A lot of people don't understand the appeal of the mathematics to others. Paul Lockhart gave many possible reasons for this in his book "A Mathematician's Lament". His main point was that we don't teach kids how to do math recreationally. For the sake of exploration and experimentation. Mathematicians love to ask questions and see where logic takes them.
There are many reasons to love math and not every math enthusiast has the same reasons. Some people may like the power math offers, the manipulation of symbols, the different perspectives of a single idea, the breadth of concepts math can express, the shortcuts that can be achieved, or seeing how ideas are related to others.
I think anyone can enjoy trying creative ideas and the feeling of being enlightened or inspired when finding a solution. Almost like a work of art, along the way you will find that certain patterns reveal themselves and pop up here and there in different situations. It may not be clear why these patterns exist but they seem meaningful and somehow fundamental.
The rest of the article will provide some games to start with and some guidelines on asking questions for yourself.
Games That You Can Do In Your Head
1. Pick a random number (or use the current time) and find it's prime factors
2. Pick a random number and find the closest prime to it
3. Drill math operations between two random numbers. (addition, subtraction, division, and multiplication)
4. Convert a random number to different bases
Other Games
1. Counting amount of all possible intersections from n lines in an m-dimensional plane. (From four lines is 1,2,3,4,5 and 6 intersections possible?)
4. Convert a number to another base
5. If we want to create a right triangle of int length n what could the length of the sides be if they also had to be natural numbers. (Sum of the squares of 2 nats is equal in n. Does a solution exist?)
6. Same as 5. but with selected amount of summands and a selected power
7. Count the number of possible equations or strings that can be made that follow a selected set of constraints
8. Partition Some set of mathematical objects
10. Define a language (or a set of possible strings) where each string corresponds uniquely to a single case of a set of real world things/situations/orientations etc. (A language to describe the different orientations 2 cups can be in on a desk). Maybe even define operations on these strings.
11. Use graphical programming (or imagine) to create shapes with interesting patterns, such as a snowflake, or flower petals.
12. Try to ask as many questions as you can about a math or logical situation, try to identify special cases and interesting situations.
13. Create A stylized secret encoding, with cool shapes or patterns or whatever. Maybe make it not look like an encoding.
14. Program the game of life with your own added rules
15. Try to create a fake proof where people have to spot the mistake
17. Given a recursive square after 4 interations, (each iteration draws a vertical and horizontal line through the middle of each square) count the number of rectangles in the image, (not the amount of squares.)
18. Create a new compex image and count the number of subshapes of a certain type with your image
19. Try to make sense of a real number interpretation of something that relies on counting numbers. (ex. 2.5 derivative, Gamma function)
20. Find Integer solutions of an equation