# ASTEROID BLASTER

Date of the Problem
September 26, 2022

In the video arcade game Asteroid Blaster, there are a total of five rounds on the first level. During each round, asteroids appear on the screen and players must blast as many as possible before time expires. To advance to the next round, a player must destroy all the asteroids in the current round. This becomes more difficult to accomplish because the number of asteroids increases with each round. The table shows the number of asteroids for each of the first three rounds. If this pattern continues, how many asteroids will there be to destroy in Round 4?

We can’t really tell much by just looking at Round 1. But for Round 2, we can see there are 8 asteroids. The pattern could be 2 × 4 = 8 or 2 × 2 × 2 = 8. Let’s look at Round 3 to get more information. Round 3 has 27 asteroids, which is either 3 × 9 or 3 × 3 × 3. It looks like the pattern is 13, 23, 33, … In which case, the next round would have 43 = 4 × 4 × 4 = 64 asteroids.

As the number of asteroids increases with each round of Asteroid Blaster, so does the amount of time within which a player must blast all the asteroids. The table shows the time limit, in seconds, for each round. If this pattern continues, what is the maximum number of seconds allotted to complete all five rounds of this level?

In the first round, there are 2 seconds. The second round has a time limit of 2 × 2 = 4 seconds. Then the third round gives the player 2 × 2 × 2 = 8 seconds. In Round 4, a player has 2 × 2 × 2 × 2 = 16 seconds. Therefore, a player will have 2 × 2 × 2 × 2 × 2 = 32 seconds to complete Round 5. To complete all five rounds, a player can take a maximum of 2 + 4 + 8 + 16 + 32 = 62 seconds.

The number of points awarded for blasting each asteroid also increases with each round. The table shows the number of points awarded for each asteroid shot per round. How many points will a player earn for completing Rounds 1 through 4 and blasting 2/5 of the asteroids in Round 5 before time expires?

Image

For blasting every asteroid in Rounds 1 through 4, a player will earn (1 × 10) + (8 × 20) + (27 × 30) + (64 × 40) = 10 + 160 + 810 + 2560 = 3540 points. The total number of asteroids in Round 5 is 5 × 5 × 5 = 125 asteroids. If a player blasts 2/5 of the asteroids in that round, that is equal to (2/5) × 125 = 50 asteroids. Since it’s Round 5, 50 points are awarded for each asteroid blasted, which is 50 × 50 = 2500 points. Therefore, the total number of points the player will earn is 3540 + 2500 = 6040 points.

Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.

Page 2 contains ONLY PROBLEMS. ♦

Math topic
CCSS (Common Core State Standard)
Difficulty