## Broken eggs problem

****

__Broken Eggs Problem__**Problem Statement**

A farmer’s inventory of eggs is destroyed due to her negligence of not securing her cargo. Luckily, the insurance company doesn't know this and all they need to know is the number of eggs so they can pay the farmer the cost of the eggs. Unfortunately, the farmer is very poor at keeping inventory of her eggs and doesn't know how many eggs she had in the back of her car. She does, however, remember how she packed her eggs. Sh remembers when she tried to pack the eggs in groups of two, there was one egg left over. The same goes when she tried to group them in threes, fours, fives, and sixes. When grouped in sevens, however, there were no eggs left over. The insurance agent sighs and takes the information to you so that you can figure out the amount of eggs.

We need to figure out how the farmer can determine the amount of eggs she had and if there is more than one possibility in the number of eggs.

**Process Description**

When looking at this problem, our group first decided to filter out all the unnecessary words from the problem statement and just have the information we needed. So we determined the number of eggs would be a multiple of 7. If the total is divided by 2,3,4,5 or 6, there would be a remainder of 1. So in order to find the answer, we would need a multiple of 7 that when divided by 2,3,4,5 or 6, has a remainder of 1. We began to pick random numbers that would fit this condition. I tried 49, 14, 21, 28, 42, 56, 63, and 84. Some numbers would have a remainder of one, but the other numbers would net a remainder greater than one. I decided this method was inefficient and would take a long time to figure out the actual number(s). This did get the ball rolling, though. What we did next, which was kind of outside of the rules, was use a program to find the answer. We had our resident coding guru, Austin, write a quick program that would help determine the answer(s). What the program did was it iterated through multiples of 7, checked those multiples for a remainder of 1 in 2,3,4,5 and 6. The numbers that met the condition were displayed. The program displayed multiple values, the lowest being 301. It also showed 721,1141,1561,1981, and 2401. So we had the solution, but we needed to figure out how we arrived at it. We noticed the difference between each term was 420. We looked at 420’s factors, which were 7,3,2,5, but that led to nothing. That did get us thinking about the factors of 720, though. We noticed that 23456=720 and since 721 was one of the solutions, we assumed we were onto something since 1 was the remainder for 1 through 6. 720 was not divisible by 7, but adding a 1 made it divisible by 7. We arrived at this because we were bouncing ideas off each other. If anyone had any idea to contribute, we would all listen and see if the idea had merit. It was very open and friendly. We kept each others ideas in check, meaning we would check to see if the idea would work. If not, we would see if we could expand any of it in order to work with the problem.

**Solution**

As stated earlier, we actually found the solution by using Austin’s custom-coded program. The lowest number we could that met all the conditions of being a number that had a remainder of 1 if divided by 2,3,4,5, and 6, but no remainder when divided by 7 was 301. Since the difference of 301 and 721 was 420, we created an equation for finding any number that meets the conditions. The equation is 301+420(x)=n where n=possible number of eggs. If we plug in any real number for x, we multiply it by 420, then add 301, we get the number of eggs. By adding 301, we make the solution divisible by 7 with no remainder. By using this equation, we found out there are many different solutions to this problem as well.

**Self-Assessment/Reflection**

What I learned from this problem is that it is vital to identify the important parts of your problem so you don’t get confused with unnecessary information. I also learned that it is a valid option to work backwards from a solution like how we did by figuring out the answer and then finding out how we can obtain this answer. I think I deserve an 8 out of 10 because I worked well with my group-mates and aided in finding the solution. For example, I transcribed the problem from the story stated to actual numbers. I also came up with the idea to use a program in helping us to find a solution. On the other hand, I didn't completely understand why the equation is what it is and not something lower. I still don’t know how to obtain 301 by using our equation. I also think I could have been more involved in the equation process by asking how it worked as my group did not communicate that part well to me.

As for a Mathematical Practice and Expectation, I feel using appropriate tools strategically was the biggest part in figuring out this problem. Using a program to do the work for us seems like it was outside the scope of the rules, but in my opinion, it feels justified. What’s the point of just trying random numbers until we get a result? We have technology that can do this quickly and we still needed to work backwards in trying to find the equation. The computer was a tool that aided us in figuring out a possible solution.