PROBLEM of the week 1
During Problem of the Week #1, we had a chance to choose from two different problem types. One was called the 1-2-3-4 problem. In the 1-2-3-4 problem, students tried to make the sums of numbers 1-25 by just using the terms 1,2,3, and 4. In order to get these sums, you could use any operation. For example, if you wanted to make the number 18, you could use the equation 4(3+1)+2 = 18. The criteria of the problem is you could only use each number once. You could also put the numbers together and use them in a double digit number in order to reach a sum. For example, 24-13= 11. Some other operations you could use were roots, exponents, factorials, and the basic operations like adding, subtracting, multiplying, and dividing. For the second problem type and the one that I attempted was called the 4 fours problem. This was a challenge of the 1-2-3-4 problem. It was very similar, but instead of using just 1,2,3, and 4, you could only use the number four in your problem, and you had to use it four times.
In order to solve this problem, I used a strategy of conjecturing and testing. A conjecture is basically an educated guess, so I used my background math knowledge in order to creatively devise the sums of the numbers 1-30. I started out with attempting the numbers in consecutive order, and this worked for numbers 1-10, but 11 was tricky. Instead of spending a lot of time on 11 I moved onto doing the other ones that were a bit easier for me. When I got to the final stage of the problem, I had three numbers left, 11,13, and 19. I am unsure of why these were so hard for me to get, but something that they all have in common is that they are all prime numbers. After being stuck for a few minutes, I asked my teacher Mr.Corner for some assistance. He played around with it for a while, and I sat back down. I told a peer of my troubles and he gave me a hint, "use factorials". After he said that, my brain sparked up with ideas. I realized by using factorials, I was able to get 11,13,and 19!
Above you will see my work for problems 1-10 . When solving this problem, something important that I learned is that you need to have a plan when solving a problem and not to just approach it without thinking. To solve the problem, I came up with a set of rules or facts that could help me solve the problem. For example, since you wanted to solve each number with exactly 4 4's, you may want to get 8 with 2 or 3 terms. This way I knew how to make other numbers that could be associated with 8. I did this for a few other numbers as well
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My solution is that at least all numbers 1-30 could be reached by using 4 fours and multiple operations. I am unsure when this perfect streak of sums stop, but at first I assumed it wasn't too long after 30. After my curiosity overwhelming me after completing the problem I found this website that had the answers from 1-100! http://mathforum.org/ruth/four4s.puzzle.html
After that point the easiest way to reach a high number with a lot options afterwords is to work from 24 after doing 4!. As you can see from 26-30 I used this strategy every single time. |
From this problem, I feel as if what I took out of it are learning how to use factorials, and working out an open ended problem. Before this problem, I never really used or understood factorials well. Now if I ever need to represent multiplication in that way, I will know how to. In order to do this work, you have to be semi organized, or everything will get all mixed up. For this problem, I would assign myself a grade of 10/10 because I went to my level of math and completed the challenge problem. It took quite a while to think up some of the equations to solve the 4 fours problem, and it was worth the time when I was able to find all the problems from 1-30. A habit of a mathematician I used when solving this problem is solve a simpler problem. Instead of looking at the problem as a whole, I broke it apart to just solve 1 number at a time. Than I used the information I had from the past numbers to solve the ones I completed in the future. Another one that I used like I mentioned before is staying organized. If my work was scattered around I might not be able to find the work for each number.