Arrange the non zero natural numbers in the form of the bottom, and in this arrangement, in what row

1.Arrange the natural numbers in the following form, in such an arrangement, in which row and column is 195 in it?

The first solution:

Look diagonally, every diagonal line.

The first line is 1 number: 1

2 numbers in the second row: 3,2

3 numbers in the third row, 4, 5, 6

The total number is 1+2+3+.

Odd diagonal rows, from bottom left to top right.

Even-numbered diagonal rows, from top right to bottom left.

The sum of each number, the position, the number of rows and the number of columns, is equal to the number of rows in the oblique row plus 1, for example: 8, in the 2nd row, in the 3rd column, it is in the oblique row, that is, 2+3-1=4 rows.

In Grade 6, you should have learned the summation of continuous natural numbers.

1+2+3+。。n=n(n+1)/2

So 195 is diagonally at the 20th.

The first 19 lines, total: 19 (19 + 1) 2 = 190 numbers.

So: 195 is the 5th number of the 20th oblique row.

The 20th oblique row, the first number is: 190+1=191, the last number is: 20(20+1) 2=210, the 20th oblique row, is an even number row, from the top right to the bottom left.

Column 195 is in: 20 - (195-191) = 16 columns.

Behavior of 195: 20 + 1 - 16 = 5

So 195 is in row 5, column 16.

The second solution:

Diagonally, Article 1, 1 number, Article 2, 2 numbers, Article N, N number.

It adds up to n(n+1) 2 numbers, the odd number is the number from the bottom left to the top right, and the even number is the number from the top right to the bottom left.

195, first find the largest full integer bar, 19 * 20 2 = 190, that is, 195, on the 20th bar, count 5 numbers from the top right to the bottom left.

So on row 5, column 16.

So 195 is in row 5, column 16.

The number in row 18, column 22, is 759. a1=1=1*2/2

a2=1+2=3=2*3/2

a3=1+2+3=6=3*4/2

a4=1+2+3+4=10=4*5/2

a20=20*21/2=210

200=210-10, then 200 is in column 11 (1+10) of row 10 (20-10).

a39=39*40/2=780

Ways to find patterns:

1. Mark the serial number: the problem of finding the law usually gives a series of quantities in a certain order, requiring us to find the general law according to these known quantities. Find out the pattern, usually the package serial number.

Therefore, it is easier to find the mystery when you compare the variables and serial numbers together.

2. Fibonacci number sequence: Each number is the sum of the first two numbers.

3. Equal difference number series method: the difference between every two numbers is equal.

4. Jumping grid method: you can look at the interval to see what is the relationship between the separated numbers, such as 14,1,12,3,10,5, the odd number of terms into an equal difference series, and the even number of terms into an equal difference series, so you should fill in 8 next.

a1=1=1*2/2

a2=1+2=3=2*3/2

a3=1+2+3=6=3*4/2

a4=1+2+3+4=10=4*5/2

a20=20*21/2=210

200=210-10, then 200 is in column 11 (1+10) of row 10 (20-10).

a39=39*40/2=780

The number in row 18, column 22, is 759.

It has been observed that the arrangement of this natural number table has the following characteristics:

Each number in the first column is perfectly squared and is exactly equal to the square of the number of rows it is in, i.e. the first number in the nth row is n 2;

The nth digit in the first row is (n-1) 2 +1;

In the nth row, from the 1st number to the nth number, decrement by 1;

In the nth column, from the 1st royal number to the nth number, the number is increased by 1 Therefore, from the 50th line above, the number in the 51st column from the left of the divine cavity should be the 50th number of the 51st column, that is, [(51-1) 2 +1]+49=2550, so the answer is: 2550

The following rules should be expressed, observe the rules: find the intersection of the first row and the first column 1, the intersection of the second row and the second column is 3, the intersection of the third row and the third column is 7, the intersection of the fourth row and the fourth column is 13, and the law is increasing by 2The intersection of the fifth row and the fifth column is 21, and according to the law, each intersection is the number of rows and the number of columns -1. For example, the intersection point of the fifth column in the fifth row is 5 4+1=21

Get: 45 44+1=1981The intersection of column 45 in row 45 is 1981

The singular column and column are from left to right through the intersection point upwards. 2013 is 32 rows up from the intersection point. 45 rows - 32 rows = 13 rows, which is 45 columns in row 13.

45th conductor line 13.

Observation is in the form of a phalanx.

After the even-numbered square, start below. If it is odd power, starting from above, the closest to 2013 is 45*45=2025 2015-2013+1=13

So the few 12 numbers are 13 rows, obviously in column 45.

**The law of natural numbers is that around a region, the numbers extend naturally. 2-4 is to make a circle around 1, 5-9 is to circle around the block area of 1243, and 10-16 is to circle around the block area of 129438567. It follows from this that the first digit of an odd row (except the first row) = the square of the number of rows (even) in the previous row + 1, and the first digit of an even row = the square of the number of rows in that row; The first digit of an odd number column (except the first column) = the square of the number of columns (even) in the previous column + 1, and the first number of the even number of columns = the square of the number of columns in that column.

It is also known that the circle of odd rows and columns is counterclockwise, and the circle of even rows and columns is clockwise. The first position of the even-numbered row and the first position of the odd-numbered column are the end points of the circle.

According to the arrangement law of natural numbers, 2011 is between 44-46, so it may be in 46 rows or 46 columns, and because the first number in the 46th row is 46 2 = 2116 and the even number of lines is clockwise, so the natural number to the right in the 46th row is decreasing, and it takes 46 * 2-1 = 91 numbers to go around a circle, and 2116-2011 = 105 >91, so 2011 is not in row 46 nor in column 46, and the first number in column 45 = 45 2=2025 and the odd number column is circled counterclockwise, so 2011 is in the 15th row below 2025 (first row), so 2011 is in 15 rows 45.

Summary. Hello, this adds up to a total of 9 numbers for every two cases, and in 2023, there are 224 of these two columns, and there are 6 left, which happens to be the first column of the next two columns, so it should be in the 6th column of the 449 row.

As shown in the figure below, the natural numbers are regularly arranged as follows, so how many rows, columns is 2023 in?

Hello, I don't see ** here, please send it again**.

Hello, this adds up to a total of 9 numbers for every two cases, and there are 224 in the return to the bridge in 2023, and there are 6 left in these two columns, which happens to be the first column of the next two columns, so it should be in the 6th column of the 449th line of Xiaoxiao.

If we call an oblique row "oblique row", then the first "oblique row" has only one number, the second "oblique row" has two numbers, and the third "oblique row" has three numbers ,..., and so on, and the sum of the rows and columns of each number of each number in the first "diagonal row" is 2, the sum of the rows and columns of each number in the second "diagonal row" is 3, the sum of the rows and columns of each number in the third "diagonal row" is 4, and the sum of the rows and columns of each number in the fourth "diagonal row" is 5,...

In addition, odd "diagonal" numbers are arranged from the bottom left to the top right, and even "oblique" numbers are arranged from the top right to the bottom left.

According to these laws, 1995 1953 42, and 1953 1 2 3 4 ....62, that is, after 1995 numbers are arranged in the 62nd "oblique row", there are still 42 numbers arranged in the 63rd "oblique row".

It can be seen that the first 1953 numbers are arranged in 62 "oblique rows", and 1953 is in the last position of the 62nd "oblique row", which is the first column of the 62nd row.

Therefore, the remaining numbers are in the 63rd "oblique row", and the number of rows and columns of each number in this "oblique row" is equal to 64, 1954 in the first column of row 63, and one number in each row after that, the number of rows is subtracted by 1, and the number of columns is added by 1, so that 1995 is in the 42nd column of the 22nd row.

The last number of rows you give should be.

17 18 19 20 21 22 23 24 25 Bar 15125 This number should be in line 123, the 241st number in line n, and the last number in line n is n

Therefore, there are 122 whole rows before 15125. The last number on line 122 is 122*122 = 14884

15125 in line 123, 15125-14884=241.

Hello, glad to answer for you.

According to the order given in the figure, we find the pattern:

1. The value of the odd-numbered row in the first column (e.g., row 2n-1) is the square of the number of rows (i.e., (2n-1) 2);

2. The value of the even-numbered column in the first row (e.g., column 2n) is the square of the number of columns (i.e., (2n) 2);

And what is sought: 2011 = 1936 + 75 = 44 2 + 75 ; or 2011=2025-14=45 2-14 (obviously 2011 is closer to 45 2, 45 2 is in the first column of row 45).

So: 2011 is in row 45, column 15.