My guess would be it's similar to the (now rather archaic) "in the least [bit]", and "at least".
Er det i det mindste muligt? - Is it in the least (bit) possible? Er det i mindst muligt? - Is it at least possible?
Need a proper Dansk speaker to give a real answer though, this is just a guess.
I am not 100% sure either, but I think that it may have to do with the difference between predicative and attributive words. In English grammar it would be the difference between "the dog is the best" and "the best dog."
Does anyone else know for sure whether I am completely off base or not?
For the "pick the words" question, I chose "is it at all possible?" which was marked wrong. I am curious what others think.
I know it doesn't use the word "least", but I think we have established a literal translation would be bad English.
Also, maybe "is it in the least bit possible?" would be a better translation if we are aiming for something close to the literal version.
dyr in Danish means animal and is related to the English word "deer" (which used to mean any small animal but now refers to a particular kind of animal).
dyr in Danish also means expensive and is related to the English word "dear" (precious, valuable, costly).
The two words used to be separate but have fallen together through sound changes in Danish -- just as they did in pronunciation in English, though English still spells them differently.