Kura Puzzles are unlocked the same way wallpapers are, which is when certain events are triggered. Here is a list of all the puzzles you can currently unlock.
Kura puzzle
#1 - You get this automatically when you go to the puzzle game
#2 - Interact with Painter-kun in the Art Gallery
#3 - When you first go to Sign World
#4 - Interact with Aoshiru at The Hospital
#5 - When you first go to the Bowling Zone
#7 - Interact with the blob NPCs in the Marijuana Goddess World
#8 - After you go up the ladder to Tapir-san's Place
#10 - When you get a certain number of effects
#11 - When you get the bug effect
#12 - Interact with Box-chan at the Eyeball Shop
#14 - Interact with the blue Beret Twin
#16 - When you use the fairy effect on Tako Otoko
#17 - When you get the spring effect
#18 - Interact with Odorika (you don't need the maiko effect)
#19 - Interact with Gakuran-kun
#20 - When you enter Space from Tapir-san's Place
#21 - When you first go to Theater World
#22 - When you go to the Tribe Settlement
#23 - When you get the Red Riding Hood effect
#24 - Cutting the rope at the well in Fairy Tale Woods and getting captured
#25 - When you encounter Koshukei-tan (Hanged Girl)
#26 - Interact with Megusuri Uri and Box-chan in Visine World
#27 - Interact with the clown that plays the accordian at the Underwater Amusement Park
#28 - When you chainsaw Painter-kun
#29 - When you get the penguin effect
#30 - After chainsawing the sane Shadow Woman in Forest World, going underground and then emerging
#32 - When you get the chainsaw effect
#33 - When you first go to Guts World
#34 - When you pass the meteor along to the Highway (near the tunnel)
#35 - After you go to sleep 30 times
#36 - When you get the fairy effect
#37 - When you use the cake effect and interact with Sweets Musume
#38 - When you chainsaw a stick person in the Penguin Game
#39 - When you use the cake effect and interact with Sweets Musume
#40 - When you spell "Urotsuki" in the Spelling Room
#41 - When entering Seishonen's room and it is glitched
#42 - Get the chasers to surround you in the Dream Apartments, then leave the room and get attacked
#43 - Interact with the blue NPC at the end of the Arrow Maze
#54 - After you wake up when you have some money