One of my favorite subjects!
First, I think the ratio of cards 1-24 as compared to cards 25-36 is estimated at around 3 to 1. The excellent research Doug Spanning, Brent Bucher, John Spano and others have done suggests there was a first printing of cards 1-24 and then a second printing of cards 1-36.
It is likely the Chicle set was printed in 12 card sheets. There are uncut sheets out there for the 1934 Sky Birds and 1933 Daredevils sets also produced by National Chicle, and both uncut sheets are 12 cards in size.
I can understand why you would think that the number on the back is an error, John. It's soooo far away from what they actually produced that it seems preposterous. I personally don't think it was an error at all. I think they were just overly ambitious and perhaps had already printed the first 24 cards that said they were producing 240 cards before realizing how difficult it was going to be to make this goal.
Note that this isn't the first time National Chicle under or over estimated the cards they would produce. I believe some of the backs of the Sky Birds cards say the set includes 48 cards. At some point they realized they were popular enough to print more so they started printing additional cards that said 144 cards were in the set and produced new cards. In the end they only printed 108 cards so didn't make this goal either! All of this, to me, is evidence that National Chicle wasn't very good at predicting their set sizes and probably screwed it up for their football set too. I can see them deciding to just go with the goal number and selling toward that, filling out the set as demand grew.
Some will argue that there weren't enough NFL players for National Chicle to produce at set of 240 cards. That's not exactly true. There were 9 NFL teams with a maximum roster size of 25. However, it was common for more than 24 players to play for a team during the year. Knowing this, perhaps National Chicle figured they would add new cards for replacement players added later in the year as they had a history of printing cards on demand. Also, the letter from National Chicle to the Eagles that was up for sale recently states that they planned to create colorized cards of each team card too. Also, with the inclusion of Knute Rockne, they were probably planning to include some other tribute cards and perhaps even some college stars if they needed to round out the set. The fact that they produced a card and a premium of Stan Kostka makes me believe they were planning to create cards for all of the players in the premium set as well as the coaches. They just ran out of money to do so. All of this makes getting to 240 cards possible, just not very realistic.
I was fortunate to pick up the National Chicle letter. The tone of this letter doesn't sound like a company looking to produce 36 cards. They were eagerly awaiting the 11 cards the Eagles had promised and reminded them that they wanted team and action images too.
jeff