These problems were solved by 8th grader Beverly W. and myself working together. The answers will be compared to the official solution published one week later.

]]>According to the published solution:

]]>- Correct
- We inadvertently converted half a foot (0.5 ft) to 5 inches instead of 6 inches, which would have given us the correct answer.
- From the last day of school, we overcounted the number of days remaining in June by one. We also carried over our incorrect answer from part two. Otherwise, we would have calculated the correct answer.

According to the published solution:

]]>- Correct
- Incorrect. I misinterpreted the problem as the number of different sequences of the last 5 fireworks out of the 15 available. Rather, those five were already determined. Therefore, instead of the number of permutations of 15 items arranged 5 at a time, the problem asks for the number of ways to arrange the remaining 10 fireworks.
- Approximately correct. Overlooking the last sentence, which asks for the answer to be rounded to the nearest tenth, I instead rounded up to the nearest whole number.

According to the published solution:

]]>- Correct
- Correct
- Incorrect. I forgot to square 5,280 in the denominator, which would have resulted in the correct answer. Furthermore, I did not conduct a reasonability check, which would have raised a red flag when considering that my answer was more than four times the surface area of the entire Earth, 1.96 million square miles.

All three answers are correct according to the published solution. For part three, my answer was expressed as 407 inches, which is approximately 34 feet.

]]>According to the published solution:

]]>- Incorrect. My solution counted permutations rather than combinations. The number of combinations is small enough that they can quickly be listed and counted.
- Correct
- Incorrect. At the time, I could not figure out how to solve the problem. In hindsight, I could have solved it by using one of my favorite techniques: a table diagram. In the image below, I constructed a 10 x 10 table to see which of the first 10 lockers are closed after the first ten students exited the school. The result is that only the lockers numbered as perfect squares were closed. Assuming that this pattern continues to the 900th locker, it is a safe guess that all lockers numbered as perfect squares will be the only ones that are closed. It is also interesting to observe that the total number of lockers, 900, is a perfect square. The square root of 900 is 30, which means there are a total of 30 perfect squares from 1 to 30.