When one is taking the question literaly, it would be eighter one of the two ice. Nowhere in the question there is a requirement for the object to move. It is questioned for the least force. So one could say, when the person slides on the ice, the force on the object should be smaller then the force on the round object on gravel.
Where is the requirement for a person to push until failure?
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Why push intil failure?
The premise of your answer (as I read it) implies that if you push until you slip (failure) you will have applied less pressure on ice than on gravel.
