Day 16: Restoring the Bridges

Previously

After carefully calculating the time it would take to traverse the islands, you discovered that the first fragment of the Veil of Wonder was already in your possession. The Guardian revealed that your next step is to travel to the Main Island and restore the missing bridges. Only then can you access the remaining islands and gather the other fragments needed to restore the Veil.

Narrative

Stepping onto the Main Island, you feel the charged energy of ancient ruins, though the island remains eerily still. The bridges to some of islands lie dormant, waiting to be restored. Suddenly, the Guardian’s voice echoes in your mind.

"I can reach you telepathically, but must conserve my strength. Deep within this island is a power source. Activate it, and the bridges will reconnect. You must find the chamber with the mechanism."

You navigate the ruins and find the central chamber — a room with four large wheels, each spinning at different speeds. Above them rests an ancient device meant to cast the beam that will restore the bridges. But the beam must be released precisely when all four wheels align at their starting positions.

"You’ll need to calculate the exact moment the wheels align," the Guardian continues. "There’s a tool there to cast the beam. You only get one chance."

With the wheels spinning in rhythm, you begin studying their motion. Precision and timing are your only hope to reignite the bridges and continue your quest for the Veil of Wonder.

Objective

Calculate the precise time when all four wheels align at their starting positions and use the provided tool to release the energy beam, restoring the bridges to the surrounding islands.

Challenge

Four wheels are spinning at different speeds. You observe the following:

The first wheel completes one full rotation every 1 minute.
The second wheel completes one full rotation every 2 minutes.
The third wheel completes one full rotation every 3 minutes.
The fourth wheel completes one full rotation every 4 minutes.

Assuming they all start at the same position, how many minutes will it take for all four wheels to align at the starting position again?

Hint 1

If you wait 4 minutes, every wheel except the third will be at the start position. How many times do you have to repeat this?

Hint 2

You need to find the Least Common Multiple of these four numbers.

Answer

Wrong answer!

Outcome

As you release the energy beam at the precise moment, the chamber hums with ancient power. The wheels slow to a halt, and the bridges connecting the islands begin to shimmer into existence, glowing faintly as they reconnect the Main Island to the others.

A rush of excitement fills you — the path ahead is open, and the journey to collect the fragments of the Veil of Wonder can finally continue. With the first fragment already in your possession from the Guardian's island, your next step is clear: to travel to the nearest island and retrieve the second fragment.