I didn't say that a proof wasn't in principle accessible to them; I said that it wasn't obvious. In particular, if the students in question didn't indicate that they were going through some such proof - and surely, previous posters would have remarked on it if they had - then I don't believe they had a proof in mind.
As I illustrated in my earlier message, the entire line of reasoning is within the concepts of integer multiplication in probably grade-2 math. So the label of "being good in math" and especially for the kids already in grade 3, should only be reserved for those knowing how and when to apply such concepts in different types of problems. And even more so, because this is a gifted forum and presumably we may steer the discussion toward gifted kids.
Then how do I know at least several Asian kids can solve this type of problems at age 5, without any help whatsoever? Simply because I asked them to explain after arriving at the correct answers. They may not have illustrated the line of reasoning as I did in my earlier message, although in a more clumsy 5-year-old manner. But as long as they can point out that each higher-value coin, can represent an exact number of each lower-value coin, and consequently the higher the value, the less number of coins are necessary, then they have implicitly discovered the "greedy algorithm" on their own. And incidentally, these kids have been studying grade-3 math at the age of 5 and apparently have mastered grade-2 math.
(In fact, if your kids go to a school where many 3rd graders would be capable of producing one, I expect many people here would be envious!)
I suppose we may accept as fact that most 3rd-graders don't master grade-2 math at all, and many people here supplement their kids with more advanced curriculum, than merely depend on the school.
