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problem of the week 1

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**1-2-3-4 puzzle**

For this problems, we had to find 1-2-3-4 equations for the numbers 1-25. A 1-2-3-4 equation is an expression that uses those digits only once, to find these equations we could use any arithmetic operation, exponents, radicals, factorials, parentheses and brackets and we could juxtapose the numbers (put them together as in: 1 and 2 to become 12).

What I did to solve this problem was to see all the combinations I could come up with and see how I could make each number. Constantly using the tool of conjecture and test I went through many, many renditions of problems trying to get the number I wanted. As I went on I began using more tools (such as factorials, brackets, etc) to achieve the number I wanted to prove.

1: 2+1x3-4

2: 3+2+1-4

3: (4-2-1)x3

4: 4x2-3-1

5: 4x2x1-3

6: 1+3+4-2

7: 31-24

8: 3+2+4-1

9: 41-32

10: 1+2+3+4

11: 42-31

12: 4x2+3+1

13: 34-21

14: (3^2)+4+1

15: 4x3+2+1

16: (4+3+1)x2

17: 4!-(2^3)+1

18: 4!-(3x2)x1

19: (4^2)+3x1

20: (4^2)+3+1

21: (4+2+1)x3

22: 3!+(4^2)x1

23: 3!+4+1

24: 4!-3+(2+1)

25: 4!+(3-2)x1

One extension to the problem I came up with is you can only repeat digits if they are exponents. This lets you use bigger numbers and allows for longer equations. Some of the answers I found for this are but are not limited to:

1: (3^2)-4x2x1

2: 21-(4^2)-3

3: (4^2)-(2^2)x(3^2)x1

4: 1+ [(2^2)-4]+3

5: (4^2)-(3^2)-2x1

...

During this problem I learned that you can use a simple amount of digits to form equations for many numbers if not all numbers, the ways you set up the equations and the mathematical tools you use in the equations will impact the final result heavily. The problem also made me think more thoroughly and have to constantly use the habit of conjecture and test. I was consistently setting up multiple equations trying to get the number I needed to find. I think out of 10, I would give myself a 9 in this problem because I was persistent with my method and I used all the tools that were at my disposal. I also found how I could add more rules to broaden the problem and allow me to think of more ways of solving the problem all while using the habit of conjecture and test.

__Process__What I did to solve this problem was to see all the combinations I could come up with and see how I could make each number. Constantly using the tool of conjecture and test I went through many, many renditions of problems trying to get the number I wanted. As I went on I began using more tools (such as factorials, brackets, etc) to achieve the number I wanted to prove.

__Solutions:__1: 2+1x3-4

2: 3+2+1-4

3: (4-2-1)x3

4: 4x2-3-1

5: 4x2x1-3

6: 1+3+4-2

7: 31-24

8: 3+2+4-1

9: 41-32

10: 1+2+3+4

11: 42-31

12: 4x2+3+1

13: 34-21

14: (3^2)+4+1

15: 4x3+2+1

16: (4+3+1)x2

17: 4!-(2^3)+1

18: 4!-(3x2)x1

19: (4^2)+3x1

20: (4^2)+3+1

21: (4+2+1)x3

22: 3!+(4^2)x1

23: 3!+4+1

24: 4!-3+(2+1)

25: 4!+(3-2)x1

__Extensions:__One extension to the problem I came up with is you can only repeat digits if they are exponents. This lets you use bigger numbers and allows for longer equations. Some of the answers I found for this are but are not limited to:

1: (3^2)-4x2x1

2: 21-(4^2)-3

3: (4^2)-(2^2)x(3^2)x1

4: 1+ [(2^2)-4]+3

5: (4^2)-(3^2)-2x1

...

__Evaluation:__During this problem I learned that you can use a simple amount of digits to form equations for many numbers if not all numbers, the ways you set up the equations and the mathematical tools you use in the equations will impact the final result heavily. The problem also made me think more thoroughly and have to constantly use the habit of conjecture and test. I was consistently setting up multiple equations trying to get the number I needed to find. I think out of 10, I would give myself a 9 in this problem because I was persistent with my method and I used all the tools that were at my disposal. I also found how I could add more rules to broaden the problem and allow me to think of more ways of solving the problem all while using the habit of conjecture and test.

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