The first part of the setting of "The Problem of Cell 13" of course is Van Dusens's home. One may imagine his laboratory where he proves "that two and two always equal four, except in unusual cases, where they equal three or five, as the case may be" and that "all things that start must go somewhere." However there is nothing in the text to support the image of the first part of the story being set in his laboratory. Part of the setting includes the era of the story; Jacques Futrelle wrote "The Problem" is 1905 and it is set in 1905.
Once the story shifts to the prison, we know that the setting is Chisholm Prison. Chisholm is in Beaufort County in South Carolina in the U.S. The setting at Chisholm begins where the three men--Ransome, Fielding, and Van Dusen--are greeted somewhere in the prison by the warden. The setting then accommodates a place to search Van Dusen. One may imagine the warden's office for these steps but, again, there is nothing in the text to identify the specific location of the greeting or the search. We do know that the next shift in setting includes jailers--who are not part of the setting--who will allow no communication from Cell13 and will report anything Van Dusen says or hands over to them:
Absolutely impossible [to communicate]. ... Not one word, directly or indirectly. ... They will report anything he might say or turn over to me anything he might give them.
The first actual description of setting comes when the warden escorts them to Cell 13. We are told the warden "led [them] back into the prison ... stopping three doors down the steel corridor," and that the warden then said, "It's only three doors back of my office." And we know there is a "heavy steel door" with double locks. We also know the setting is bestrewn with scurrying rats: "dozens of them."
Finally, once Van Dusen is tucked away and quietly thinking, we are given some broader understanding of the setting. Chisholm Prison--large, four stories, and made of granite--is surrounded on all sides by open space and bounded by a "wall of solid masonry eighteen feet high." The prison is completely isolated--there are no signs of civilization near it at all, just "acres of open space." We are also told that a river lay beyond the wall nearest to Cell 13: Van Dusen knew this because of the sound of a motor boat and the presence of a river bird in the air. Further, he discerned that between the river and the wall was a playground where children were playing baseball: "came the shouts of boys at play and the occasional crack of a batted ball." These details ofsetting and a few others regarding paint and an outmoded pipe system were later confirmed by one of the jailers.
Other essential aspects of the setting are that his window was only three to four feet above the ground and there were seven doors to pass before getting out of the prison: "seven doors to be overcome before one could pass from Cell 13 into the outer world, a free man." In addition, though there was an opening of about two inches below the heavy steel door, the rats, when frightened, got out of the cell some way other than under the door--there was another egress in the cell, no matter how small.