Photos taken by David Pleacher: Portland Head, Pemaquid Point Light, Bass Harbor Head Light
On the coast, there are three lighthouses.
The first light shines for 3 seconds then it is off for three seconds.
The second light shines for 4 seconds then it is off for 4 seconds.
The third light shines for 5 seconds then it is off for 5 seconds.
All three lights have just come on together.
When is the first time that all three of the lights will be off together?
When is the next time that all three lights will come on at exactly the same moment?
Solution to the Problem:
The first time that all three of the lights will be off together is at six seconds.The next time that all three lights will come on at exactly the same moment is after 120 seconds (or at the beginning of the 121st second).
Look at the chart below to see that six seconds is the first time that all three lights will be off.
To determine the next time that all three lights will be on, find the least common multiple of 6, 8, and 10 seconds.
6 = 2 x 3
8 = 2 x 2 x 2
10 = 2 x 5
Hence, you will need three 2s, one 3, and one 5:
2 x 2 x 2 x 3 x 5 = 120.
But 120 seconds represents the end of the cycle and all lights would be off during that second. So the lights would all be on at the beginning of the 121st second.
James Alarie sent in the following picture to help visualize what is happening. Note that all lights are on at the 25th second but the question was when all three lights will come on at exactly the same moment, and that doesn't happen until the 121st second.
Brijesh Dave sent in the following solution:
Correctly solved by:
| 1. James Alarie | Flint, Michigan |
| 2. Rob Miles | Northbrook, Illinois |
| 3. Sreeroopa Sankararaman | Singapore, Singapore |
| 4. Brijesh Dave | Mumbai City, Maharashtra, India |
| 5. Ivy Joseph | Pune, Maharashtra, India |
| 6. Kimberly Howe | Vienna, Virginia |
| 7. Kelly Stubblefield | Mobile, Alabama |
| 8. Travis Riggs | Strasburg, Virginia |