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The base nine numbers $17_9=16$ and $71_9=64$ are both squares, while $14_{12}=16$ and $41_{12}=49$ are both squares.
About a quarter of bases have a pair of two-digit squares that are the reverse of each other, and some bases have several pairs.
How do I calculate the exact proportion of bases that have a pair like that?

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Does the claim that "about a square of bases" apply as a probabilistic limit, or just to small bases that you have considered? –  Calvin Lin Jun 30 '13 at 14:26
    
Also, do you allow the pair $(11_8,11_8)$? $11_8=9$ and both are reverse to each other. –  Tomas Jun 30 '13 at 14:40
    
@CalvinLin, that was up to about 1000. It looks quite steady at about 25 percent. –  Michael Jul 1 '13 at 1:13
    
@Tomas, I excluded your example. –  Michael Jul 1 '13 at 1:15

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