Six must be in the bottom row because the only possible differences are 1, 2, 3, 4 ,and 5, so 6 cannot be in the first or second row.
Five cannot be at the top because if 5 was at the top then the second row would be 6-1, but 6 must be on the third row.
Four cannot be at the top because if it were then the second row would have to be 5-1, but if that were so then 6-1 would have to be in the third row, which uses 1 twice.
We determined that 4, 5 and 6 cannot all be in the third row because that would mean 2 and 1 would be in the second row which doesn't give you a difference of 3 for three to be in the top row.
We say there are only four solutions.
1
3-4
5-2-6
This is the only solution with 1 in the first row. We know this because we tested all the possibilities for the second row.
2-1 cannot be because we already used 1.
3-2 doesn't work because 4, 5, and 6 cannot be in the third row.
5-4 doesn't work because you would need 6-1 in the third row but we have already used 1.
6-5 doesn't work because 6 must be in the last row.
2
5-3
6-1-4
This is the only solution with 2 in the top row.
3-1 cannot be in the second row because that would leave 4, 5, and 6 in the third row.
4-2 doesn't work because we have already used 2.
6-4 doesn't work because 6 must be in the third row.
3
4-1
2-6-5
3
5-2
1-6-4
These are the only two solutions with 3 in the top.
The only other possible pair to give a difference of 3 is 6-3 which doesn't work in the second because we have already used 3 and 6 must be in the third row.
We found four solutions using 1 to 10.
3
5-2
4-9-7
6-10-1-8
3
7-4
2-9-5
8-10-1-6
4
5-1
2-7-6
8-10-3-9
4
2-6
5-7-1
8-3-10-9
We believe that these are all the solutions for 1-10.
Some rules we determined were:
10 and 8 must both be in the 4th row.
9 and 1 cannot be side by side in the third row.
1, 2, 5, 6, 7, 8, 9, and 10 cannot be in the top row.