POW #2: How Many Egg-xactly?
The “How Many Egg-xactly?” problem covered concepts like finding similarities between numbers, and applying certain rules to those commonalities. It also involved a lot of logical thinking, meaning that every aspect of the problem had to be taken into consideration. Once I found the answer, I also had to take it a step further to discover other solutions along the way, which made the task even more strenuous. However, once I was on the right track, it was smooth sailing throughout the rest of the process.
Part 1: Problem Statement
In this problem, I had to find how many eggs Dave spilled from the jars in his cart after hitting a pothole. Dave knew that when he put the eggs in groups of two, there was one egg left over. This pattern was also consistent when he organized the eggs in groups of three, four, five and six; there was always one left over. However, when he put those eggs in groups of seven, there were no eggs left over. My job was to answer the question; how many eggs did Dave have before the spill?
Part 1: Problem Statement
In this problem, I had to find how many eggs Dave spilled from the jars in his cart after hitting a pothole. Dave knew that when he put the eggs in groups of two, there was one egg left over. This pattern was also consistent when he organized the eggs in groups of three, four, five and six; there was always one left over. However, when he put those eggs in groups of seven, there were no eggs left over. My job was to answer the question; how many eggs did Dave have before the spill?
Part 2: Process
To start on this task, I had to recognize that the number of eggs he had was a multiple of seven, and a multiple of one, two, three, four, five and six when subtracted by one. This pertains to the aspect of the problem related to the leftover egg. The only scenario in which there was not a leftover egg was when they were grouped in sevens. Therefore, the number I needed to find was one less than a multiple of seven. The first step I took to finding the “magic number” was by taking what I already knew, and narrowing down my search. I knew that this number I was looking for needed to be divisible by numbers one through six. Since it is a given that a number divided by one equals itself, I could eliminate the number 1 from my search. I also recognized a correlation between the numbers two, three and six. As you will see in the diagram to the right, I tried to come up with a way to represent the fact that the numbers two and three were factors of six. This piece of information was crucial. Since the eggs were grouped in twos, threes and sixes with one egg left over, I knew that the number of eggs in those scenarios would be exactly the same. |
If you were to group the eggs in three groups of two, or two groups of three, or even one group of six, the number of eggs would be exactly all the way around. The number six was basically a quicker and simpler way to find the number of eggs that I was looking for, since it contained both two and three as factors. As a result, I could limit my search to finding a number that was one less than a multiple of seven that only had to be divided by the numbers four, five and six.
From here, I looked at the division rules surrounding each one of these numbers. Here were the rules I jotted down regarding the numbers four through six.
From here, I looked at the division rules surrounding each one of these numbers. Here were the rules I jotted down regarding the numbers four through six.
- Four - The last two digits are divisible by 4
- Five - The last digit is 0 or 5
- Six - The number is divisible by both 2 and 3
As soon as I found that the number sixty worked in this table, I immediately moved on to the next step. Since sixty was divisible by numbers two through six, I knew the answer I was looking for was a multiple of sixty, with the addition of the left over one. To display my thoughts on this, I set up a multiples table in order to solve the problem. As you will find below, I tested each number from sixty up to when I found an answer, and if when it was added to one it would be divisible by seven.
I had my answer.
Dave had 301 eggs in his jars before he spilled them.
Part 3: Solution
The first step I took to double-check and make sure that my answer was valid was to confirm that three hundred could be divided by numbers two through six.
Dave had 301 eggs in his jars before he spilled them.
Part 3: Solution
The first step I took to double-check and make sure that my answer was valid was to confirm that three hundred could be divided by numbers two through six.
At this point, I knew for a fact that 301 was the correct solution. It met all of the requirements set in the problem description; it could be split up into groups of one, two, three, four, five and six with one egg left over, and it was also a multiple of seven. Now it was a matter of expanding on that answer and trying to find a pattern or expression that could help me find other solutions. I realized that sixty was my interval in trying to get to a number that when added to one would equal a multiple of seven. In addition, I knew that sixty (when added to one) was not divisible by seven. Using this information, I could find the least common multiple of both seven and sixty to find a number that could be divided by every number from one to seven.
Four hundred twenty was going to be the interval I used between 301 and any number coming after it. Since it was both a multiple of seven and sixty, I knew I had to be on the right track. To check my reasoning, I created another table (below) to see if I could find other solutions tied into this pattern.
And it worked! I previously had described the “magic number” as being my answer, (301), but it actually turned out to be 420. That special number was the one that could help me solve for any other number, and a pattern emerged. With the data above, I was also able to formulate an equation, assuming that a variable x was the solution number, and a variable y was the answer:
I subtracted 119 after the 420x because it is the difference between 420 and 301. Without the negative 119 there, the first solution would technically be 420, even though it is 301. By plugging in the solution numbers from the table above, the equation I created can solve any problem along this pattern.
All in all, I felt like I persevered more than ever through this problem to find the answer, yet it was a successful process. I was able to find an answer, an interval, and an equation that could help solve similar problems like this in the future, which I was very proud of. My method was very consistent over the course of my experience with the egg problem, which helped me to stay on track, even when I thought I was lost.
Reflection
The “How Many Egg-xactly” problem was crucial in my ability to understand concepts surrounding multiples of numbers, and what traits certain numbers share. I examined mathematical rules pertaining to each number from one to seven in order to find the correct answer in this situation. I was able to identify difference in specificity of rules, (the rule for five being more particular than the rules for four and six), and I could apply those rules to my estimations. I also learned how similar numbers can be just by tweaking them a little bit. Three hundred was divisible by every number from one through six, but it was not divisible by seven. By adding just one to three hundred, it became a multiple of seven. Frankly, this was mind-blowing, that a number of three digits strong could be changed to fit the problem so effortlessly. I did not have to do any fancy calculations to make it a multiple of seven, or make my brain explode trying to come up with a formula. All I had to do was add one.
I honestly believe that I deserve a 10 out of 10 for my work on this problem, for my thoroughness and all of the effort I put into solving it. I found a unique way to approach the problem, and I also came up with an equation to apply to the problem at the end. Once I sorted out the information given in the problem, I was able to represent my thinking through a colorful diagram, and I displayed the process of my work in a step-by-step format. I found the correct answer, which is a statement to how diligent I was in trying to locate a solution. My explanations were very clear and detailed, and I spent countless hours trying to finetune my work into the best possible shape it could be in. (And when I say countless hours, I really mean countless hours.)
The most prominent Habit of a Mathematician shown in my work was Be Systematic. I broke down the problem in its entirety into steps. The first step I took was to familiarize myself with mathematical rules, and then I organized what I knew and what I did not know into tables. By taking a more calm, easy approach to the problem, I believe that I was most efficient in finding the correct solution. I also looked for small differences between numbers, (e.g., 300 vs. 301) and what those distinctions meant. Three hundred was divisible by numbers one through six, while 301 was divisible by seven. Without my strategy of trying to recognize how these numbers contrasted, I would not have noticed the similarities between 300 and 301. Each one of the steps I took to find the final answer was both purposeful and meaningful to solving the problem, and I was happy with the path I took.
All in all, I felt like I persevered more than ever through this problem to find the answer, yet it was a successful process. I was able to find an answer, an interval, and an equation that could help solve similar problems like this in the future, which I was very proud of. My method was very consistent over the course of my experience with the egg problem, which helped me to stay on track, even when I thought I was lost.
Reflection
The “How Many Egg-xactly” problem was crucial in my ability to understand concepts surrounding multiples of numbers, and what traits certain numbers share. I examined mathematical rules pertaining to each number from one to seven in order to find the correct answer in this situation. I was able to identify difference in specificity of rules, (the rule for five being more particular than the rules for four and six), and I could apply those rules to my estimations. I also learned how similar numbers can be just by tweaking them a little bit. Three hundred was divisible by every number from one through six, but it was not divisible by seven. By adding just one to three hundred, it became a multiple of seven. Frankly, this was mind-blowing, that a number of three digits strong could be changed to fit the problem so effortlessly. I did not have to do any fancy calculations to make it a multiple of seven, or make my brain explode trying to come up with a formula. All I had to do was add one.
I honestly believe that I deserve a 10 out of 10 for my work on this problem, for my thoroughness and all of the effort I put into solving it. I found a unique way to approach the problem, and I also came up with an equation to apply to the problem at the end. Once I sorted out the information given in the problem, I was able to represent my thinking through a colorful diagram, and I displayed the process of my work in a step-by-step format. I found the correct answer, which is a statement to how diligent I was in trying to locate a solution. My explanations were very clear and detailed, and I spent countless hours trying to finetune my work into the best possible shape it could be in. (And when I say countless hours, I really mean countless hours.)
The most prominent Habit of a Mathematician shown in my work was Be Systematic. I broke down the problem in its entirety into steps. The first step I took was to familiarize myself with mathematical rules, and then I organized what I knew and what I did not know into tables. By taking a more calm, easy approach to the problem, I believe that I was most efficient in finding the correct solution. I also looked for small differences between numbers, (e.g., 300 vs. 301) and what those distinctions meant. Three hundred was divisible by numbers one through six, while 301 was divisible by seven. Without my strategy of trying to recognize how these numbers contrasted, I would not have noticed the similarities between 300 and 301. Each one of the steps I took to find the final answer was both purposeful and meaningful to solving the problem, and I was happy with the path I took.