# Happy April 1 Day

Posted by: Gary Ernest Davis on: April 1, 2013

The 4! = 24 distinct fractions whose base 4 representation consists of permutations of the digits (0, look 1,2,3) repeated:

$frac{9}{85} = 0.0123ldots _4$
$frac{2}{17} = 0.0132ldots _4$
$frac{13}{85} = 0.0213ldots _4$
$frac{3}{17} = 0.0231ldots _4$
$frac{18}{85} = 0.0312ldots _4$
$frac{19}{85} = 0.0321ldots _4$
$frac{5}{17} = 0.1023ldots _4$
$frac{26}{85} = 0.1032ldots _4$
$frac{33}{85} =0.1203ldots _4$
$frac{36}{85} = 0.1230ldots _4$
$frac{38}{85} = 0.1302ldots _4$
$frac{8}{17} = 0.1320ldots _4$
$frac{9}{17} = 0.2013ldots _4$
$frac{47}{85} = 0.2031ldots _4$
$frac{49}{85} =[0.211ldots _4$
$frac{52}{85} =0.2130ldots _4$
$frac{59}{85} = 0.2301ldots _4$
$frac{12}{17} = 0.2310ldots _4$
$frac{66}{85} = 0.3012ldots _4$
$frac{67}{85} = 0.3021ldots _4$
$frac{14}{17} = 0.3102ldots _4$
$frac{72}{85} = 0.3120ldots _4$
$frac{15}{17} = 0.3201ldots _4$
$frac{76}{85} = 0.3210ldots _4$

These are all fractions of the form $frac{3a}{4^4-1}=frac{3a}{255}=frac{a}{85}$, namely those fractions $frac{a}{85}$ with numerators 9, 10, 13, 15, 18, 19, 25, 26, 33, 36 ,38, 40, 45, 47 ,49 ,52, 59 ,60 66, 67, 70, 72, 75 and 76.