This was a rather delightful mystery. It is complicated, and there are numerous characters. I didn't figure out the solution in advance, but looking back I could see that Mitchell provides the clues. She offers plenty of red herrings.
I've read a number of books about convent life and schools during this periods, and the characterization of this one is decidedly different. Mitchell's convent is a warm and benevolent place. The children receive plenty of affection from the nuns, and students and nuns alike are generally happy. Even the orphans who are being groomed for domestic service content. That's a far cry from the cold and miserable convent schools that appear in so many other early-twentieth century novels and memoirs. Admittedly, Mitchell is primarily interested in crafting a mystery, and that she does well.
Gladys Mitchell, St. Peter's Finger (orig. 1938) ISBN: 0312001924