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Answer: 83421096        

The winner is Dave Kirkham of Frenchs Forest, New South Wales, Australia.

Worked solution

Only when E = 0, 1, 8, or 9 does a number beginning with E, when squared, also have the possibility of beginning with E.

Clearly, E ≠ 0 here, and if E = 1 then ENID2 will have seven digits, so E = 8 or 9.

If E = 8, then D = 0, which implies I = 0, so E = 9.

Since EWAN2 begins with T and E = 9, then T= 8.

If a number ends with N and when squared also ends with N, then N = 0, 1, 5, or 6.

ENID2 begins at least 9090, so EN ≥ 95, so N = 5 or 6.

If N=5, AN2 ends with 25 whereas EN = 95. Hence, N=6.

EWAN2 ends 96, so AN = 36 or 86. Since T = 8, A = 3.

9600