Question 977106: In the regional power station there are a thousand switches on the main console and these are numbered in order of 1-1000. An engineer is employed to control the switches. At the beginning of the shift, the engineer switches on all of the 1000 switches. Sometime later he switches off all the switches with even numbers. A little while later he changes the state if every 3rd switch (3, 6, 9, etc). In other words he changes the switch to “ON” if it was “OFF” and vice versa. Still later, he changes the state of every 4th switch, then every 5th switch and so on until the end of his shift by which by which time has visited the main console 1000 times in all to attend to the switches.
Find with a full explanation which switches will be in the “ON” position at the end of his shift.
Answer by rajagopalan(174)   (Show Source):
You can put this solution on YOUR website! Now this is a very tricky problem & needs to be elaborated for a logical answer.
All switches are toggle switches.
1 Pressing all switches, all are on.on List 1,2,3,4,5,6,7,8,9
2 Pressing 2,4,6,8...leaves all even switches off..0n list 1,3,5,7,9
3 Pressing 3,6,9 etc bring life to some even switches. onlist 1,5,7,
4 pressing 4,8,12 etc bring life to 1,4,..
thus indicating switches pressed odd number of times will be on
First Switch to be on is 1
Next odd touch goes to 4 ( 1,2,4)
Next odd touch goes to 9 ( 1,3,9)
Next odd touch goes to 16 ( 1,2,4,8,16)
So squares of natural numbers receive odd touches
they are..1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625 676 729 784 841 900 961
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