Once you’ve beat the game on Normal, “Flashback” to Case 7 – Solomon. This case has a Kaleidoscope mini game right near the beginning. The puzzle is right after moving the boxes with the crane. Don’t forget to select Insane difficulty before attempting the mini game. If you don't beat the best time, just restore the checkpoint and try again.
This is one of the few puzzles that you can walk up to but not select the game immediately. Instead, walk up so you can see it but DO NOT press . Pause the game and take a long look to get a plan together, then unpause and start it up.
Also the final puzzle in Case 5 - Joseph, has 4 kaleidoscope mini games in it where you can try this.
Thanks to Tommycbs for the tip.