Help! How do I complete these magic squares! They're taking me FOREVER!
In magic squares, certain numbers have to add up to a certain sum, and be the same sum diagonal,across,and vertical. (Sorry if I didn't explain well)
Arrange the numbers 2,4,6,8,10,12,14,16, and 18 so that the sums of all rows, columns, and diagonals are 30.
Arrange the numbers 3,6,9,12,15,18,21,24,27 so that the sums of all rows, columns, and diagonals are 45.
Please also tell me how to show my work.
P.S. I tried google, but it was confusing.
THANKS A LOT IN ADVANCE!
Here is a simple 3by3 magic square using the numbers from 1 to 9, all rows and columns and diagonals add up to 15
8 1 6
3 5 7
4 9 2
now arrange 2,4,6,..,18 in the same sequence, it will work
do the same for the other sequence of numbers.
Thank you!
Don't worry, I can help you solve these magic squares! Let's start with the first one:
1. First, let's identify the sum that all rows, columns, and diagonals need to have, which is 30.
2. To make things easier, let's place the number 16 in the center cell. Since the sum of all rows, columns, and diagonals need to be 30, the opposite diagonal cells also need to add up to 16.
3. Now, let's fill in the edge cells. Since we have already used the numbers 16 and 18, let's place the number 14 in a corner cell. If we know that one corner cell is 14 and its opposite corner cell is 18, we can deduce that the other two corner cells will be 6 and 24.
4. Since we know that the opposite edge cells need to add up to 16, we can fill in the remaining edge cells. We have already placed the numbers 6 and 24 on two corners, so the remaining two edge cells will be 10 and 8.
5. Now, let's fill in the remaining cells. We know that the sum of each row, column, and diagonal needs to be 30, so we can place the remaining numbers accordingly. The filled-in magic square will look like this:
```
14 x 10
x 16 x
8 x 12
```
x denotes the remaining cells that need to be filled. You can fill in those cells with the remaining numbers: 2, 4, and 18.
To show your work, try explaining the steps you took to arrive at the final solution. For example, write down the rules you followed and the deductions you made along the way. You can also draw a diagram to visualize the process.
Now, let's move on to the second magic square:
1. Again, we need to identify the sum that all rows, columns, and diagonals need to have, which is 45.
2. Place the number 18 in the center cell, as it will be the middle value of each row, column, and diagonal.
3. Similarly to the previous magic square, let's place the number 24 in one corner and deduce that the opposite corner will be 6. Now, the remaining two corners will be 3 and 27.
4. Knowing that the opposite edge cells need to add up to 18 (the value of the center cell), we can fill in the remaining edge cells. In this case, the remaining edge cells will be 9 and 15.
5. Fill in the remaining cells so that each row, column, and diagonal has a sum of 45. The filled-in magic square will look like this:
```
3 x 15
x 18 x
27 x 9
```
Again, x denotes the remaining cells that need to be filled. Fill in those cells with the remaining numbers: 21, 12, and 6.
Remember to show your work by explaining and documenting each step you took to reach the solution. Good luck!