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Using the numbers 1,2,3,4,5,6,7,8,9 only once in the complete set, Give me a set of perfect squares.

Once you have found the set, you have the liberty to use all these numbers without a comma and give me a nine digit perfect square featuring all of the nine digits...JUST ONCE!

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We have, for instance, the following set of squares:

\begin{array}{c}25 = 5^2\\784 = 28^2\\1369 = 37^2\end{array}

As for the nine digit perfect square, there are several. Below, they are listed in 'reverse' increasing order.

\begin{array}{l}549386721 = 23439^2\\735982641 = 27129^2\\385297641 = 19629^2\\529874361 = 23019^2\\245893761 = 15681^2\\382945761 = 19569^2\\697435281 = 26409^2\\597362481 = 24441^2\\627953481 = 25059^2\\672935481 = 25941^2\\653927184 = 25572^2\\326597184 = 18072^2\\537219684 = 23178^2\\847159236 = 29106^2\\842973156 = 29034^2\\923187456 = 30384^2\\412739856 = 20316^2\\589324176 = 24276^2\\139854276 = 11826^2\\215384976 = 14676^2\\375468129 = 19377^2\\361874529 = 19023^2\\743816529 = 27273^2\\615387249 = 24807^2\\157326849 = 12543^2\\587432169 = 24237^2\\254817369 = 15963^2\\152843769 = 12363^2\\523814769 = 22887^2\\714653289 = 26733^2\end{array}

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  • $\begingroup$ ... and how is the first spoiler related to the second? $\endgroup$ – Will Mar 8 '16 at 9:03
  • $\begingroup$ @Fimpellizieri That is right. Another instance for the first part may be 9,81, 324 , 576. $\endgroup$ – Prashant Mar 8 '16 at 9:03
  • $\begingroup$ @Will The question has two parts. The first part (answered by the first spoiler) asks for a set of positive integer squares in which each of the digits $1$ through $9$ appears exactly once. The second part (answered by the second spoiler) asks for a nine-digit square number in which each of the digits also appears exactly once. $\endgroup$ – Fimpellizieri Mar 8 '16 at 16:46

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