In Time Trials the puzzles are in random order and you can't choose them yourself. You can however select 'Give-Up' in the pause menu to force another puzzle. There are 64 possible puzzles, organized here by the first number of the first column.
First Number: 1[edit]
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Bird
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Buzzy Beetle
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Corn
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Grim Reaper
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Pineapple
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Shrimp
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Tulip
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Witch
First Number: 2[edit]
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Bi-Plane
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Dirk
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Flower
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Tea Cup
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Tortoise
First Number: 3[edit]
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Angler
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Banana
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Blooper
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Cat
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Chick
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Crane
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Fish
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Lock
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Ostrich
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Rocket
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Sheriff
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White Eye
First Number: 4[edit]
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Alarm Clock
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Bee
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Bow
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Candy
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Cheep-Cheeps
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Clay Image
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Hippopotamus
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Koala
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Koopa Troopas
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Man in the Moon
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Mouse
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Ram
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Ring
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Sake Bottle
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Scooter
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Strawhat
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Wooden Horse
First Number: 5[edit]
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Butterfly
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Feather
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Friendly Dog
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Giraffe
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Horse
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Hedgehog
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Kangaroo
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Pegasus
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Pig
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Snake
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Water Melon
First Number: 6[edit]
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Football
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Monkey
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Rabbit
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Scorpion
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Teruterubouzu
First Number: 7[edit]
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White Lion
First Number: 8[edit]
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Dump Truck
First Number: 10[edit]
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Rhinoceros
First Number: 11[edit]
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Socks
First Number: 12[edit]
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Elephant
First Number: 15[edit]
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Woodpecker