# Given a string which was produced by mixing up a string of digits, find the original digits

I encountered the following problem:

Given a string which was produced by mixing up a string of digits (0-9), for example: "otetwonhree" was produced by "onetwothree"~123, find the original digits (without importance of order). in our example we would return "123". Another example: "oneoneeon", we return "111".

My question is, is there necessarily a unique way to interpret such a string? meaning, could there be a string with 2 different correct interpretations? I feel like it is complicated to show that (if we know that there isn't such a string, then the solution is easy, otherwise the problem doesn't really have a solution).

More generally, given some codes $$\sigma_1, ..., \sigma_n$$ instead of digits, each has a unique fitting string of letters, could we decide if any string which was created similarly as in the original problem, has a unique interpretation? It's easy to find some codes for which the answer would be no, which is why I find this problem also difficult.

I'll mention I haven't studied any computability courses yet so I am mainly looking for an answer rather than a full solution (but feel free to post such if it is of any interest to anyone else), this problem just made me curious. If there is a solution which could be somewhat understood with only basic algorithms background I wouldn't mind that.

The letters in the garbled message come from the number of times a code was used. In a forward way: if the code for "7" occurs $$x_7$$ times we have $$2x_7$$ e's, and $$x_7$$ n, s, v each.
For decoding we in reverse try to determine the number $$x_i$$ of times that code for "i" is present, for each $$i=0,1,\dots,9$$.
As an example, the total number of e's in the message equals $$\#e = x_0+x_1+2x_3+2x_7+x_8+x_9$$. The number of f's equals $$\#f = x_4+x_5$$.
I will not do this for every symbol, but after doing this we get a system of equations from which the numbers $$x_0,\dots,x_9$$ can be determined (if the system has no dependencies). Note that for some letters we might be lucky, like $$\#z = x_0$$.