Question

The Temple of Heaven in Beijing is a circular structure with three concentric tiers. At the center of the top tier is a round flagstone. A series of concentric circles made of flagstone surrounds this center stone. The first circle has nine stones, and each circle after that has nine more stones than the last one. If there are nine concentric circles on each of the temple's three tiers, then how many stones are there in total (including the center stone)?

Short answer

Answer: There are a total of 1216 stones in the Temple of Heaven, including the center stone.

Step-by-step solution

Calculate the number of stones in each circle

In each circle, the number of stones increases by 9. Therefore, we can find the number of stones in each circle using the arithmetic progression formula: A_n = A_1 + (n-1)d Where A_n is the number of stones in the nth circle, A_1 is the number of stones in the first circle (9 stones), n is the circle number, and d is the difference between the number of stones in each circle (9).

Calculate the stones in 9 concentric circles

We will now find the number of stones in each of the 9 concentric circles, using the arithmetic progression formula from Step 1: Circle 1: A_1 = 9 + (1-1)9 = 9 Circle 2: A_2 = 9 + (2-1)9 = 18 Circle 3: A_3 = 9 + (3-1)9 = 27 Circle 4: A_4 = 9 + (4-1)9 = 36 Circle 5: A_5 = 9 + (5-1)9 = 45 Circle 6: A_6 = 9 + (6-1)9 = 54 Circle 7: A_7 = 9 + (7-1)9 = 63 Circle 8: A_8 = 9 + (8-1)9 = 72 Circle 9: A_9 = 9 + (9-1)9 = 81

Find the total number of stones in all 9 concentric circles

Now we can add the number of stones in each concentric circle to find the total number of stones for a single tier: Total stones = A_1 + A_2 + A_3 + A_4 + A_5 + A_6 + A_7 + A_8 + A_9 Total stones = 9 + 18 + 27 + 36 + 45 + 54 + 63 + 72 + 81 Total stones = 405

Calculate the total number of stones for all 3 tiers

Since there are 3 tiers in the Temple of Heaven, each with 9 concentric circles, we can multiply the total number of stones in a single tier by 3 to find the overall number of stones: Total stones for 3 tiers = 3 * 405 = 1215

Include the center stone in the count

Finally, we need to include the center stone in our count. Since there is only one center stone, we simply add it to the total number of stones for all 3 tiers: Total stones (including center stone) = 1215 + 1 = 1216 The Temple of Heaven in Beijing has a total of 1216 stones, including the center stone.

Key concepts

Concentric Circles

Concentric circles are a set of circles that share the same center. Imagine drawing a small circle and then drawing a bigger one around it, without changing the point at which your compass needle is placed. In the Temple of Heaven, these circles are formed by flagstones that increase in number with each level.

The arrangement starts with the innermost circle having a few stones, and each subsequent circle has more stones than the previous one.

• The number of stones in each new circle grows consistently by nine stones in this scenario.
• This means if the first circle has 9 stones, the next will have 18 stones, followed by 27 stones, and so on.

These concentric circles add a beautiful symmetry and structure to the design of the temple's tiers.

Algebra

Algebra is like a language of symbols, variables, and mathematics. It allows us to solve problems by finding the relationships between quantities. In the context of the Temple of Heaven, algebra helps in determining the number of stones in each concentric circle using a formula.

We use the arithmetic progression formula: \[A_n = A_1 + (n-1)d\] where:

• \(A_n\) is the number of stones in the nth circle.
• \(A_1\) is the number of stones in the first circle (9 stones).
• \(n\) is the circle number.
• \(d\) is the constant difference in stones between each circle (9 in this case).

With this formula, we can calculate the stone count for any circle quickly and efficiently, allowing for seamless transitions between mathematical concepts and practical applications.

Temple of Heaven

The Temple of Heaven, located in Beijing, China, is a masterpiece of architecture and symbolism. As a UNESCO World Heritage site, it holds a great deal of cultural significance. The design is an excellent example of how architecture can represent concepts—like symmetry and harmony—through its circular patterns.

The temple itself consists of three main tiers, each defined by concentric circles of flagstones. These circles aren't just artistic; they symbolize concepts like eternal life and the cosmos, reflecting ancient Chinese philosophy. Understanding the structure of these circles not only helps one appreciate the temple's beauty but also the mathematical ingenuity involved in its creation.

Mathematical Problem Solving

Mathematical problem solving involves a structured approach to breaking down complex questions into more manageable parts. Let’s walk through how we achieve this step-by-step in the exercise about the Temple of Heaven.

• First, we calculate how many stones each circle contains by applying the arithmetic progression formula.
• Next, we sum up the stones across all circles to get a total for one tier.
• Then, multiply this total by the number of tiers (3 in this case) to account for the entire temple.
• Finally, don't forget to include the single center stone that is counted in the total tally.

The beauty of mathematical problem solving is in its methodical and logical nature, ensuring every step is covered, leaving no room for error. This structured problem-solving process helps students tackle similar educational tasks more confidently.