Variation on A Thousand Points of Light A thousand simple on-off lamps are lined

Question

Variation on A Thousand Points of Light A thousand simple on-off lamps are lined up and are initially off. Person 1 flips every switch once. Person 2 flips every second switch 2 times. Person j flips every jth switch j times. The last one flips switch 1000 one thousand times. Which lamps are on when it is over? Give a precise description of this set of lamps as simply as you can(*). (*) Degrees of Simplicity of your Final Description: Just OK: Using the word divisor in your final description. A bit better: Referring to prime factors or exponents in your final answer, but not divisor. Best: Not using any of the words divisor, prime, factor, power, or exponent.

Explanation / Answer

Here we first think of a person who operates which bulb, now given information it is cleared that second person will do all the even number switches and accordingly say #100 person will operate all the bulbs that end with two zeros. Now question is that who will operate , for example, bulb number 44 :

Persons numbered: 1 & 44, 2 & 22, 4 & 11, ...........

or we conclude that all the factors ( numbers that divide 44) of 44 will be in pair . That shows that every person who switches a bulb on, there will be another person with him, who will switch it off. so that the bulb will again remain in its previous original position after this execution.

Now question arises that why are not all the bulbs off ?

Let us consider about bulb 16 :-

The factors are: 1, 2, 4,8 and 16
OR 1 & 16, 2 & 8 and 4

In this example previously the pairs cancelled , if 2 flicks the switch one way then 8 would surely flick it back but in case of 4, there is no other person to reverse his operation so that its operation will remain as it is that will change the original condition and this will be true for all the perfect sqaure numbers.

Now we have that starting from 1 to 1000 ( 1,4,9,16,25,36,49,64,81,100,121,169,196,225,256,289,324,400,441,484,529,576,625,676,729,784,841,900,961) there are total 31 perfect squares that means in the last total 31 bulbs will remain on.

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