N is a 9-digit perfect square, with no leading zero, which is constituted by each of the nonzero base ten digits from 1 to 9 occurring exactly once.

Determine the probability that the positive square root of N is a positive integer having the form AB0CD, where each of A, B, C and D represents a different base ten digit from 1 to 9.

As a bonus, what is the probability, if the base ten digits represented by A, B, C, D are not necessarily different?

Note: The "0" inclusive of the string "AB0CD" is the digit zero, and not the letter O.

A simple solution here is to find the 9-digit perfect squares and count them, find their square roots and count the number having the form AB0CD where A, B, C and D represent distinct digits.

The list of the 30 pandigital perfect squares and their square roots is as follows:

PANDIGITAL ROOT
139854276  11826
152843769  12363
157326849  12543
215384976  14676
245893761  15681
254817369  15963
326597184  18072
361874529  19023
375468129  19377
382945761  19569
385297641  19629
412739856  20316
523814769  22887
529874361  23019
537219684  23178
549386721  23439
587432169  24237
589324176  24276
597362481  24441
615387249  24807
627953481  25059
653927184  25572
672935481  25941
697435281  26409
714653289  26733
735982641  27129
743816529  27273
842973156  29034
847159236  29106
923187456  30384

Of the above list, their are five square roots in the form AB0CD, four of which where each of A, B, C and D represents a different base ten digit. These are as follows...:

326597184  18072
361874529  19023
529874361  23019
627953481  25059 
842973156  29034

For square root, 25059, B = C, therefore we exclude counting it in the determination of the probability where each of A, B, C and D represents a different base ten digit from 1 to 9.
The probability is 4/30 = 2/15 = .1333....

For where the base ten digits represented by A, B, C, D are not necessarily different, the probability is 5/30 = 1/6 = .1666....

Posted by Dej Mar on 2010-07-04 13:02:53