Almost every formula, idea, and trick that Agatha Christie used in her detective fiction works proved to be entirely successful and won her an enormous reading public. Making use of nursery rhymes was one such formula. Nursery rhymes can reawaken the sense of wonder, mystery and enchantment in any reader. They also can carry symbolic levels of meaning, and some are allegories.
In this her 1953 offering she makes use of the nursery rhyme "Sing A Song Of Sixpence". Appropriately it is one of her Miss Marple books. Although her elderly spinster sleuth has little to do here, and is late making her appearance, it is she who perceives and urges the significance of the nursery rhyme. "Don't you see, it makes a pattern to all this."
The murders occur in the disfunctional family of Rex Fortescue, a financier, and the action occurs in his London office and in the family home, Yew Tree Lodge. The opening chapters are wonderfully engaging. Agatha Christie, when she took the trouble, could sketch characters vividly. Amongst all of them in this book, there are not more than a handful of suspects. To compensate, Mrs Christie throws in buckets full of red herrings.
You'll enjoy the puzzle, and having innumerable theories suggested and dismissed. The solution, when it comes, however, is no more plausible than is the likelihood of a blackbird pecking off a maid's nose.
If you can obtain the unabridged reading of the book by Rosemary Leach, your enjoyment will be enhanced. Rosemary Leach is unusually skilled at "doing" the voices of a large cast of characters, male and female.