Digits reversed | a^2 + b^2 + c^2 = x^2 + y^2 + z^2

 
Take, for example, the numbers   77   and   89.
 

77 \; = \; 4^2 \; + \; 5^2 \; + \; 6^2 \; = \; 8^2 \; + \; 3^2 \; + \; 2^2

4 \; + \; 5 \; + \; 6 \; = \; 15   ……….   8 \; + \; 3 \; + \; 2 \; = \; 13

 

combining digits on the right and left-hand side of the equal sign and obtain

                                    digits reversed
48, 53, 62   ——>   84, 35, 26
48, 52, 63   ——>   84, 25, 36
43, 58, 62   ——>   34, 85, 26
43, 52, 68   ——>   34, 25, 86
42, 58, 63   ——>   24, 85, 36
42, 53, 68   ——>   24, 35, 86

then

48^2 \; + \; 53^2 \; + \; 62^2 \; = \; 84^2 \; + \; 35^2 \; + \; 26^2 \; = \; 8957
48^2 \; + \; 52^2 \; + \; 63^2 \; = \; 84^2 \; + \; 25^2 \; + \; 36^2 \; = \; 8977
43^2 \; + \; 58^2 \; + \; 62^2 \; = \; 34^2 \; + \; 85^2 \; + \; 26^2 \; = \; 9057
43^2 \; + \; 52^2 \; + \; 68^2 \; = \; 34^2 \; + \; 25^2 \; + \; 86^2 \; = \; 9177
42^2 \; + \; 58^2 \; + \; 63^2 \; = \; 24^2 \; + \; 85^2 \; + \; 36^2 \; = \; 9097
42^2 \; + \; 53^2 \; + \; 68^2 \; = \; 24^2 \; + \; 35^2 \; + \; 86^2 \; = \; 9197

 
Similarly,

89 \; = \; 3^2 \; + \; 4^2 \; + \; 8^2 \; = \; 2^2 \; + \; 6^2 \; + \; 7^2

combining digits on the right (2, 6, 7) and left-hand side (3, 4, 8)

3 \; + \; 4 \; + \; 8 \; = \; 2 \; + \; 6 \; + \; 7 \; = \; 15

32^2 \; + \; 46^2 \; + \; 87^2 \; = \; 23^2 \; + \; 64^2 \; + \; 78^2 \; = \; 10709
32^2 \; + \; 47^2 \; + \; 86^2 \; = \; 23^2 \; + \; 74^2 \; + \; 68^2 \; = \; 10629
36^2 \; + \; 42^2 \; + \; 87^2 \; = \; 63^2 \; + \; 24^2 \; + \; 78^2 \; = \; 10629
36^2 \; + \; 47^2 \; + \; 82^2 \; = \; 63^2 \; + \; 74^2 \; + \; 28^2 \; = \; 10229
37^2 \; + \; 42^2 \; + \; 86^2 \; = \; 73^2 \; + \; 24^2 \; + \; 68^2 \; = \; 10529
37^2 \; + \; 46^2 \; + \; 82^2 \; = \; 73^2 \; + \; 64^2 \; + \; 28^2 \; = \; 10209

 

Find more examples.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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