1) What balloons will be red?
Balloons that are perfect squares will be red. In other words, balloons that have an odd number of factors.
2) What balloons will be blue?
Balloons that have an even number of factors will be blue.
3) How many balloons will be red?
25 ballons will be red
4)How many balloons will be blue?
600 balloons will be blue.
This problem, like the locker problem, requires one to find the pattern of what balloons will be red and what balloons will be blue. Similar to the locker problem, this problem uses a large enough number so that individuals are forced to find the pattern instead of just simply writing out all the balloons and solving through a visual. This problem, similar to the locker problem, requires students to notice a pattern that connects with the concept of factors.
To find this solution, individuals may write out say the first 10-20 balloons and solve through this visual, noticing a pattern. Through the visual of this they may begin to write out the numbers which produce red balloons and the numbers which will produce blue balloons. Once working for a while the goal is for students to come to find that the balloons which will be red are the numbered balloons of perfect squares. Therefore, balloons that have an odd number of factors. They then will also realize that all the remaining balloons are blue and are the balloon numbers that have an even amount of factors.
Given the pattern, individuals then can find out how many balloons will be red by finding the highest squared number that goes into 625. This number happens to perfectly be 25. As a result, it is known that within 625 there are 25 perfect squares and therefore 25 of the balloons will be red and the remaining 600 will be blue.