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(*). Next, think about which powers of primes will be on or off at the end. Then consider products of prime powers..... (*) 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

Number of times a lamp is flipped will be the number of factors of the lamp number. Now if the number of factors is even then the final state of lamp will be same as the initial state otherwise it'll be opposite.

For eg. if lamp is numbered 12, then it will be visited 6 times because it has 6 factors (1,2,3,4,6,12) which is an even number, and hence the lamp will be in same state as initial (closed).

Any Number say XX can be factorized and represented as the product of power of all prime factors in the following pattern - 2a3b5c...2a3b5c...

For eg. 1212 can be represented as 223150...223150...

Number of factors of XX will be (a+1)(b+1)(c+1)..(a+1)(b+1)(c+1).. (All the possible ways of choosing factors)

Now only if all the powers a,b,c..a,b,c.. are even, all the components of product (a+1)(b+1)(c+1)..(a+1)(b+1)(c+1).. will be odd and hence the product will be odd. Therefore, all the perfect squares will have even powers and eventually will have odd number of factors.

Initially, all the lamp are off. Hence, at the end of the process, all the lamps with a number which is not a perfect square will have the same state i.e. closed.

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