TITLE: Smallest pandigital number perfect square
QUESTION [5 upvotes]: What is the smallest $9$-digit number that has all the digits from $1$ to $9$ exactly once and is also a perfect square? 

Please give me a method that doesn't involve programming.

REPLY [2 votes]: Just look up A036744, the penholodigital squares.
The possible solutions are
139854276, 152843769, 157326849, 215384976, 245893761, 254817369,
326597184, 361874529, 375468129, 382945761, 385297641, 412739856,
523814769, 529874361, 537219684, 549386721, 587432169, 589324176,
597362481, 615387249, 627953481, 653927184, 672935481, 697435281,
714653289, 735982641, 743816529, 842973156, 847159236, 923187456  
With square roots
11826, 12363, 12543, 14676, 15681, 15963,
18072, 19023, 19377, 19569, 19629, 20316,
22887, 23019, 23178, 23439, 24237, 24276,
24441, 24807, 25059, 25572, 25941, 26409,
26733, 27129, 27273, 29034, 29106, 30384