Problem of the Week #2 - 1-2-3-4 problem
You’ve seen that you can change the meaning of an arithmetic expression by inserting or removing parentheses. Of course, another way to change the meaning of an expression is to rearrange its terms.
This problem is about using the digits 1, 2, 3, and 4, in any order you choose, to create arithmetic expressions with different numerical values according to the rules for order of operations.
For this problem, a 1-2-3-4 expression is any expression written using each of these digits exactly once, according to the following rules:
Your Task:
Your task in this problem is to create as many 1-2-3-4 expressions as you can for each of the numbers from 1 to 25. Remember: In every case, the expression must use each of the digits 1, 2, 3, and 4 exactly once.
Write Up
Problem Statement
-The question is asking to try to find the most expressions for the numbers 1-25 but you can only use the numbers 1-2-3-4. Also you can only use 1-2-3-4 once in every expression. You can do Multiplication,Division,Parenthesis and even Juxtapose. This is how I understood the problem and interpreted it.
Process Description
-Our group first tried to figure out the easiest expression for the numbers, like adding 1+2+3+4 we figured out the basic and obvious expressions first. We then tried to figure out the harder ones like 15,16, and 17. Honestly I was just putting 1-2-3-4 in random orders and hoped to get the answer. Our group decided that we all should try to figure out the expression for one number so we all contributed and combined all our ideas to try to figure out the harder numbers, Our method worked and got to solve some expressions out for 18,15, and 16 but we were still stuck on trying to figure out 17. We went to see other groups ideas and tried to build off of there ideas to figure out the only number we still hadn’t figured out. Finally, after some brainstorming with our partners around us we came up with an expression that fit 17 which was 34÷ 2×1=17. Although our group had a tough time trying to figure out the expressions for the numbers, cooperation really help us out and made it easier.
Extensions
-Some extensions that our decided to come up with is trying to figure out the expressions out for bigger numbers like 40,51,54 and 84. They all still follow the rule of only using 1-2-3-4 in every expression and only once.
Expressions for each one:
40-12×3+4
51-12×4+3
54-13×4+2
84-14×2×3
Solution
Reflection
- For this problem of the week I used many Habits of Mathematician but there were two that I used the most while trying to figure out different expressions for each number. The Habits of Mathematician that I used were Conjecture and Test and Collaborating & Listening. These were crucial to trying to solve the problem and making it easier on ourselves. We collaborated a lot while trying to come up with all the different kind of expressions. I Conjectured and Test mostly throughout the beginning to figure out all the basic expressions for the numbers. These Habits of Mathematician were very important and help us a lot.
You’ve seen that you can change the meaning of an arithmetic expression by inserting or removing parentheses. Of course, another way to change the meaning of an expression is to rearrange its terms.
This problem is about using the digits 1, 2, 3, and 4, in any order you choose, to create arithmetic expressions with different numerical values according to the rules for order of operations.
For this problem, a 1-2-3-4 expression is any expression written using each of these digits exactly once, according to the following rules:
- You may use any of the four basic arithmetic operations—addition, subtraction, multiplication, and division (according to the order-of-operations rules). For example, 2 + 1 • 3 – 4 is a 1-2-3-4 expression for the number 1 (since 2 + 1 • 3 – 4 = 1).
- You may use exponents. For example, 23 – 4 – 1 is a 1-2-3-4 expression for the number 3.
- You may use radicals. For example, is equal to 3, so is a 1-2-3-4 expression for the number 6.
- You may use factorials. For example, 4! means 4 • 3 • 2 • 1, so 3 + 4! + 1 – 2 is a 1-2-3-4 expression for the number 26.
- You may juxtapose two or more digits (that is, put them next to each other) to form a number such as 12. For example, 43 – 12 is a 1-2-3-4 expression for the number 31.
- You may use parentheses and brackets to change the meaning of an expression. For example, according to the rules for order of operations, 1 + 4 • 32 is a 1-2-3-4 expression for the number 37. You can add parentheses and brackets to get [(1 + 4) • 3]2, which is a 1-2-3-4 expression for the number 225.
Your Task:
Your task in this problem is to create as many 1-2-3-4 expressions as you can for each of the numbers from 1 to 25. Remember: In every case, the expression must use each of the digits 1, 2, 3, and 4 exactly once.
Write Up
- Problem statement - re-write the question in your own words
- Process Description - describe and show how you and your group attempted to solve the problem
- Extensions - invent some extensions of variations of the problem
- Solution - describe your final solution to the problem
- Reflection - reflect on how you used two habits of mathematicians to solve the problem.
Problem Statement
-The question is asking to try to find the most expressions for the numbers 1-25 but you can only use the numbers 1-2-3-4. Also you can only use 1-2-3-4 once in every expression. You can do Multiplication,Division,Parenthesis and even Juxtapose. This is how I understood the problem and interpreted it.
Process Description
-Our group first tried to figure out the easiest expression for the numbers, like adding 1+2+3+4 we figured out the basic and obvious expressions first. We then tried to figure out the harder ones like 15,16, and 17. Honestly I was just putting 1-2-3-4 in random orders and hoped to get the answer. Our group decided that we all should try to figure out the expression for one number so we all contributed and combined all our ideas to try to figure out the harder numbers, Our method worked and got to solve some expressions out for 18,15, and 16 but we were still stuck on trying to figure out 17. We went to see other groups ideas and tried to build off of there ideas to figure out the only number we still hadn’t figured out. Finally, after some brainstorming with our partners around us we came up with an expression that fit 17 which was 34÷ 2×1=17. Although our group had a tough time trying to figure out the expressions for the numbers, cooperation really help us out and made it easier.
Extensions
-Some extensions that our decided to come up with is trying to figure out the expressions out for bigger numbers like 40,51,54 and 84. They all still follow the rule of only using 1-2-3-4 in every expression and only once.
Expressions for each one:
40-12×3+4
51-12×4+3
54-13×4+2
84-14×2×3
Solution
- For our solution we got all the expressions from 1-25 and found multiple on each of the numbers. I found multiple for each numbers but these are the ones I first came up with but there are other expressions you could use for each number.
Reflection
- For this problem of the week I used many Habits of Mathematician but there were two that I used the most while trying to figure out different expressions for each number. The Habits of Mathematician that I used were Conjecture and Test and Collaborating & Listening. These were crucial to trying to solve the problem and making it easier on ourselves. We collaborated a lot while trying to come up with all the different kind of expressions. I Conjectured and Test mostly throughout the beginning to figure out all the basic expressions for the numbers. These Habits of Mathematician were very important and help us a lot.