CathyW, the logic I used takes advantage of the shape of the pieces to force placements in particular columns. When I look at the puzzle, I see 9 pieces that touch each other along an upper-left to lower-right diagonal. Each of these 9 pieces has a "left" column, a "middle" column, and a "right" column. R5-7 C4 for example is the "left" part of the cruical piece for Hint 1.

In any two pieces that are touching each other (and by touching, I mean sharing 8 cell edges, not just barely touching at the top and bottom of the piece), if the same number occurs in the "left" part of those pieces, then it must occur in the "left" part of all 9 pieces. Similarly, if the same number occurs in the "right" part of those two touching pieces, then it must be in the "right" part of all 9 pieces (the rule does not apply to the "middles"). The reasoning follows:

Consider two touching pieces that both have the number X in the "left" column of those pieces. Now, there is a piece that connects to the left of those two pieces. The X in the two existing pieces blocks the column for the "middle" and "right" part of that piece, so the X in the third piece must be in the "left" as well. As this is on a torus, the logic extends to all 9 pieces.

Let me explain then the placement of the 4 in R2C6. The only other placements of the 4 in that piece are in C4 on the "left" of that piece. This would then force all 4's to be in the "left" because of the 4 at R8C5 in the "left" of the connecting piece. But, looking at R1C1, you see this is a problem.

Here's a detailed reasoning. If you place a 4 in C4 (R5-R7), then in the next piece to the left, you'll see that the only places remaining for a 4 are in C3 (R4 or R5) as both C4 and C5 already have 4's. In the next piece, the 4 must now be in C2 (R3-R5) as both C3 and C4 already have 4's. But now there is no place to put the 4 in the next connecting piece. Because of the 4 in R1C1, in the "middle" of that piece, you cannot put a 4 in C4 at R5-R7. So R2C6 is 4.

The 4 is rather hard to see, I admit. The number I saw very easily is that a 9 cannot be at R4C7. Put the 9 in at R4C7 and then start placing 9's in connecting pieces. You'll hopefully see how the 9's begin to propagate in the "right" of the pieces to form a diagonal, which then causes trouble 5 pieces away. Once you can exclude 9 from R4C7, you can place 9 at R7C5.

Once you have that 9 placed (and eventually the 4 from hint 1), you can really start to solve the puzzle. The early placements I got are almost exactly the same as the ones r.e.s posted so just try to get those logically if you are stuck at where to go.