Core formula:
1. The total number of people in a square matrix = the square of the number of people on the outermost side (the core of the square matrix problem)
2. The number of people on each side of the outermost layer of the phalanx = (the total number of people in the outermost layer of the phalanx is 4)+ 1.
3. The total number of people in the outer layer of the phalanx is 2 more than that in the inner layer.
4. The total number of people removed in one row and one column = the number of people removed in each side × 2- 1.
Example 1 School students form a square, and the outermost number is 60. How many students are there in this phalanx?
A.256 people B.250 people C.225 people D. 196 people (2002 class A real question)
Analysis: The core of the phalanx problem is to find the number of people on each side of the outermost layer.
According to the relationship between the number of people around and the number of people on each side, we can know that:
Number of people on each side = number of people around ÷4+ 1, and the number of people on each side of the outermost layer of the phalanx can be calculated, then the total number of people in the whole phalanx queue can be calculated.
Number of people on each side of the outermost square: 60÷4+ 1= 16 (person)
The number of students in the whole phalanx is 16× 16=256 (people).
So, the correct answer is A.
The athletes who participated in the group gymnastics competition in the middle school sports meeting lined up in a square queue. If this square queue is to be reduced by one row and one column, it will be reduced by 33 people. How many athletes are there in the group gymnastics performance?
The analysis figure below shows a square queue with five rows and five columns. As can be seen from the figure, the number of people in each row and column of the square is equal; The number of people on each side of the outermost layer is 5, and going to a row or a column means going to 9 people, so the following formula can be obtained:
Total number of people with one row and one column removed = number of people removed on each side × 2- 1.
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
Analysis: The core of the phalanx problem is to find the number of people on each side of the outermost layer.
The number of people who removed a row and a column in the original question is 33, so the number of people who removed a row (or a column) is = (33+ 1) ÷ 2 = 17.
The total number of people in a square matrix is the square of the number of people on the outermost sides, so the total number of people is 17× 17=289 (people).
The following exercises are for everyone:
1. Xiaohong first used up all the five cents of a regular triangle she had saved, and then changed it into a square, which happened to be used up. If each side of a square uses 5 coins less than each side of a triangle, the total value of all nickels in Xiaohong is:
A. 1 yuan B.2 yuan C.3 yuan D.4 yuan (2005 Central Zhenti)
2. The guard of honor was arranged in a square, and several people were arranged for the first time. As a result, there were more people 100; The second time, there were 3 more people in each row and 29 fewer people in each column than the first time. What is the total number of honor guards?
Answer: 1. C 2。 500 people.