H μαγεία των Αριθμών

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


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



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



Brilliant, isn't it?
And finally, take a look at this symmetry:


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



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
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8^2



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)



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


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
14 + 28 + 57 = 99
28 + 57 + 14 = 99
42 + 85 + 71 = 99 + 99
57 + 14 + 28 = 99
71 + 42 + 85 = 99 + 99
85 + 71 + 42 = 99 + 99



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



3 2 6 + 4 5 1 = 7 7 7
2 6 4 + 5 1 3 = 7 7 7
6 4 5 + 1 3 2 = 7 7 7
+ + + + + +
4 5 1 + 3 2 6 = 7 7 7
5 1 3 + 2 6 4 = 7 7 7
1 3 2 + 6 4 5 = 7 7 7
3 2 6 + 4 5 1 = 7 7 7
2 6 4 + 5 1 3 = 7 7 7
6 4 5 + 1 3 2 = 7 7 7
+ + + + + +
4 5 1 + 3 2 6 = 7 7 7
5 1 3 + 2 6 4 = 7 7 7
1 3 2 + 6 4 5 = 7 7 7
= = = =
7 7 7 7 7 7
7 7 7 7 7 7
7 7 7 7 7 7





3 x 37 = 111 and 1 + 1 + 1 = 3
6 x 37 = 222 and 2 + 2 + 2 = 6
9 x 37 = 333 and 3 + 3 + 3 = 9
12 x 37 = 444 and 4 + 4 + 4 = 12
15 x 37 = 555 and 5 + 5 + 5 = 15
18 x 37 = 666 and 6 + 6 + 6 = 18
21 x 37 = 777 and 7 + 7 + 7 = 21
24 x 37 = 888 and 8 + 8 + 8 = 24
27 x 37 = 999 and 9 + 9 + 9 = 27




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

7 x 7 = 49
67 x 67 = 4489
667 x 667 = 444889
6667 x 6667 = 44448889
66667 x 66667 = 4444488889
666667 x 666667 = 444444888889
6666667 x 6666667 = 44444448888889
etc.

4 x 4 = 16
34 x 34 = 1156
334 x 334 = 111556
3334 x 3334 = 11115556
33334 x 33334 = 1111155556
etc.