During a five-day festival, lanterns are lit at the beginning of each evening's celebration. They are lit for two hours each evening and then extinguished. All lanterns are new at the start, and each lantern can burn for a total of 6 hours. On the first day, one lantern is lit, on the second day two lanterns are lit, and so on, adding one more lantern each day. Traditionally, one new lantern is started each day. Fiona wonders if fewer lanterns could be used if it is not necessary to use a new lantern each day. If the lanterns are all the same at the start, for how long must each lantern be able to burn?

To solve the problem, we need to analyze the lantern usage over the five days of the festival. ### Known Conditions: 1. Each lantern can burn for a total of 6 hours. 2. Lanterns are lit for 2 hours each evening. 3. On Day 1, 1 lantern is lit; on Day 2, 2 lanterns are lit; on Day 3, 3 lanterns are lit; on Day 4, 4 lanterns are lit; and on Day 5, 5 lanterns are lit. 4. A new lantern is traditionally started each day. ### Step-by-Step Analysis: 1. **Calculate Total Lantern Usage:** - **Day 1:** 1 lantern for 2 hours - **Day 2:** 2 lanterns for 2 hours each (total 4 hours) - **Day 3:** 3 lanterns for 2 hours each (total 6 hours) - **Day 4:** 4 lanterns for 2 hours each (total 8 hours) - **Day 5:** 5 lanterns for 2 hours each (total 10 hours) Now, let's calculate the total hours of lantern usage over the five days: \[ \text{Total hours} = 2 + 4 + 6 + 8 + 10 \] \[ \text{Total hours} = 30 \text{ hours} \] 2. **Determine the Number of Lanterns Used:** - If we use \( n \) lanterns, each lantern must be able to burn for a total of \( x \) hours. - The total burn time of \( n \) lanterns is \( n \cdot x \). We need this total burn time to be at least equal to the total hours of usage: \[ n \cdot x \geq 30 \] 3. **Calculate the Minimum Burn Time per Lantern:** - If we use the maximum number of lanterns lit on any day, which is 5 (on Day 5), we can set \( n = 5 \): \[ 5 \cdot x \geq 30 \] Dividing both sides by 5 gives: \[ x \geq 6 \text{ hours} \] ### Conclusion: To ensure that fewer lanterns can be used without needing to start a new lantern each day, each lantern must be able to burn for at least **6 hours**. This means that if each lantern can burn for 6 hours, we can use a total of 5 lanterns over the five days, which matches the total burn time required.