 2
7 = 3 + 2 + 2
7 = 3 + 3 + 1
8 = 2 * 2 * 2
8 = 2 * 3 + 2
8 = 2 + 2 * 3
8 = 2 + 3 * 2
8 = 2 + 3 + 3
8 = 3 * 2 + 2
8 = 3 * 3 - 1
8 = 3 + 2 + 3
8 = 3 + 3 + 2
9 = 1 * 3 * 3
9 = 2 * 3 + 3
9 = 3 * 1 * 3
9 = 3 * 2 + 3
9 = 3 * 3 * 1
9 = 3 * 3 / 1
9 = 3 / 1 * 3
9 = 3 + 2 * 3
9 = 3 + 3 * 2
9 = 3 + 3 + 3
Conclusion:
The challenge statement is true.
