Rules for Figures (Arrangements of Equilateral Triangles and Squares)
When figures are arranged according to a fixed rule, patterns also appear in the number of objects or items used.
The key to solving these problems is to find “how many are added when the number increases by 1.”
What You Will Learn
When equilateral triangles are arranged horizontally using matchsticks,
the numbers increase by 2 each time: the first uses 3 sticks, the second uses 5 sticks, the third uses 7 sticks, and so on.
This is an arithmetic sequence, so the number of sticks in the nth figure can be written as an equation:
nth figure = 3 ＋（n－1）× 2 ＝ 2n ＋ 1
Example
Squares are arranged in a single horizontal row using matchsticks. Express the number of matchsticks used in the nth figure as an equation, and find the number of sticks used in the 10th figure.
Solution:
① First, count the number of sticks.
1st: 4 sticks, 2nd: 7 sticks, 3rd: 10 sticks, …
② We can see that the number increases by 3 each time (common difference = 3).
③ The first figure uses 4 sticks, and 3 sticks are added (n－1) times, so:
nth figure = 4 ＋（n－1）× 3 ＝ 3n ＋ 1
④ Calculate the number of sticks in the 10th figure.
3 × 10 ＋ 1 ＝ 31 sticks
Squares are arranged in a single horizontal row using matchsticks. Express the number of matchsticks used in the nth figure as an equation, and find the number of sticks used in the 10th figure.
Solution:
① First, count the number of sticks.
1st: 4 sticks, 2nd: 7 sticks, 3rd: 10 sticks, …
② We can see that the number increases by 3 each time (common difference = 3).
③ The first figure uses 4 sticks, and 3 sticks are added (n－1) times, so:
nth figure = 4 ＋（n－1）× 3 ＝ 3n ＋ 1
④ Calculate the number of sticks in the 10th figure.
3 × 10 ＋ 1 ＝ 31 sticks
Key Points
Remember
- First, make a table and check “how many are added when the number increases by 1.”
- If the amount added is the same each time, you can use the equation for an arithmetic sequence.
- Focusing on the “sides shared with neighboring figures (the sides that are connected)” makes it easier to create the equation.
Practice Problems
- Arrange equilateral triangles horizontally using matchsticks. Express the number of matchsticks used in the nth figure as an equation, and find how many sticks are needed for the 8th figure.
- Arrange squares in n rows and n columns. When there is 1 row and 1 column, there is 1 square; when there are 2 rows and 2 columns, there are 4 squares. Express the number of squares in n rows and n columns as an equation, and find how many squares there are in 5 rows and 5 columns.
Answers and Explanations
- Answer: Number of matchsticks in the nth figure = 2n+1. 8th figure = 17 sticks
Explanation: 1st: 3 sticks, 2nd: 5 sticks, 3rd: 7 sticks, … This is an arithmetic sequence with a common difference of 2. The first triangle uses 3 sticks, and each subsequent figure increases by 2 sticks because one side is shared. nth figure = 3+(n-1)×2 = 2n+1. 8th figure = 2×8+1 = 17 sticks. - Answer: Number of squares in n rows and n columns = n². In 5 rows and 5 columns, there are 25 squares
Explanation: 1 row and 1 column = 1 = 1², 2 rows and 2 columns = 4 = 2², 3 rows and 3 columns = 9 = 3², and so on. n rows and n columns = n². 5 rows and 5 columns = 5² = 25 squares.