A week ago, I found a partial skeleton of a mammal on a rocky shore in Biddeford, ME. The skull was damaged and missing some teeth, and I could only locate half of the mandible (lower jaw), but I decided to take some time to determine what kind of mammal had died.
New England mammals can be classified into one of seven orders (see Wikipedia for a complete list). Let's narrow the search down one order at a time.
Start by estimating the total skeleton length. At about 2', that rules out small mammals like Insectivores (order Soricomorpha) and Bats (order Chiroptera), and large mammals, like Deer, Moose, and Whales (order Cetartiodactyla).
Next, let's examine the teeth and their placement. The upper jaw (maxilla) has eight back teeth (four on each side) separated by a sizable gap from two front incisors (or in this case, holes where the teeth would be) for a total of 10 teeth. The absence of carnassial teeth rules out a Carnivore (order Carnivora). Lagomorphs (order Lagomorpha) have four upper incisors, so they are out, too. And this isn't an Opposum (order Didelphimorphia), who would have a total of 50 teeth (26 of which are upper), more than any other North American mammal.
That leaves just one order: Rodentia, or the Rodents. Most rodents are too small to be candidates, so we can narrow our search down to the four largest rodent species: Common Muskrat, American Beaver, Woodchuck, and Porcupine. The skull's total length of 3.75" is too large for Common Muskrat (~2.25-2.75") and is a bit small for American Beaver (~4-6"). Both Woodchuck (~3-4") and Porcupine (~3.5-4.75") have skull lengths that fit, but Woodchucks have 12 upper teeth, whereas Porcupines have 10. Therefore, this skull belonged to a North American Porcupine (Erethizon dorsatum).
Source: Total skull lengths were approximated from Mark Elbroch's Animal Skulls: A guide to North American species (2006). This book is a great resource to help solve an animal skull mystery.