100 Doors in a Row
Problem: you have 100 doors in a row that are all initially closed. you make 100 passes by the doors starting with the first door every time. the first time through you visit every door and toggle the door (if the door is closed, you open it, if its open, you close it). the second time you only visit every 2nd door (door #2, #4, #6). the third time, every 3rd door (door #3, #6, #9), etc, until you only visit the 100th door.
question: what state are the doors in after the last pass? which are open which are closed?
Solution
For example, after the first pass every door is open. on the second pass you only visit the even doors (2,4,6,8…) so now the even doors are closed and the odd ones are opened. the third time through you will close door 3 (opened from the first pass), open door 6 (closed from the second pass), etc..
question: what state are the doors in after the last pass? which are open which are closed?
solution: you can figure out that for any given door, say door #42, you will visit it for every divisor it has. so 42 has 1 & 42, 2 & 21, 3 & 14, 6 & 7. so on pass 1 i will open the door, pass 2 i will close it, pass 3 open, pass 6 close, pass 7 open, pass 14 close, pass 21 open, pass 42 close. for every pair of divisors the door will just end up back in its initial state. so you might think that every door will end up closed? well what about door #9. 9 has the divisors 1 & 9, 3 & 3. but 3 is repeated because 9 is a perfect square, so you will only visit door #9, on pass 1, 3, and 9… leaving it open at the end. only perfect square doors will be open at the end.
Tags: amp, divisor, door, doors, end, EXAMPLE, initial state, odd ones, pair, pass, perfect square, Problem, question, Row, second time, solution, square, state, third time, time, toggle
You can comment below, or link to this permanent URL from your own site.
September 20, 2011 at 1:56 am
Acquiring examine this I assumed it absolutely was really useful. I value you taking some time and hard work to place this short article collectively. I when once more discover myself investing approach to considerably time the two looking at and commenting. But so what, it had been nonetheless worthwhile!
September 20, 2011 at 9:18 am
Saved as being a favored, I genuinely like your web site!
September 20, 2011 at 6:33 pm
Hi there, I believe your internet site may be acquiring browser compatibility concerns. When I seem at your internet site in Safari, it appears very good but when opening in World wide web Explorer, it’s got some overlapping. I just wished to provide you with a fast heads up! Other then that, wonderful website!
September 21, 2011 at 8:00 am
Hi there , I do imagine this can be a fantastic weblog. I stumbled on it on Yahoo , i’ll arrive back again as soon as yet again. Funds and liberty could be the finest strategy to alter, may possibly you be prosperous and aid others.