I found this solution on the Gamespot forums from a user named KIEFER2008. I do NOT take credit for this solution, but it helped me get the achievement in less than twenty minutes
You can access this puzzle game AFTER you complete the main game. Go back to Banjo's house in the starting area (Spiral Mountain), and standing at the edge of the carpet, look up at the picture of Bottles by pressing 'Y'. The puzzle game should start at that point. Now, on to the main solution:
(Original Gamespot link:
http://www.gamespot.com/xbox360/action/banjokazooiela/show_m...)
Now onto the solution! I have created 5 grid images (i didn't do one for the 1st and 2nd puzzle, they're easy) that show you where each jigsaw piece belongs on the board. Ill give you an example here.
NOTE: you will need to rotate the pieces into position yourself with the right analog stick but it really isn't much trouble. The bits sticking out on some pieces should always be facing the right and the top if there are 2 sticking out.
The board and jigsaw pieces like this at the start of each game.
.......1...2...3...4...5...6
20..X...X...X...X...X...7
19..X...X...X...X...X...8
18..X...X...X...X...X...9
17..X...X...X...X...X..10
16.15.14.13.12.11
The numbers 1-20 represent a unused jigsaw piece and the X's represent an empty board slot.
Fill each slot using the Jigsaw pieces in numeric order (clockwise) so 1 then 2 then 3 then 4 etc.
The 3rd puzzle should start like this.
............2...3...4...5...6
20..X...X...X...X...X...7
19..1...X...X...X...X...8
18..X...X...X...X...X...9
17..X...X...X...X...X..10
16.15.14.13.12.11
See that the first jigsaw piece slots into that empty slot.
To finish the 3rd puzzle the jigsaw piece's need to go in this order.
....6....16.....4...18....3
....1.... .5.....2....8....14
...19...20....9....7....11
...12...17...15..10..13
Here are the links to the grid images and i hope this has helped some of you. Sorry im not that great at explaining but you should all hopefully understand.
Puzzle 3:
Puzzle 4:
Puzzle 5:
Puzzle 6:
Puzzle 7: