I thought the game was cool so I built a little version of it here:
The only possible move is to add 1 and 3. Can't ever split a 2 as that would result in [1, 1], can't split the 5 because that would be [4, 1] or [3, 2], can't split the 3, as that would be [2, 1] or [1, 2], can't add the 5 to anything as that would be larger than 5.
It also violates Rule 1 because one of the splits is larger than the original number.
https://bewelge.github.io/aNumberGame/ (does not format well on mobile)
Example for sequence 5,2,7
Though I say the rule about repeated numbers is a bummer. I can see things might get easier (also it might be a more procedurally easy problem to solve) if this is allowed
Basically I don't think it's a "productive" complication
Like this:
https://blaise.gg/number_game/index.html?numbers=1%7E4%7E6%7...
[n, n-1]
[n-3, 3, n-1]
[n-3, 2, 1, n-1]
[n-3, 2, n]
[n-1, n]
[n, n-1]
[2, n-2, n-1]
[2, n-3, 1, n-1]
[2, n-3, n]
[n-1, n]
[3,4]
--> [1,2,4] --> 4 can't split because 2/2 and 3/1 are invalid
--> 4 can't initially split because 2/2 and 3/1 are both invalid
[5,6] --> [5,4,2] --> [5,1,3,2] --> [6,3,2] --> [6,5]
[7,8] --> [7,3,5] --> [7,1,2,5] --> [8,2,5] --> [8,7]
When you add more numbers, you need more wiggle room. So, [4,5,6] is problematic and probably [5,6,7].
[4,5,6]
--> [3,1,5,6] --> 6 can't break into 3, nor 5/1. It can do 4/2, but then 5 can't split. 5 also can't split.
--> [4,2,3,6] --> [4,2,3,1,5] --> [6,4,5] --> 4 can't split into 2/2. It can split to 3/1 but then 5 can't split
--> 6 can't initially split at all because 3/3, 4/2, and 5/1 are invalid
Even if [6,7,8] has a solution, I'm sure [6,7,8,9] does not.
6,7,8,9
6,3,4,8,9
6,3,4,8,2,7
9,4,8,2,7
9,4,8,2,1,6
9,4,3,5,2,1,6
9,7,5,2,1,6
9,7,5,3,6
9,7,8,6
9,5,2,8,6
9,5,2,1,7,6
9,5,3,7,6
9,8,7,6
6,7,8
6,7,5,3
6,7,1,4,3
6,8,4,3
6,8,7
6,3,5,7
6,2,1,5,7
8,1,5,7
8,6,7
8,4,2,1,6
8,4,3,6
8,7,6