There are 100 light bulbs that are connected to 100 different light switches. 10
ID: 3273124 • Letter: T
Question
There are 100 light bulbs that are connected to 100 different light switches. 100 different people will be turning these lights on and off. Your job is to determine what lights are on and what lights are off after all 100 people are finished with the lights. Use the table below to keep track of the progress. The table is just for #'s 1 - 30. Then when we finish the first 30 people we can predict what the results would look like for 100 people. The first person enters the control room to find all of the lights flipped to OFF. This person then turns all 100 lights ON. The second person enters the control room and flips every two switches. So he hits switch 2, 4, 6, 8, 10, 12, ..., 98, 100. These even switches were ON so they are now flipped to OFF. Be careful because here is where it starts getting tricky. From here on out there will be no pattern to what switches are on and what switches are off... The third person enters the control room and flips every three switches. So he hits switch 3, 6, 9, 12,15, 18, ..., 96, 99. He switched all of the ON switches to OFF and all of the OFF switches to ON. Continue this problem for #'s 4 - 30 1. After going through the first 30 people, what light bulbs were left on? What light bulbs were left off? 2. For the first 100 people, how many switches would still be on? What switches are they? 3. What is the rule or pattern for the switches left on? 4. When did you notice this pattern and why do you think it works. 5. How did your group set up your table. What was your strategy of recording on/off light bulbs and how did you approach the problem? 6. If there were 1,000,000 switches and 1,000,000 light bulbs would bulb number 628,973 be on or off at the end of this experiment? Why? How did you figure it out? 7. If there were 1,000,000 switches and 1,000,000 light bulbs would bulb number 123,904 be on or off at the end of this experiment? Why? How did you figure it out? 8. How did your group work together? What were your strengths and weaknesses as a group?Explanation / Answer
after 2nd person all the ON switches are 1,3,5,---,99 and
all the OFF swithches are 2,4,6,8,---,100
when 3rd person enters all the odd switches divisible by 3 will be off and all the even switches divisible by 3 will be ON. So we need to find the numbers which divide the switch's number(i.e number of factors for each number except1 &2)
if you notice all the perfect square numbers only have odd number of factors ex., 25's factors 1,5,25
16's factors 1,2,4,8,16 while all the rest of the numbers have even number of factors
So ans for the following questions.
1)after first 30 persons :
1 ON,2 Off,3Off,4ON,5Off,6Off,7Off,8Off,9ON,--- since we can not count and if you have noticed a pattern I've explained the same below
set1:1,4,9,16,25 (Perfect square till 30)
set2:31,32,33,34,35, 37, 38,39,---(All the numbers except perfect square after 30 i.e. from 31 to 100 the only Off will be 36,49,64,81,100)
So set 1,2 will be ON rest will remain OFF
2)For the 1st 100 people the following lights will be on (all the perfect Square nos.) 1,4,9,16,25,36,49,64,81,100
3)Rule: all the perfect Square nos. as they have only odd number of factors
4)As I knew that all the perfect square numbers have odd factors I observed immediately when I thought of changing the state of only those lights which are factors of that person number
5)For your answer
6)628973 is not a perfect square number and so we can say that it will remain Off
7)123904 is a perfect square number (i.e. 352*352) and so it will be ON
8)For you to answer
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