moorfield wrote:Under the tree this Christmas the three moorfield juniors will have twelve presents each, of different sizes, to open.
Usually there is a flurry of them all being opened at the same time, but this year we've decided to be a little more orderly and let them open one,
more than one, or all of their presents in turn, provided that they start from the smallest up to the largest.
How many different ways are there of opening all the presents?
Let me pose an extra puzzle: how many different ways are there to reasonably interpret that puzzle?
After looking at just its first paragraph, I can see at least two different ways to interpret it that both look reasonable to me, and arguably a third. If I call the sizes of the first moorfield junior A1, A2, A3, ... A12, those of the second B1, B2, B3, ... B12 and those of the third C1, C2, C3, ... C12, they are:
1) The moorfield juniors have twelve presents each, and the sizes of the 36 presents are all different - i.e. all of the Ais, Bis and Cis are different from each other.
2) The moorfield juniors have twelve presents each, and the sizes of each junior's twelve presents are all different from each other - i.e. all of the Ais are different from each other, all of the Bis are different from each other, and all of the Cis are different from each other, but it's possible that an Ai is the same as a Bi, an Ai is the same as a Ci, or a Bi is the same as a Ci.
3) The moorfield juniors have twelve presents each, and there are twelve different sizes of present, with each junior having one present of each size - i.e. all of the Ais are different from each other, and the collection of all the Bis and the collection of all the Cis are each a rearrangement of the collection of all the Ais. I don't think this is as natural an interpretation, but I'm not entirely convinced it can be ruled out...
Moving on, the second paragraph doesn't seem clear about whether the smallest->largest condition applies to each junior independently, so that it's OK for junior A to open a present of size X before junior B has opened a present of size Y < X, or whether it applies to them all together, so that the same is not OK. And there are various other somewhat unclear points, such as who chooses how many presents are to be opened and when they make that choice: I think the most natural interpretation is that each junior chooses the number they will open each time their turn comes around, but there are certainly others, e.g. that moorfield says at the start of each round "This round, I'll let you each open only one present" or "This round, I'll let you each open more than one present" or "This round, I'll let you each open all your presents".
There are quite a few combinations of those (and other) interpretations of various aspects of the question, and they lead to quite a few different answers - more than I really care to take the time to post!
Gengulphus