The first stage of research: this method of filling in forms can be divided into even-numbered cells and odd-numbered cells. Take even grids as columns, such as 1- 16. The sum of all numbers/line number = the value of each line, (1+16) *16/2 =136, 136/4=34, so the number of each line should be 34. 34 is an even number. Start looking for the combination of four numbers in 1- 16 and get 34. Note that 34 is an even number, so four values must be a combination of two even numbers+two odd numbers or four even numbers or four odd numbers. You can simply combine first, 1, 2, 15,16; 3、4、 13、 14; 5、6、 1 1、 12; 7、8、9、 10。 Fill in the number of 16 first. Get 1 2 15 16.
3 4 13 14
5 6 1 1 12
7 8 9 10
Then adjust by selecting the value of the vertical row, fix any row and adjust the position of the numbers in other rows, only change the columns and keep the rows unchanged. These numbers are represented by Xij, where I is the number of rows and j is the number of columns. Then when adjusting, only the value of j is changed, and the value of I is not changed. If the first column is fixed, that is, x11,X 12=2, X 13= 15, and x14 =/kloc-0. The principle is that changed columns in previous rows cannot be changed. I started to change the second line, and I chose to use 14 for 3 to get 1 2 15 16.
14 4 13 3
5 6 1 1 12
7 8 9 10
Of course, you can also exchange 3 and 13, so I won't introduce it. Next, adjust the third line, which means that 5 and 12 cannot be interchanged, because the previous changes have been made. Select 5 and 1 1 to get 1 2 15 16.
14 4 13 3
1 1 6 5 12
7 8 9 10
Next, there is only one adjustment method to adjust the fourth row, that is, the exchange of 7 and 8, because the second row has already gone through the first and fourth columns, and the third row has gone through the first and third columns, so only the first and second columns are left. The adjusted value is 1 2 15 16.
14 4 13 3
1 1 6 5 12
8 7 9 10
As you can see, the number1+14+11+8 = 34 in the first column meets our requirements. The second column is adjust in this way, and that adjusted column remain unchanged. After the last column adjustment
1 2 15 16
14 13 4 3
1 1 12 6 5
8 7 9 10。
Next, keep the rows and columns unchanged and adjust the diagonal value, that is, move the rows or columns as a whole.