# Comment On Nerds, Jocks, and Lockers

Mr. Zargas was the zany math teacher at Cliffmont High that everyone seemed to love. Whether you were a nerd or a jock, he made mathematics interesting, challenging, and fun to learn. That, in and of itself, was impressive enough, but Mr. Zargus took it one step further. When it came time for his frequent "Mathematical Battle of Wits," he would let the jocks use their brawn instead of their brains. The nerds never stood a chance, especially when it came to his "locker challenge." [expand full text]
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### Re: Nerds, Jocks, and Lockers

2009-08-05 10:14 • by Márton Balassa (unregistered)
 Delphi solution. The function returns an array of integers. The Nth element of the array is 0 if the Nth locker is closed, and X if it's opened, where X was the number of toggles. ```function Lockers(const Count: Integer): TIntegerDynArray; var I, K: Integer; begin SetLength(Result, Count); for I := 1 to Count do begin Result[I - 1] := 0; K := Count; while K > 0 do begin if I mod K = 0 then Inc(Result[I - 1]); Dec(K); end; end; for I := 0 to Count - 1 do if Result[I] mod 2 = 0 then Result[I] := 0; end; ```

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:15 • by pjt33
 Scot:The locker toggled the most is the locker whose prime factors can create the greatest number of combinations. (locker ... combinations ... get it?). At first, I think 64 might be a good choice, but (2,2,2,2,2,2) creates only 6 combinations. If I replace a 2 with a 3, I describe locker 96, and I can now create 11 combinations. Applying my knowledge of cribbage, I see that I can replace three 2s with two 3s to describe locker 72 (2,2,2,3,3), which also creates 11 combinations. 12, not 11 - you're probably failing to count 1. Also, the smallest number with 12 factors is 60 (and since it's the largest highly composite number smaller than 100, the next being 120, 12 factors is indeed the most possible).

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:15 • by Daniel (unregistered)
 Sinclair ZX81 / Timex TS100 (via emulator):

### C# with yield-keyword, without multiplication, variables renamed => obscured by design

2009-08-05 10:15 • by Rhombicosidodecahedron (unregistered)
 public static IEnumerable GetOpenLockers(int num0) {int num1=0,num2=-1;while((num1+=num2+=2)<=num0)yield return num1;} // You can use it like this /* foreach(int i in GetOpenLockers(100)) Console.WriteLine(i); // Or with descending and using System.Linq; Array.ForEach(GetOpenLockers(100).OrderByDescending(i => i).ToArray(), i => Console.WriteLine(i)); */

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:16 • by Mamont (unregistered)
 sub getOpenLockers { my @opened; my \$last = 1; my \$total = shift; my \$skip = 2; for (my \$i = 0; \$i < \$total; \$i++) { push @opened,\$i+1; \$i = \$i+\$skip; \$skip = \$skip+2; } return @opened; }

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:16 • by Welbog
 reallyAmazed:simply - squares of prime numbers (or perfect squares) I'm amazed with amount of i^2 solutions. Think a bit longer before posting.16 is not the square of a prime number, yet it remains open. The i^2 solution is common because it's correct.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:16 • by micksam7 (unregistered)
 Quicky crappy php using square-checking ~ ```function locker(\$in) { \$out=""; for(\$t=1;\$t<=\$in;\$t++) \$out .= pow(\$t,.5) == round(pow(\$t,.5)) ? "{\$t}, " : ""; return substr(\$out,0,-2); } ``` locker(1000); = 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

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:19 • by lolwtf
 #include int main(int argc, char **argv) { printf("4 9 16 25 36 49 64 81\n"); return 0; } Totally optimized.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:20 • by Tim (unregistered)
 This Befunge program works OK on 2 and 10. Output is reversed though. I'm currently testing it on 100. ```v ;//00 - num lockers; v ;//10 - current step; v ;//20 - current locker; v ;//x1 - locker: 0 if closed, 1 if open; >25*:*00pv ; #00 = 100 ; v < >010pv ; #10 = 0 ; v < >v ; do { // set x1 to 0; >00g1-00pv ; #00 -= 1 ; v < >000g1pv ; #[#00]1 = 0 ; v < >00g0`v; } while (#00 > 0) ; v < ^_v v < >25*:*00pv ; #00 = 100 ; v < >v ; do { // b = 0 .. #00; >10g1+10pv ; #10 += 1 ; v < >020pv ; #20 = 0 ; v < >v ; do { // c = 0, b, 2b .. #00; >20g10g+20pv ; #20 += #10 ; v < >20g1g!20g1pv ; #[#20]1 = !#[#20]1 ; v < >00g20g`v; } while (#00 > #20) ; v < ^_v v < >00g10g`v; } while (#00 > #10) ; v < ^_v v < >v ; do { // print the results ; >00g1-00pv ; #00 -= 1 ; v < >00g1g.v ; #[#00]1 = 0 ; v < >00g0`v; } while (#00 > 0) ; v < ^_v v < >@ ; exit() ; ``` ; ; denote pseudo-code, sort of

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:22 • by noway! (unregistered)
 In BASICish (num being number of lockers): For i=1 to sqrt(num) print i*i next

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:24 • by Charles400
 If I got this as an interview question, I'd go postal. Promise.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:24 • by Børge N (unregistered)
 Some ancient guy has already figured out the solution to the "number of toggles" question.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:24 • by Nathon (unregistered)
 Sorta brute force-ish... ```def testy (number): skip = 3 sum = 3 step = 1 while True: if number <= sum: return step skip += 2 sum += skip step += 1 ```

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:24 • by Tama (unregistered)
 Technically, this is incorrect; more than one factor may be repeated. To prove that a number has an odd number of factors if and only if it is a perfect prime: Let x be an integer, x = x1^a1 * x2^a2 * ... * xn^an. For a given factor, there are (a1 + 1) ways to choose at what power x1 will be, (a2 + 1) ways to choose at what power x2 will be, and so on. The number of distinct factors for x is thus p = (a1 + 1) * (a2 + 1) * ... * (an + 1). Now, p is odd if and only if all of (a1 + 1), (a2 + 1), ..., (an + 1) are odd, or equivalently, if and only if a1, a2, ..., an are even. That last condition is equivalent to saying that x is a perfect prime.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:24 • by MichaelM (unregistered)
 ```#!/usr/bin/env ruby def getOpenLockers(lockers) (1..lockers).map {|locker| locker if(Math.sqrt(locker) % 1 == 0)}.compact end puts getOpenLockers(ARGV[0].to_i).join(', ') ``` YAY!

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:26 • by Shoko (unregistered)
 public class LockersApp { /** * @param args */ public static void main(String[] args) { // TODO Auto-generated method stub double sqrts = Math.sqrt(Double.parseDouble(args[0])); for (double i = 1; i <= sqrts; i++) { System.out.print(new Double(Math.pow(i, 2.0)).intValue() + ","); } } }

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:26 • by Tama (unregistered)
 Welbog:The open lockers are perfect squares. The reasoning is simple: non-perfect squares have an even number of factors. The jocks toggle a number for every factor it has. Therefore every door should be toggled an even number of times and should therefore end closed (the starting state). However, perfect squares have an odd number of unique factors, because one of the factors is repeated. Because of this, perfect squares are toggled for their square root which has no pair to reverse the toggle, so the door ends up open (the opposite state it started in). Make sense? Technically, this is incorrect; more than one factor may be repeated. To prove that a number has an odd number of factors if and only if it is a perfect prime: Let x be an integer, x = x1^a1 * x2^a2 * ... * xn^an. For a given factor, there are (a1 + 1) ways to choose at what power x1 will be, (a2 + 1) ways to choose at what power x2 will be, and so on. The number of distinct factors for x is thus p = (a1 + 1) * (a2 + 1) * ... * (an + 1). Now, p is odd if and only if all of (a1 + 1), (a2 + 1), ..., (an + 1) are odd, or equivalently, if and only if a1, a2, ..., an are even. That last condition is equivalent to saying that x is a perfect prime.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:27 • by BobbyBob (unregistered)
 from math import sqrt def IsSquare(n): if sqrt(n) == int(sqrt(n)): return True return False def GetOpenLockers(n): open_lockers = [] for v in xrange(n): if IsSquare(v + 1): open_lockers.append(v + 1) return open_lockers

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:29 • by reallyAmazed (unregistered)
 Ok ,sorry, i will state it a bit more clearly: for any natural power of a square of a prime number (as an example 36 is 6*6 but it stays closed)

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:30 • by Ramūns (unregistered)
 More JS: ```function (n) { if (n <= 0) { return []; } var ret = [1]; for (var i = 4, plus = 5; i <= n; i += plus, plus += 2) { ret[ret.length] = i; } return ret; }```

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:31 • by port (unregistered)
 I'm really amazed you can't even use the demonstration at the top of the page before you start spouting off.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:35 • by belgariontheking
 It's all the square numbers. I've done this before, but with 1000 and only an imaginary hallway filled with lockers.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:39 • by mpcube
 A PHP Solution... ``` ```

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:41 • by BobbyBob (unregistered)
 You're still wrong: 4 is not prime, but 16 remains open. Here's an optimized (O(sqrt(N)) solution: ```def GetOpenLockersOsqrtn(n): open_lockers = [] for v in xrange(int(sqrt(n))): open_lockers.append((v+1) * (v+1)) return open_lockers ```

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:44 • by shane blake (unregistered)
 some pretty simple sql... more brute force than nerdy, but it works... declare @count int set @count = 100 declare @lockers table( id int identity, state bit ) insert into @lockers ( state ) values ( 0 ) while ( (select max(id) from @lockers) < @count ) insert into @lockers ( state ) values ( 0 ) while ( @count > 0 ) begin update @lockers set state = abs(state - 1) where id % @count = 0 set @count = @count - 1 end select * from @lockers where state = 1

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:46 • by Rob (unregistered)
 Untested and messy php ``` ```

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:46 • by pjt33
 reallyAmazed:Ok ,sorry, i will state it a bit more clearly: for any natural power of a square of a prime number (as an example 36 is 6*6 but it stays closed) 1,2,3,4,6,9,12,18,36 9 factors Think about it this way: given integer n (> 0) find every pairwise factorisation (a,b) such that a <= b with a == b iff a^2 = n. Call the set of such factorisations S (so S = {(a,b) | ab = n && a <= b}). Then the set of factors of n is F = {a | Exists b : (a,b) in S} U {b | Exists a : (a,b) in S}. That's a union of two sets which are the same size, so if the two sets are disjoint then |F| is even. The intersection of the two sets is the set of integers x such that Exists b : (x,b) in S and Exists a : (a,x) in S. By definition of S, xb = n and ax = n, so x = 0 (impossible since we said n > 0) or a=b. But, again by definition of S, x <= b and a <= x, so since b=a we have x <= a <= x, or x = a. Therefore the intersection of the two sets is the set of integers x such that x^2 = n. Therefore if n is not a square number the two sets are disjoint and |F| is even. If n is a square number the two sets have sqrt(n) in common, and |F| is odd. Edit: ok, I'm missing a step. I should also have shown that if (a,b) in S and (a,c) in S then b=c, and similarly if (a,b) in S and (c,b) in S then a=c. Trivial, since division by a non-zero real is well-defined.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:48 • by MisterCheese (unregistered)
 I see... the ones that remain open have been hit by 1, themselves, and their square root, plus pairs of factors. Eg 36 is hit by 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6. The closed ones don't have a positive integer square root so are hit an even number of times. Now I guess I need to work out how to extract my head from the toilet bowl...

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:48 • by Zack (unregistered)
 Man, you would think the nerds from one year would notice the answer is all primes and tell the younger generation so they could show up the jocks next year and tell Zargus the answer while the jocks tired themselves out slamming lockers.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:50 • by kennytm
 C++, by Brute force. ```#include #define DOORCOUNT 100 template struct DoorColor { typedef typename DoorColor::color color; typedef typename DoorColor::anticolor anticolor; static void print(); }; template struct DoorColor { typedef typename DoorColor::anticolor color; typedef typename DoorColor::color anticolor; static void print(); }; template struct DoorColor { typedef char color; typedef struct { char x[256]; } anticolor; static void print(); }; template struct DoorColor { typedef char color; typedef struct { char x[256]; } anticolor; static void print(); }; template void DoorColor::print() { std::printf("Door %d is %s.\n", Lock, sizeof(typename DoorColor::color) == 1 ? "closed" : "open"); } int main () { // Excel was used to generate this part. DoorColor<1>::print(); DoorColor<2>::print(); DoorColor<3>::print(); DoorColor<4>::print(); DoorColor<5>::print(); // ... you get the idea ... DoorColor<96>::print(); DoorColor<97>::print(); DoorColor<98>::print(); DoorColor<99>::print(); DoorColor<100>::print(); return 0; } ```Addendum (2009-08-05 10:57):Edit: The 3rd struct should be ```template struct DoorColor { typedef struct { char x[256]; } color; typedef char anticolor; static void print(); }; ``` To enforce the 1st door to be open initially. Yeah, templates are crazy.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:52 • by David Kowis (unregistered)
 ```#!/usr/bin/ruby # def getOpenLockers(total) open = [] (1..total).each do |num| sqr = num ** 2 if(sqr <= total) open << sqr elsif sqr > total break end end open end puts "100: #{getOpenLockers(100)}" ``` There's a ruby solution. Works to find all the open lockers, will have to figure out which ones are opened most often.

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:52 • by Anonymous (unregistered)
 I like the concept of coding challenges but so far they have all been the same - provide an algorithm to solve some well known mathematical problem. How about a challenge that consists of a bit more than just solving known equations? What's that? We can suggest our own programming challenges? Well OK then!

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:52 • by John (unregistered)
 Might be worth al least putting some form of color behind the title. White-on-white doesn't work out too well...

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:57 • by egilhh (unregistered)
 Ada (Quick and dirty, no validation of input) ```with Command_Line; with Ada.Text_IO; with Ada.Numerics.Generic_Elementary_Functions; procedure Lockers is package Float_Numerics is new Ada.Numerics.Generic_Elementary_Functions(Float); Input : Integer := Integer'Value(Ada.Command_Line.Argument(1)); begin for i in 1..Integer(Float_Numerics.Sqrt(Float(Input)) loop Ada.Text_IO.Put(Integer'Image(i**2)); end loop; end Lockers; ``` -- egilhh

### Re: Nerds, Jocks, and Lockers

2009-08-05 10:59 • by jonnyq
 In the same vein at the .NET IEnumerable, using Javascript 1.7 generators, and without brackets since that works too =) You'll need a recent firefox with a special flag to the script tag to use generators. function l(i) for(j = 1; j < Math.floor(Math.sqrt(i)); j++) yield i*i; and getting it is fun too: var opens = [i for each (i in l(100))]; javascript is fun =)

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:01 • by Fernando (unregistered)
 Stupid short solution in PHP: ``````

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:01 • by dkf
 reallyAmazed:Ok ,sorry, i will state it a bit more clearly: for any natural power of a square of a prime number (as an example 36 is 6*6 but it stays closed)I admire your mathematical skills! Now do it for an irrational number of lockers, so that your reasoning powers can be truly matched...

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:04 • by BJ Homer (unregistered)
 In Python: [code] count = int(sys.argv[1]) open_lockers = [] for i in range(1, count+1): factors = [j for j in range(1, i+1) if i%j == 0] if len(factors) % 2 == 1: open_lockers.append(i) print open_lockers [code] You know, just in case they decide to change the rules on square numbers or anything.

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:09 • by indrora (unregistered)
 ok. This one is easy once you get the fact its a square system. the algorithm is simple: `for(int l=1;l*l<=n;l++) {Console.WriteLine(" locker "+(l*l) + " is open" ); }` C#: Given that n is an integer representing the number of lockers. replace with your language of preferences form. Example, the basic syntax on a TI-89 (the only decent calculator) would be: ```Prgm lsol(n) For i,1,sqrt(n) if i^2 <= n Disp i^2 endif EndFor EndPrgm ``` its not brute force, and they can show it scales for however many lockers. :) Simple, elegant, and in its own simple way a brute force. Captca: Esse (Hey, Esse you want some algorithms? )

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:11 • by amischiefr
 The Java way to find the most toggled ``` private static ArrayList toggledTheMost(int n) { int arr[] = new int[n]; ArrayList list = new ArrayList(); for(int i = 1; i <= n; i++) { for(int j = i ; j <= n ; j = j + i) { arr[j-1]++; } } int toggled = 0; for(int i = 0; i < arr.length ; ++i) { if(arr[toggled] < arr[i]) toggled = i; if(arr[i]%2 != 0) list.add(i+1); } System.out.println("The locker that was toggled the most was locker: " + (toggled+1)); return list; } ```

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:12 • by Miquel Fire (unregistered)
 I noticed an interesting pattern with the brute force method. When you get to the halfway mark, you're basically doing the single locker toggle.

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:13 • by azredwing (unregistered)
 sub lockers{ push @_, \$_**2 for 1..shift; return @_; } Someone mentioned that doing the logic makes the programming "less fun" - they'd rather brute-force the program by actually flipping all 100 doors so many times - to make pretty code or something. The way I see it is if you can optimize a problem away beforehand, and your code ends up being an efficient one-liner, that's WAY better than trying to "beautifully" make a giant solution that somehow account for "edge cases" when it's clear there will never be any. Anyone can program; not everyone can think logically, and that's the difference between mediocre and good programmers, or even good and great.

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:15 • by Matt Flowers (unregistered)
 A simple solution is to just sequentially add up all of the odd numbers. Each number in the series is an open locker. Start with 1. Add 3 to get 4 Add 5 to get 9 Add 7 to get 16 Add 9 to get 25 etc.

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:15 • by DaveGamble
 C, 66chars: void f(int l){for(int i=0,j=0;++i<=l;i+=(j+=2))printf("%d,",i);} I'm claiming extra credit for not using multiplication, and it being deeply, deeply ugly code.

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:17 • by monkey (unregistered)
 Who is Paula and why is she brill(i)ant?

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:19 • by DaveGamble
 Matt Flowers: A simple solution is to just sequentially add up all of the odd numbers. Each number in the series is an open locker. Start with 1. Add 3 to get 4 Add 5 to get 9 Add 7 to get 16 Add 9 to get 25 etc. Yep. What you're seeing is the identity: (n+1)^2 = n^2 + 2n + 1 and you're subtracting off the n^2. Hence you get: 1 - 2*1+1 = 3 2 - 2*2+1 = 5 3 - 2*3+1 = 7 as you'd expect. That's the trick my horrific C experience uses.

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:20 • by Capt. Obvious
 I'm amazed at the number of brute force methods. I'm surprised at the nerds in the story as well. If you cannot solve a problem mathematically, optimize the numeric method. To wit, they could have used a bike to traverse the lockers. Or 100 boxes on a sheet of paper.

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:20 • by Reverend Gonzo (unregistered)
 Groovy def getOpenLockers(int n) { return (1..Math.sqrt(n)).collect {it**2} } println getOpenLockers(100)

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:20 • by curtmack
 That guy in Nethack is totally a jock. And may have a lot of identical twins. ```#!/usr/bin/perl \$n = \$ARGV[0]; foreach \$s (1..\$n) { for (\$i = \$s-1; \$i<\$n; \$i += \$s) { \$h[\$i] = !\$h[\$i]; } for (\$i = 0; \$i<\$n; \$i++) { print (\$h[\$i] ? "'" : "+"); } print "\n"; for (\$i = 1; \$i<=\$n; \$i++) { print ((\$i % \$s || \$i+1<\$s) ? " " : "@"); } print "\n\n"; }``` This is what it looks like when run: ```curtmack@cardboardbox:~\$ ./hall.pl 10 '''''''''' @@@@@@@@@@ '+'+'+'+'+ @ @ @ @ @ '+++'''+++ @ @ @ '++'''''++ @ @ '++'+'''+' @ @ '++'++''+' @ '++'+++'+' @ '++'+++++' @ '++'++++'' @ '++'++++'+ @```

### Re: Nerds, Jocks, and Lockers

2009-08-05 11:20 • by pjt33
 DaveGamble:C, 66chars: void f(int l){for(int i=0,j=0;++i<=l;i+=(j+=2))printf("%d,",i);} I'm claiming extra credit for not using multiplication, and it being deeply, deeply ugly code. Why not `void f(int l){for(int i=1,j=1;i<=l;i+=(j+=2))printf("%d,",i);}`