## May 2018

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My May diary included the following brainteaser.

On the coast there are three lighthouses.

The first light shines for 3 seconds, then is off for 3 seconds.

The second light shines for 4 seconds, then is off for 4 seconds.

The third light shines for 5 seconds, then 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?

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**• Solution**

Lots of scope for elementary confusions here between:

- measuring
*versus*counting, i.e.the instant*t*= 1*versus*"the first second" (which runs from*t*= 0 to*t*= 1)

and

- the lights
*being*off (or on)*versus*the lights*coming*off (or on).

A diagram tells the tale. Taking the lights as having come on
together at time *t* = 0, they will first *be* off together from *t* = 5 to *t* = 6. They will next *come*
on at exactly the same moment at *t* = 120.

As a supplementary we might turn the two questions round:

When is the first time that all three of the lights willcomeoff together?

When is the next time (i.e. after the first three seconds) that all three lights willbeon at exactly the same moment?