Four 4-digit perfect squares are arranged one above the other, in ascending order. No two digits within any one of these squares is the same. If you sum the digits of each of these four squares, each result is the same sum. Also, the sum of the digits of each column formed by placing these digits above one another is the same. There may or may not be repeated digits within a column.
What are the 4-digit squares?
I also wondered at first why Charlie specified "in ascending order" since the same set of four squares would work in any permutation. I assume it was so that there would be a unique answer. In any case, I believe brianjn gives that answer upside down, since a square "above" is to be "in ascending order" -- otherwise he has the same set I proposed. My set does make zero the top digit in column three. While similar puzzles and many alphametics may specify "no leading zero" for a number, Charlie is clear that the columns are to be regarded as sets of four digits, NOT as single numbers (square or otherwise -- only criterion being that each sums to the same value as the other rows and columns).
(Regarding the posting in another current puzzle that 0!=1, I found that Wikis agree with Charlie's explanation, though I think "undefined" would be better -- as in division by zero -- though I see the MS Calculator gives 1 as the value, and also shows factorials for fractions which converge to 1 as the fraction approaches zero. Not sure what factorials of negatives would be.
Interpretation
