I'll give it a try and let you do the explanation, ok?!

05 46 15 56 25 66 35 76 45
54 14 55 24 65 34 75 44 04
13 63 23 64 33 74 43 03 53
62 22 72 32 73 42 02 52 12
21 71 31 81 41 01 51 11 61
70 30 80 40 09 50 10 60 20
29 79 39 08 49 18 59 19 69
78 38 07 48 17 58 27 68 28
37 06 47 16 57 26 67 36 77

TO : All Those Who Are Interested

There are 4 methods of positioning the numbers 1~81 to land them such that 369 is the all-round total. Again middle number 41 multiply by 9.

I think Pretty Futari employs the upward-diagonal sequence beginning with 1 and upwards until 9; and continues in that format until number 81 is reached. See illustration (may be distorted) below.

Next, highlight the 9 by 9 squares with the centre number 41 in the middle. Then, move the numbers outside the 9 x 9 box towards the opposite direction into the box, each time counting 9 squares. Do this both vertically and horizontally.

09
08 18
07 17 27
06 16 26 36
05 46 15 56 25 66 35 76 45
04 54 14 55 24 65 34 75 44 4 54
03 13 63 23 64 33 74 43 03 53 63
02 12 62 22 72 32 73 42 02 52 12 62 72
01 11 21 71 31 81 41 01 51 11 61 71 81
10 20 70 30 80 40 9 50 10 60 20 70 80
19 29 79 39 8 49 18 59 19 69 79
28 78 38 7 48 17 58 27 68 28 78
37 6 47 16 57 26 67 36 77
46 56 66 76
55 65 75
64 74
73

ps: As expected the figure is distorted. It should look like the diagram of a rhombus instead of like a loud speaker.

My method is quite similar. I use the downward sequence. See illustration below.



37 30 23 16 09 74 67 60 53
54 38 31 24 17 01 75 68 61
62 46 39 32 25 18 02 76 69
70 63 47 40 33 26 10 03 77
78 71 55 48 41 34 27 11 04
05 79 72 56 49 42 35 19 12
13 06 80 64 57 50 43 36 20
21 14 07 81 65 58 51 44 28
29 22 15 08 73 66 59 52 45

To Geoette

Thanks for revealing a second method of solving it.
Mine is much easier but hard to explain.
Mind showing us the rest of the four you mentioned?

Sorry; should read "mine is easier but hard to explain".