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The poem is the whole puzzle here. It's a riddle to solve.The order of the poem verses is given by the number of ivy leaves on the cards.The poem's narrator is one number who is talking about another number.
The two numbers in the poem are integers ('whole integral mates are we').'None can come between' means the numbers are consecutive integers.'She sits upon my shoulder' is a reference to a particular kind of mathematical notation.
The puzzle's picture is Ivy, which could be read as a Roman numeral.Interestingly, there is only one three-digit number you can get by raising one integer to the power of another consecutive (one step higher or lower) integer.The numbers are 5 and 4. 5 to the 4th power is 625, which is the number you're seeking.