1 * 8 + 1 = 9
12 * 8 + 2 = 98
123 * 8 + 3 = 987
1234 * 8 + 4 = 9876
12345 * 8 + 5 = 98765
123456 * 8 + 6 = 987654
1234567 * 8 + 7 = 9876543
12345678 * 8 + 8 = 98765432
123456789 * 8 + 9 = 987654321

1 * 9 + 2 = 11
12 * 9 + 3 = 111
123 * 9 + 4 = 1111
1234 * 9 + 5 = 11111
12345 * 9 + 6 = 111111
123456 * 9 + 7 = 1111111
1234567 * 9 + 8 = 11111111
12345678 * 9 + 9 = 111111111
123456789 * 9 +10= 1111111111

9 * 9 + 7 = 88
98 * 9 + 6 = 888
987 * 9 + 5 = 8888
9876 * 9 + 4 = 88888
98765 * 9 + 3 = 888888
987654 * 9 + 2 = 8888888
9876543 * 9 + 1 = 88888888
98765432 * 9 + 0 = 888888888

1 * 1 = 1
11 * 11 = 121
111 * 111 = 12321
1111 * 1111 = 1234321
11111 * 11111 = 123454321
111111 * 111111 = 12345654321
1111111 * 1111111 = 1234567654321
11111111 * 11111111 = 123456787654321
111111111 * 111111111=123456789 87654321

9 * 1 = 9 | 9 + 0 = 0
9 * 2 = 18 | 1 + 8 = 9
9 * 3 = 27 | 2 + 7 = 9
9 * 4 = 36 | 3 + 6 = 9
9 * 5 = 45 | 4 + 5 = 9
9 * 6 = 54 | 5 + 4 = 9
9 * 7 = 63 | 6 + 3 = 9
9 * 8 = 72 | 7 + 2 = 9
9 * 9 = 81 | 8 + 1 = 9
9 * 10 = 90 | 9 + 0 = 9

12345679 * 9 = 111111111 (9*1)
12345679 * 18 = 222222222 (9*2)
12345679 * 27 = 333333333 (9*3)
12345679 * 36 = 444444444 (9*4)
12345679 * 45 = 555555555 (9*5)
12345679 * 54 = 666666666 (9*6)
12345679 * 63 = 777777777 (9*7)
12345679 * 72 = 888888888 (9*8)
12345679 * 81 = 999999999 (9*9)

11^0 = 1
11^1 = 1 1
11^2 = 1 2 1
11^3 = 1 3 3 1
11^4 = 1 4 6 4 1

1 = 1^2
1 + 3 = 4 = 2^2
1 + 3 + 5 = 9 = 3^2
1 + 3 + 5 + 7 = 16 = 4^2
1 + 3 + 5 + 7 + 9 = 25 = 5^2
1 + 3 + 5 + 7 + 9 + 11 = 36 = 6^2
1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 = 7^2

142857 *1 = 142857
142857 * 2 = 285714
142857 * 3 = 428571
142857 * 4 = 571428
142857 * 5 = 714285
142857 * 6 = 857142

142 + 857 = 999
285 + 714 = 999
428 + 571 = 999
571 + 428 = 999
714 + 285 = 999
857 + 142 = 999

1 * 7 + 3 = 10
14 * 7 + 2 = 100
142 * 7 + 6 = 1000
1428 * 7 + 4 = 10000
14285 * 7 + 5 = 100000
142857 * 7 + 1 = 1000000
1428571 * 7 + 3 = 10000000
14285714 * 7 + 2 = 100000000
142857142 * 7 + 6 = 1000000000
1428571428 * 7 + 4 = 10000000000
14285714285 * 7 + 5 = 100000000000
142857142857 * 7 + 1 = 1000000000000
1428571428571 * 7 + 3 = 10000000000000
14285714285714 * 7 + 2 = 100000000000000
142857142857142 * 7 + 6 = 1000000000000000
1428571428571428 * 7 + 4 = 10000000000000000
14285714285714285 * 7 + 5 = 100000000000000000
142857142857142857 * 7 + 1 = 1000000000000000000