A problem is a situation that is hard to deal with. The word comes from a Greek word meaning an "obstacle" (something that is in your way). If someone has a problem, they have to find a way of solving the problem. The way to solve it is called a solution.
"John has locked his car keys inside his car so that he cannot get at them. John has a problem".
We can also talk about a child with "behaviour problems". That would mean they act in ways that hurt and upset other people or themselves or cause too much disruption.
Sometimes people describe the ways in which people with learning difficulties and autism interact with their own bodies and the environment as behaviour problems because they find it disturbing, but this is controversial. These do not cause hurt and are not meant to cause upset but are seen as self-expression or self-soothing by the adult or child acting in these ways- such as rocking themselves or repeated actions like moving doors and drawers or echoing the movement in the environment, such as the movement of a flag in the wind. The argument is that it is other people's problem if self-expression or self soothing that hurts no-one is something which other people find disturbing.
"Communication problems" are when people struggle to talk in a way that helps and feels good. Communication problems come from many kinds of problems, body problems such as with the ears, learning and mental problems, being in a new country or an ethnic minority or using special science words. For example a person who is deaf has communication problems because he cannot hear the spoken words most people use, but this is different when he is in an environment where everyone can use sign-language well, or if he can speak and others allow him to lip read. Those with developmental delay or learning disability or learning differences or speaking in a second language may be said to have "communication problems". Problems in relationships in families and between friends are often called "communication problems". People who must work together but use different words, for example science words and simple words, to talk about the same things, they are also said to have communication problems.
Here is an example of a mathematical problem: "John is three times as old as Mary. In three years time he will be twice as old as Mary. How old are John and Mary?" (9 and 3)
Children often like to give one another problems that can be solved by "lateral thinking". This means using the imagination rather than strict logic. Here is an example:
"Peter, Ruth, Samuel and Jessica live in the same house. Peter and Ruth went out. When they returned they found Jessica lying dead, surrounded by glass. They were sure Samuel had done it. Why did they not call the police?" The answer is: Samuel was the cat and Jessica was a goldfish.