Determine the respective values of SQUARE and CUBE, given that SQUARE is a perfect cube and CUBE is a perfect square. Each of the capital letters in bold denotes a different decimal digit from 0 to 9 and neither C nor S can be zero.

Below is a table of all available 6 digit squares which do not contain a repeated digit as well all 4 digit cubes which also contain unique digits(values were created within a spreadsheet and then edited to leave just these).

S  Square   C  Cube  C  Cube  C    Cube  C  Cube  
59 205379  32 1024  51 2601  66    4356  84 7056
66 287496  33 1089  52 2704  69    4761  86 7396
69 328509  36 1296  53 2809  71    5041  87 7569
73 389017  37 1369  54 2916  72    5184  89 7921
75 421875  42 1764  55 3025  73    5329  93 8649
76 438976  43 1849  57 3249  74    5476  95 9025
84 592704  44 1936  59 3481  78    6084  96 9216
88 681472  48 2304  61 3721  79    6241  98 9604
93 804357  49 2401  64 4096  82    6724  99 9801

In a suitably constructed spreadsheet, of which this is a "copy":

S Q U A R E B C
3 2 8 5 0 9 4 1   [input line]
                            
                           
S Q U A R E
3 2 8 5 0 9       [Test of Square was here]*
                           
   C U B E       
   1 8 4 9      [Test of Cube was here]*

The respective squares were added one by one into the input line under the letters "SQUARE".
Those values were echoed beneath the line of letters "SQUARE".
The letters "U" and "E" were echoed beneath the letters of the line "CUBE".
It was then just a matter of finding a cube from the above table that corresponded to the U=8 and E=9.

* The Test cells provided verification but were included before the above (and edited) table was constructed.

Posted by brianjn on 2009-08-04 21:54:28