Three men went to a hotel to rent a room, the cost of the room was $30. Each man paid $10 to the bellboy and proceeded to their room. After a little while the bellboy realized that there was a special on rooms that night and the price for the men's room should have been $25. On his way to the men's room to give them back $5, he was puzzled how he was going to split $5 as he had no change. He decided he would give them each $1 and keep the remaining $2 for himself. So each man originally paid $10, but after the bellboy gave each man $1 back, each man paid $9. 9 x 3 = $27 plus the $2 the bellboy put in his pocket equals $29. The original price for the room was $30. Where did the last dollar go? Which shows that even though something sounds logical, it might not be. There is no "last dollar". There is a fallacy in the thought process and calculations. The $2 should be subtracted from the $27 to show what ended up being paid for the room. Or $30 - $1 - $1 - $1 - $2 = $25 Probably the most difficult recently solved problem was Fermats enigama. Which involved cubes. It has taken many years and the consolidation of the work of dozens of people to resolve.
