The issue with multiple solutions that I could see is that (no offense to elem. teachers) but some of them may take the answer out of the book and unless the book notes multiple solutions I could see my kid coming home and saying her teacher told her she was wrong at which point I would end up having to conference with them via email about multiple solutions in some nice way. I'm sure I would end up being tactless but sometimes teachers will go strictly by the book if they aren't comfortable with a subject. Kids at that age are likely to believe their teacher that they were wrong even when they did it correctly (just different)
There are no multiple solutions; there's only one correct answer, that it's impossible to tell what coins each has. Someone could elaborate on that by explaining why, but an answer with a single particular combination of coins that fits the numeric relationships of the problem is no more correct than an answer that no combinations fit those relationships.
So for a child who finds one of the combinations, they've done some valuable work, but not finished the problem. Any acceptance of a partial answer harms the student's ability to correctly understand the problem as framed, and in math problems I think it's very important to be precisely correct and to understand exactly what's being asked.
From that aspect there are multiple harmful aspects to this problem, assuming the teacher takes any combination that fits as a correct answer (or worse, only one of them):
* Students are encouraged in fuzzy, incorrect thinking about the call of the problem, when it could be a useful opportunity to develop critical thinking abilities
* Students are encouraged to quit easily after finding a seeming answer, instead of developing the ability to discern between a situation where they have to disprove other possible answers and when the first is good enough; in this scenario they'll learn to never even question whether they're being asked to find a single answer or any that fits
* A student may be called incorrect, when finding one combination of coins which fits and thinking she's found the answer-- when of course she would be incorrect, but given the wrong answer as to why
* A student may be crushed after feeling elated over finding a winning combo, only to be told that the real answer is "no answer can be given"-- though of course feeling gypped by a trick question, and adjusting to be more precise in one's thinking, is far better than the alternatives
* A student who actually finds the correct answer will almost certainly go unrewarded, as his teacher might say, "Hm, that's interesting" and mention it in conversation, but no changes in instruction will be made; worst-case, that student will be "behind" the class time-wise, and perhaps interrupted and not allowed to finish, as he works toward the correct solution instead of quitting halfway through with the "right" answer
I could come up with scads more blathery ways of explaining the massive harm from this innocuous-seeming little coin problem, but hopefully you get the idea.
