(I) Determine all possible base ten 8-digit perfect square(s) N, with no leading zeroes, such that the number obtained by increasing the second digit (reading left to right) of N by 1 is also a perfect square.

(II)Determine all possible base ten 4-digit perfect square(s) N, with no leading zeroes, such that the number obtained by inserting two zeroes between the thousands digit and the hundreds digit of N is also a perfect square.

**** Calculators are allowed, but no computer programs.

(In reply to answers by Dej Mar)

Must be just a typo, but 7569 = 87^2, not 86^2.

Some variations, found via forbidden tree of knowledge:

If wrap-around of digits is allowed so that 9 is incremented to 0, then

29702500 =  5450^2 and 20702500  = 4550^2

If, however, when you do this you also carry the 1 from what's actually a 10, you get:

99500625 = 9975^2 and   100500625 = 10025^2

 

Posted by Charlie on 2011-10-27 18:31:11