White mates in 20 moves with the pawn, without capturing any of the black pawns.
On 1858-07-17, two short solutions (both in 10 moves) to the problem are printed. The second of these solutions is by E. B. Cook, and is followed by the comment:
In short, as the author of this solution remarked to the [Clipper] Editor, “the ‘Egg’ was rotten and wouldn’t stand Cooking.”
On its own, this should probably not be taken as more than a double pun, but it may have helped reinforce the idea that the term ‘cook’ was in some way connected with the name of E. B. Cook, even though other solvers had also identified the lack of soundness.
See also Edward Winter’s C. N. 4341 for additional info.