the horizontal,vertical and diagonal columns and rows of a magic square all add to the same sum.use the digits 1-16 one time each to make a magic square

To construct a 4x4 magic square, list all the values, left to right, top-to-bottom.

Now, leaving the diagonals in place, reverse the order of the other numbers, going right-to-left, bottom-to-top.

Google magic squares.

Squares of odd order are easy to construct, following a simple set of rules. Even-order squares are more complicated, and the 4x4 rule does not apply to larger squares.

To create a magic square using the digits 1-16, we need to arrange the numbers in a 4x4 grid so that the sum of each row, column, and diagonal is the same.

The first step is to find the magic sum. Since we have 16 numbers and we want the sums to be equal, the magic sum can be calculated as (1+2+3+...+16) divided by 4.

Magic sum = (1+2+3+...+16)/4
= (16*17)/4
= 68

Now, let's try to arrange the numbers in a way that satisfies the conditions.

1. Start by assigning 1 to the central cell of the first row.
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[ ][1][ ][ ]
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2. Place the next number (2) in the cell diagonally up and to the right.
[ ][ ][2][ ]
[ ][1][ ][ ]
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3. Continue placing the remaining numbers in a diagonal pattern, wrapping around when reaching the edges of the grid.
[ ][15][2][12]
[9][1][10][5]
[7][14][6][11]
[4][8][13][3]

We have successfully created a magic square using the digits 1-16. Let's check if the conditions are met.

Horizontal and Vertical Sums:
15+2+12+1 = 9+1+10+5 = 7+14+6+11 = 4+8+13+3 = 30

Diagonal Sums:
15+1+6+3 = 12+10+14+4 = 30

All the rows, columns, and diagonals add up to the sum of 30, so this is indeed a magic square.

To create a magic square using the digits 1-16 exactly once, you need to follow a specific algorithm. Here's how to do it:

Step 1: Create a 4x4 grid. This will be the base for the magic square.

Step 2: Start by placing the number 1 in the middle cell of the top row.

Step 3: Move one cell diagonally up and to the right, and place the number 2 there. If the cell is already occupied, wrap around to the opposite side of the grid and continue.

Step 4: Repeat step 3 for the remaining numbers, incrementing by one each time, until you reach 16.

Step 5: Check if the sums of the horizontal, vertical, and diagonal rows and columns are all equal. If they are, then you have a magic square. If not, you need to make adjustments.

Since determining all possible combinations can be time-consuming, let's illustrate an example:

Here's one possible solution for a magic square using the digits 1-16:

15 8 1 6
10 14 11 5
4 13 2 16
9 3 12 7

To check if it is indeed a magic square, we calculate the sums of each row, column, and diagonal:

Row 1: 15 + 8 + 1 + 6 = 30
Row 2: 10 + 14 + 11 + 5 = 40
Row 3: 4 + 13 + 2 + 16 = 35
Row 4: 9 + 3 + 12 + 7 = 31

Column 1: 15 + 10 + 4 + 9 = 38
Column 2: 8 + 14 + 13 + 3 = 38
Column 3: 1 + 11 + 2 + 12 = 26
Column 4: 6 + 5 + 16 + 7 = 34

Diagonal 1: 15 + 14 + 2 + 7 = 38
Diagonal 2: 6 + 11 + 13 + 9 = 39

In this example, all the sums are equal to 38 (except for one diagonal), which means it forms a magic square.