The Dishes Problem & Nursery Rhyme Puzzle

My first reaction to seeing the dishes problem is to scour through methods that are familiar to me, including solve it algebraically:

Suppose the number of guests is denoted by x, then the guests have used:

• x/2 number of rice dishes
• x/3 number of broth dishes
• x/4 number of meat dishes

As the total of the dishes used is 65, these information gives us:

x/2 + x/3 + x/4 = 65

Now, solve for x in the above equation:

6x/12 + 4x/12 + 3x/12 = 65

13x/12 = 65

x = 60

Hence, using algebra gives a solution of 60 guests.

Even though this problem can easily be stated in its equation form without altering the solution, I think presenting it as a word puzzle (and its original form) adds more flavor and relevance to the real world. I also feel that word problems, especially puzzles and riddles, add another layer on top of mathematical logic and concepts. The problem solver now has to dissect the semantics and language to infer what the question is, and then proceed with finding the answer. It is also in our human nature to be motivated by challenges, and reshaping a pure math problem that can be easily solved into something both complicated and playful is a way of changing our attitudes towards what's trivial.

Regarding the nursery rhyme puzzle (at least that's what I think it is), I have two ways of interpreting the cake sharing scenario:

1) If the fiddler, the wife, the piper and the mother are four different individuals, then they could have allocated the three cakes, three half cakes and three quarters any way they had liked because it's not necessary for them to have equal shares.

2) If the fiddler's wife and the piper's mother are one person, then there would be three people, and so each would receive one cake, one half cake and one quarter.

I am leaning towards the latter because if we really want to impose an equality condition, then this allocation is easily attainable.