Solution

Did you say the answer was 20 seconds - the LCM of 1, 2, 4 and 5? I think it is not 20 but 30 seconds. It is the one second for which the words glow, that is causing all the mischief.

Just for the ease of understanding, imagine that each word immediately upon lighting up lets off a small beep. The first word, therefore, gives a beep on lighting up. It remains glowing for one second and, then, goes off for one second. Then, it glows again after making a beep. So, the interval between the two beeps is two seconds.

Similarly, the second word gives a beep on lighting up; remains glowing for one second; and, then, goes off for two seconds. It lights up again, giving a beep. In this case, the interval between two beeps is three seconds.

Likewise, the interval between two beeps by the third word will be five seconds, and by the fourth word will be six seconds.

When will all the four words light up simultaneously? Obviously that will happen when you hear them beep at the same time.

Just as an example, consider only the first two words: "New" beeps every two seconds; so, starting from 0 (when the swith is pressed), it will glow at 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ..... seconds. Similarly "Pankaj", which beeps every three seconds, will glow at 0, 3, 6, 9, 12, 15, 18, 21, .... seconds. These two words will glow simultaneously when they beep together. That will happen at 0, 6, 12, 18, ... seconds. So, they glow together at intervals of 6 seconds. But what is 6? It is the LCM of 2 and 3.

Now, similarly, if we consider the first three words: "New" beeps every 2 seconds, "Pankaj" beeps every 3 seconds, and "Super" beeps every 5 seconds. They will beep together every "t" seconds where "t" is the LCM of 2, 3, 5. LCM of 2, 3, 5 is 30. So, the three words will glow together every 30 seconds.

Finally, we consider all the four words. In this case, we have to consider the LCM of 2, 3, 5 and 6, which is also 30. So, all the four words glow together at intervals of 30 seconds.

Just for fun, let's work out other combinations: "New" and "Pankaj" will glow simultaneously every 6 seconds (LCM of 2 and 3); "New" and "Super" will glow simultaneously every 10 seconds (LCM of 2 and 5); "New" and "Shoppe" will glow simultaneously every 6 seconds (LCM of 2 and 6); "Pankaj" and "Super" will glow simultaneously every 15 seconds (LCM of 3 and 5); "Pankaj" and "Shoppe" will glow simultaneously every 6 seconds (LCM of 3 and 6); "New", "Pankaj" and "Super" will glow simultaneously every 30 seconds (LCM of 2, 3 and 5); "New", "Pankaj" and "Shoppe" will glow simultaneously every 6 seconds (LCM of 2, 3 and 6); "New", "Super" and "Shoppe" will glow simultaneously every 30 seconds (LCM of 2, 5 and 6); "Pankaj", "Super" and "Shoppe" will glow simultaneously every 30 seconds (LCM of 3, 5 and 6); and, of course, all the four words will glow simultaneously every 30 seconds as we have already determined.

You don't think I am right? Okay, I have tried to create a table. The time under the "Time" column is in seconds; I am not interested in the hour and minute part. Let's say I press the switch (at 00 seconds) that causes all the words to glow at once. For one second after that (00 - 01 seconds) all the words will glow together, and then go off. After one second, only the word "New" will light up for one second (02-03 seconds) while the rest will be off. From 03 to 04 seconds, the first word goes OFF for a second and "Pankaj" lights up. That is how it continues. This is what I get: