7. 6666. Eliot could only figure out the code from the information
that Drew says is sufficient if all the digits in that number are
the same; otherwise, he could not know in what order the digits occur.
Furthermore, since he cannot determine the number from her first
statement about it, the product cannot be unique and so the number
cannot be 1111, 5555, 7777, 8888, or 9999. Of the remaining
possibilities only 6666 meets the requirement that the first digit
of the square of the product of its digits should be odd:
6x6x6x6 = 1296, squared = 1679616.