Puzzle Types

Here are a list of tags I'm using to denote specific puzzle types. Not having much background in brainteasers except an enjoyment in trying to solve them, the classifications are not very scientific, but should give you an idea of what is involved.
For some puzzle types I am using the terms as put forth by tvtropes which appears to maintain a very nice list.
I've also tagged each puzzle with their Picarat value. Generally, the more Picarats you can gain from solving a puzzle, the harder it is.

Story puzzles [tagged puzzles]
Like other Professor Layton games, most puzzles are found by talking to people or examining the landscape. These puzzles tend to have only a small connection to the larger plot of the game. However, Story Puzzles are integral to the plot and must be solved directly in order to advance. For example, when you arrive in Mist Haley and want to cross a bridge, it is tied off with ropes. The ropes become a puzzle you must solve in order to move into the town proper. These puzzles are not numbered or saved in the Puzzle Diary, so I'll note them as I run into them. Also, you don't get picarats for solving them :(

Secret puzzles [tagged puzzles]
Puzzles you find throughout the game which are not part of the general progression of puzzles (they are often found hidden in places like trees or cabinets or other parts of the background.)

Broad puzzle types
Language puzzles [tagged puzzles]
Puzzles that involve wordplay, manipulating letters, etc.

Visual puzzles [tagged puzzles]
Puzzles where you are given an image and told to solve something based on the image.

Math puzzles [tagged puzzles]
Puzzles where you must do some mathematical calculation to find the answer.

Logic puzzles [tagged puzzles]
Puzzles where you must use logic/reasoning to deduct an answer.

Visual puzzle types
Arrangement [tagged puzzles]
Visual puzzles where you must arrange pieces to form a whole. I tend to think of these things as Tetris puzzles, since a lot of them involve arranging oddly shaped boxes

Attention to detail [tagged puzzles]
Visual puzzles where you must decide the answer based on subtle visual cues in one or more images.

Breakup [tagged puzzles]
A visual puzzle that involves cutting or breaking up one shape into multiple other shapes. Often the new shapes must be all the same, or all different. Usually the pieces are also arranged with grids.

Connect the dots [tagged puzzles]
Visual puzzles where you need to draw lines or follow lines already drawn to find the answer. For example, you might be given a knot and need to figure out which of the ropes is not tied to the others; or you may be asked to draw lines connecting fence posts in order to form pens around individual animals.

Map [tagged puzzles]
A visual puzzle that involves finding locations on a map.

Route [tagged puzzles]
A visual puzzle involving directions or routes, such as finding the most efficient path through a maze.

Sliding [tagged puzzles]
A visual puzzle where you must slide around different size/shaped blocks, usually in order to move one specific block to a different position. One particular version is known as a Klotski puzzle.

Clock [tagged puzzles]
A visual puzzle (or numerical/mathematical) involving a clock. It may be an analog or digital clock face, so you may be manipulating digits or clock hands, or other clock-related parts.

Math puzzle types
...tbd...

Logic puzzle types
Fox, Chicken, Grain [tagged puzzles]
A logic puzzle where you must figure out how to transport a number of items (generally animals/food) from one side of a river to the other under specific conditions. Generally, you can only transport a small number of items at a time, and leaving certain items together while you head to the other side of the river results in loosing one of the items (usually due to the animals eating each other.) More information at tvtropes.

Calendar [tagged puzzles]
Calendar logic puzzles involve time as a particular condition, usually in the form of days, but sometimes in the form of hours or weeks.