In this Problem of the Week, called the 1-2-3-4 Puzzle, we had to figure out equations to equal each digit from 1 to 25 using only 1, 2, 3, and 4. We could add those four numbers up, subtract them, multiply them, divide them, square root, use factorials, put them together to make numbers like 12, and any combination of these. The expression needed to have all four numbers, though. No more, no less. An example of a 1-2-3-4 expression that I used is (4+1)-2+3=6.
To solve this problem, I mostly just experimented with combinations of numbers that I thought of, then found out what they equalled to. For instance, randomly creating the equation 34-12, then working it out in my head and finding that it equalled 22. Sometimes, I would go backwards, and try to figure out the equation coming from the end product, though this really only happened once I neared the end of the problem, and had only a few numbers that needed to be given equations. An example of this is the number 20. I knew I needed 20, so I subtracted 23, and 4-1 (3).
Some of the numbers were easier to find equations for than others, though. I had a very easy time coming up with equations for 3, 13, 15, and 22. I don’t know why this is, but it piqued my curiosity. Some numbers I had a harder time with were 7, 17, and 19. To overcome this, I would sometimes just take a break, think about something else, then come back to the problem. Often, that helped get my ideas flowing again.
My solution (attached to this write up) only contains the equations I found, and the numbers they correspond to. I found more equations for some numbers than I did for others, but each number from 1 to 25 has at least one equation. I know my solution is correct because all of the equations work, and equal the number they are supposed to correspond to.
To solve this problem, I mostly just experimented with combinations of numbers that I thought of, then found out what they equalled to. For instance, randomly creating the equation 34-12, then working it out in my head and finding that it equalled 22. Sometimes, I would go backwards, and try to figure out the equation coming from the end product, though this really only happened once I neared the end of the problem, and had only a few numbers that needed to be given equations. An example of this is the number 20. I knew I needed 20, so I subtracted 23, and 4-1 (3).
Some of the numbers were easier to find equations for than others, though. I had a very easy time coming up with equations for 3, 13, 15, and 22. I don’t know why this is, but it piqued my curiosity. Some numbers I had a harder time with were 7, 17, and 19. To overcome this, I would sometimes just take a break, think about something else, then come back to the problem. Often, that helped get my ideas flowing again.
My solution (attached to this write up) only contains the equations I found, and the numbers they correspond to. I found more equations for some numbers than I did for others, but each number from 1 to 25 has at least one equation. I know my solution is correct because all of the equations work, and equal the number they are supposed to correspond to.
To explore this problem further, I think it would be interesting to look at numbers from 1 to 100, or possibly negative numbers. Also, the already provided “Four Four’s” extension looked interesting. In that problem, all of the same basic rules from this problem applied, except for the fact that you had to use four 4’s, rather than 1, 2, 3, and 4. I think it would be interesting to see if this works with numbers besides 4, and if not, what makes 4 so special.
Doing this problem, I used the Habits of a Mathematician experiment through conjecture, and stay organized. I used experiment through conjecture because of all of the guessing and checking I did though the problem. This was useful because it really kick started my thinking with the problem, then helped to continue it. I used stay organized because of the way I organized my thoughts as I was working. The one drawback to the way I organized my paper was that I didn’t write down my thought process as much as I would have liked to. I just wrote down the solution to my thoughts, rather than everything I was thinking about with the problem. A way I could help fix that (with anything I work on, not just math) is to have two pieces of paper, one for jotting down anything I think of, and one that looks a little better, with the solutions on it.
I think the work I did on this problem deserves a 9/10. I did all of the required work, and found some extra equations, but I think I could have found more equations (at least two) for each number. I also could have explored one of the extensions I talked about earlier.
Doing this problem, I used the Habits of a Mathematician experiment through conjecture, and stay organized. I used experiment through conjecture because of all of the guessing and checking I did though the problem. This was useful because it really kick started my thinking with the problem, then helped to continue it. I used stay organized because of the way I organized my thoughts as I was working. The one drawback to the way I organized my paper was that I didn’t write down my thought process as much as I would have liked to. I just wrote down the solution to my thoughts, rather than everything I was thinking about with the problem. A way I could help fix that (with anything I work on, not just math) is to have two pieces of paper, one for jotting down anything I think of, and one that looks a little better, with the solutions on it.
I think the work I did on this problem deserves a 9/10. I did all of the required work, and found some extra equations, but I think I could have found more equations (at least two) for each number. I also could have explored one of the extensions I talked about earlier.