N is a 9digit 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 9digit 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 20100704 13:02:53 