On a flat plane, arrange nine dots in such a way that you can draw ten straight lines. Each line must be unique and must pass through exactly three of the dots. As a starting example, arranging the dots in a 3x3 square would satisfy eight lines (three vertical, three horizontal, two diagonal). There is, to my knowledge, exactly one solution which permits ten lines.  

Construct a cube and label each corner with a whole number such that each corner's number is either a factor or multiple of the numbers on the adjacent corners (diagonals not counting). If the corners are not adjacent, their numbers must not by factors or multiples of each other. Many solutions exist, along with a general solution.

Extension: Following the same rules, label a simple pentagon.

Take the numbers from 1-12. Remove numbers one at a time, crossing out the number as well as any factors of that number in the list. You may only remove a number if it has at least one factor still remaining on the board. What is the highest sum of numbers that you can select?

Extension: Take a larger list. A suggested difficult problem is 1-50.

Break the number 20 into any number of whole-number partitions (2 tens, 20 ones, etc). What is the largest product you can make by multiplying those partitions together? 

Extension: Remove the restriction that partitions must be of whole numbers.