This problem allows for multiple possible solutions, and will require students to think about some assumptions they will have to make to solve this problem. You can easily vary the difficulty of the problem by changing the number of people invited to the party.
Ellen is planning a party for her friends. She has invited 100 of them, but she doesn’t know exactly how many of her friends will attend. She wants to put out tables for her friends, and she wants to put enough chairs at each table so that none of her friends has to sit alone. Assume that her friends will fill up each table as they arrive. How many tables should she put out, with how many chairs at each table?
I’m pretty sure that it is impossible. Here is my proof. Set aside one table that has x seats. So, all the other tables can hold 100-x people all together. Then, if 100-x people arrive and then just 1 more, the 1 more person is doomed to sit in the last table.
Yeah, it is impossible, but children don’t know that before they start working on the problem. I’m absolutely fine with students working on problems which are impossible to find an exact solution for, particularly if they can “prove” in some way why it doesn’t work. Also, it may be that students can find solutions which almost work. Not every problem needs to have a solution to be a source of rich mathematical thinking. Look at the Konisberg bridge problem for example, which led Euler to developing graph theory to prove that it was impossible to solve.