Devise an efficient method of finding consecutive squares which have the same digits.

Find such pairs up to 1000000.

Feel free to go further.

Well, if n^2 and (n+1)^2 have the same digits, then they have they same value mod 9, and their difference = 0 mod 9.

Their difference = 2n+1, so n = 4 mod 9.

So we only need to investigate every 9th n between 1 and 1000, starting with 4, for a possible 111 values. Not very efficient, so I want to see what somebody else comes up with.

Using Excel on these 111 values, the only ones I notice are:

n n^2 (n+1)^2

-- ------ ---------

13 169 196

157 24649 24964

913 833569 835396

This was based on manual inspection, so I might have missed some.