Consider an equilateral triangle with points located at each vertex and at each midpoint of a side. (See picture.) This problem uses the set of numbers {1, 2, 3, 4, 5, 6}. Place one number at each point. Call the sum of the three numbers along any one side (two vertices and one midpoint) a “Side Sum.” There are arrangements of the numbers so that the sum of the numbers along any side is equal to the sum of the numbers along each of the two other sides. We will call this arrangement an Equal Side Sum solution.
1) Show that more than one Equal Side Sum solution exists.
a. For which numbers are Equal Side Sum solutions possible? (Show by giving examples for each Equal Side Sum solution that is possible.) Comment on how you obtained these solutions.
b. What is the smallest number for which there is an Equal Side Sum solution? Why?
c. What is the largest number for which there is an Equal Side Sum solution? Why?
2) Is there more than one Equal Side Sum solution for the same number? (To answer this question, you will need to be precise as to what you mean when you say that two Equal Side Sum solutions are the same or are different.) Explain your answer.
Extra Credit: It is possible to generalize The Triangle Game to create a similar game involving other polygons. Describe such a game. Are you able to find any solutions to your new game? Worth up to two extra credit points
Two whole numbers are less than 10 and grèater than 0.whats the difference between their product and their sum.
To answer the question, let's go step by step:
1) Show that more than one Equal Side Sum solution exists.
a. For which numbers are Equal Side Sum solutions possible?
To find Equal Side Sum solutions, we need to place the numbers {1, 2, 3, 4, 5, 6} in the equilateral triangle such that the sum of the three numbers along any side is equal to the sum of the numbers along each of the other two sides.
One possible Equal Side Sum solution is:
1
3 5
2 4 6
The sum of the numbers along each side is:
Side Sum = 1 + 3 + 2 = 6
Side Sum = 4 + 1 + 5 = 10
Side Sum = 6 + 3 + 4 = 13
Another possible solution is:
2
5 4
3 1 6
The sum of the numbers along each side is:
Side Sum = 2 + 5 + 3 = 10
Side Sum = 1 + 2 + 6 = 9
Side Sum = 4 + 3 + 1 = 8
To obtain these solutions, we can start by placing the numbers {1, 2, 3} on one side of the triangle, and then assign the remaining numbers {4, 5, 6} to the opposite side. The vertex and midpoint with the same index will be assigned the same number.
b. What is the smallest number for which there is an Equal Side Sum solution? Why?
The smallest number for which there is an Equal Side Sum solution is 4. This is because to have three sides with equal sums, each side must have a sum of at least 1 + 2 + 3 = 6. So, the smallest number we can place on a side is 4, which results in a side sum of 4 + 1 + 1 = 6.
c. What is the largest number for which there is an Equal Side Sum solution? Why?
The largest number for which there is an Equal Side Sum solution is 6. This is because the sum of the numbers {1, 2, 3} is 6, and each side of the equilateral triangle must have a sum equal to this value. So, the largest number we can place on a side is 6, resulting in a side sum of 6 + 1 + 1 = 8.
2) Is there more than one Equal Side Sum solution for the same number?
No, there is only one Equal Side Sum solution for each number. If the sum of the numbers {1, 2, 3} is fixed, all possible placements of numbers on the sides will lead to unique solutions. Two Equal Side Sum solutions are considered different if the numbers are placed in different positions within the triangle.
Extra Credit: It is possible to generalize The Triangle Game to create a similar game involving other polygons. Describe such a game. Are you able to find any solutions to your new game?
The generalized game can involve any regular polygon, where the numbers are placed on the vertices and midpoints of the sides. The objective is to find arrangements of numbers such that the sum of the numbers along any side is equal to the sum of the numbers along each of the other sides.
For example, let's consider a regular hexagon. We have the numbers {1, 2, 3, 4, 5, 6}. Placing the numbers as shown below can result in an Equal Side Sum solution:
1 4
5 2 3 6
The sum of the numbers along each side is:
Side Sum = 1 + 5 + 2 = 8
Side Sum = 4 + 1 + 3 = 8
Side Sum = 6 + 2 + 4 = 12
Side Sum = 3 + 5 + 6 = 14
However, finding solutions for different polygons can be more complex and may require specific strategies or algorithms.
To solve this problem, let's go step by step.
1) Show that more than one Equal Side Sum solution exists.
To demonstrate that there are multiple Equal Side Sum solutions, we need to find at least two arrangements where the side sums are equal.
Let's consider the set of numbers {1, 2, 3, 4, 5, 6} and try out different placements.
a) For which numbers are Equal Side Sum solutions possible? (Show by giving examples for each Equal Side Sum solution that is possible.) Comment on how you obtained these solutions.
To find examples of Equal Side Sum solutions, we need to experiment with different number placements.
One possible arrangement is:
```
2
3 4
1 5 6
```
In this arrangement, the side sums are:
2 + 3 + 1 = 6
3 + 4 + 5 = 12
1 + 5 + 6 = 12
Another possible arrangement is:
```
4
1 6
3 2 5
```
In this arrangement, the side sums are:
4 + 1 + 3 = 8
1 + 6 + 2 = 9
3 + 5 + 2 = 10
By exploring different number placements, we can find a variety of Equal Side Sum solutions.
b) What is the smallest number for which there is an Equal Side Sum solution? Why?
The smallest number for which there is an Equal Side Sum solution is 3. This is because to form a triangle, we need a minimum of three numbers. By placing 1, 1, and 1 in the triangle, we have an Equal Side Sum solution. The side sums will be equal to 3.
c) What is the largest number for which there is an Equal Side Sum solution? Why?
The largest number for which there is an Equal Side Sum solution is 6. This is because the numbers in the set are limited to {1, 2, 3, 4, 5, 6}. By placing 6, 6, and 6 in the triangle, we have an Equal Side Sum solution. The side sums will be equal to 18.
2) Is there more than one Equal Side Sum solution for the same number? Explain your answer.
To determine if there are multiple Equal Side Sum solutions for the same number, we need to define what we mean by "the same" solutions. Assuming that "the same" means having the same values in the same positions (ignoring rotational symmetry), then there is only one Equal Side Sum solution for each number.
However, if we consider different arrangements as distinct solutions, then there can be multiple Equal Side Sum solutions for the same number. For example, for the number 12, we can have the following arrangements:
```
4 5
1 6 or 1 4
3 2 5 3 2 6
```
Both of these arrangements have a side sum of 12.
Extra Credit: Generalizing the Triangle Game to other polygons.
We can create a similar game involving other polygons by extending the concept of Equal Side Sum to those shapes. Let's consider the Square Game as an example.
In the Square Game, we have a square with points located at each vertex and at each midpoint of a side. We will also use the set of numbers {1, 2, 3, 4, 5, 6}.
The goal is to place one number at each point in such a way that the sum of the numbers along any one side (two vertices and one midpoint) is equal to the sum of the numbers along each of the other three sides.
To find solutions to the Square Game, we can follow a similar approach to the Triangle Game. Experiment with different number placements and check if the side sums are equal for all four sides.
Finding solutions to the Square Game requires careful consideration of number placements to ensure the equal side sum condition is satisfied.
As for finding solutions to the Square Game, I'm unable to provide them within the scope of this explanation. However, solving the Square Game could be a fascinating challenge.