#113 - "The Pet Hotel" (DON'T READ BELOW IF YOU DON'T WANT THE ANSWER)
You have to move matchsticks and spell out "cat." You start with OA|, and the idea is that you move the two right-most vertical matchsticks over the farther letter. See below:
._...._.._
|..|..|.|..|
|_|...-|..|
which becomes (you remove two from the O, put one in the middle to make an A, and then the other above on the right to make a T)
._...._..__
|.....|.|..|
|_...|-|..|
But I kept making it
._....._..__
|......|.|..|
|_|...|-|..|
See, you could move two matches. So what I kept doing was moving one from the O into the A, and then I moved the already-existing top part of the T. I kept thinking "Well, will the game recognize the C?" After that failed, I tried it again by using the bottom part of it, but that didn't work either.
#55 - "The Odd Sandwich"
I solved this one completely by accident. You're supposed to make a horizontal cut, which makes a shape like a house with a pointy top, and then a side with an indention, which ends up making a full rectangle. But I reasoned that you could make a vertical cut on the right side. This would chop off the tips, forming small triangles. Then you could put them in the bottom right corner of the sandwich and in the top indention, which would also form a rectangle. The bottom tip ends up being a right triangle, so it fits neatly in the corner. The top part does the same. I don't actually know if it would work, but the answer was only the number of cuts, not how they were oriented. Besides, it was hard to visualize since you can't get a real feel for how large the sandwich is versus the box it's going in.
#88 - "In a Hole"
This is a simple one about how to get a ball out of a hole. The answer is a word - you type out "water." The idea is that you flood the hole, forcing the ball upward, as it will float.
But see, the condition for the answer is "a five letter word for something that's all around you and very common." I don't know if anyone has ever heard of Mindtrap, but it's a boardgame full of these kinds of question (some of them probably overlap between
Layton). Anyway, there is a variation of this problem about a ping-pong ball in a PVC pipe, and it says "A six year old figured out how to retrieve it. What did he do?" The answer is that he pees into the pipe, which causes the ball to float.
Which means I wanted to write in "urine" for the answer for that puzzle. I didn't, but I tried it just now while writing this post, and it didn't work. Besides, I guess they didn't have artwork to depict that.