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Topic: Mastermind Puzzles (6 Colours) (Read 1992 times) |
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daemonturk
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Mastermind Puzzles (6 Colours)
« on: Feb 21st, 2009, 4:12am » |
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You may need to familiarize yourself with the game of Mastermind (Logic/Deduction type boardgame).
http://en.wikipedia.org/wiki/Mastermind_(board_game)
I'm not sure if this belongs in pure math, but I need help.
MARKERS:
W = Correct colour, Wrong Position
B = Correct colour, Correct Position
(Becareful with the rules about corresponding markers with duplicate pegs)
Colours: 6
1. If the first guess is BBCC, and the feedback is WWW, how many remaining possible combinations are there?
2. If the first guess is BBCG, and the feedback is WWBB, how many remaining possible combinations are there?
3. If the first guess is BCGR, and the feedback is WWWB, how many remaining possible combinations are there?
4. First Guess: BCGR, Feedback: WWBB, Remaining Possible Combinations: ?
Please help me find the answers to all these questions, and if possible, explain how you solved them.
To help you guys further, I will put an example question and answer.
Example Question and Solution
First Guess: BBBC, Feedback: B
Solution:
If we assume Black is the one in the correct position, the other three colours MUST be wrong entirely because there was no 'W's in the feedback. Hence, the other 3 slots can be any of 4 colours (because they can't be Black or Cyan).
So we end up with (3 x 4^3) because there were 3 Blacks.
If we assume Cyan is correct in position, the first 3 slots are entirely wrong, and hence could be any of the 5 colours (for black is wrong). Hence, 5^3.
The total possible combinations remaining out of the total 1296, is (3 x 4^3) + 5^3 = 317.
Thanks a lot guys!
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tohuvabohu
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Re: Mastermind Puzzles (6 Colours)
« Reply #1 on: Feb 21st, 2009, 1:03pm » |
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1. If the first guess is BBCC, and the feedback is WWW, how many remaining possible combinations are there?
Possibilities: swap bb with cc, then replace any peg with one of the 4 other colors.
?CBB 4
C?BB 4
CC?B 4
CCB? 4
2. If the first guess is BBCG, and the feedback is WWBB, how many remaining possible combinations are there?
Possibilities: every possible swap of two pegs except swapping the two B’s
BBGC
CBBG
GBCB
BCBG
BGCB
3. If the first guess is BCGR, and the feedback is WWWB, how many remaining possible combinations are there?
Possibilities: two rotations possible for each of 4 choices of the one stationary peg
BGRC
BRCG
GCRB
RCBG
CRGB
RBGC
CGBR
GBCR
4. First Guess: BCGR, Feedback: WWBB, Remaining Possible Combinations: ?
Possibilities: every possible swap of two pegs
CBGR
GCBR
RCGB
BGCR
BRGC
BCRG
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daemonturk
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Posts: 46
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Re: Mastermind Puzzles (6 Colours)
« Reply #2 on: Feb 21st, 2009, 6:43pm » |
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Very nice.
How about:
First Guess: BBCG, Feedback: WWW
(Challenging)
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towr
wu::riddles Moderator
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Re: Mastermind Puzzles (6 Colours)
« Reply #3 on: Feb 22nd, 2009, 8:25am » |
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It's fairly easy to write a small program to give you the result. And since 64 isn't that much, you don't even need to take a smart approach to it. Just running through all possible sequences and checking which match works.
Code:<script>
function match(p, g, ccwp, ccrp)
{
var i,j,s,t;
s = (p[0]==g[0]) + (p[1]==g[1]) + (p[2]==g[2]) + (p[3]==g[3]);
p.sort(); g.sort();
i=j=t=0;
while(i < 4 && j < 4)
{
if(p[i]==g[j])
i++, j++, t++;
else if(p[i]<g[j])
i++;
else
j++;
}
return (t==ccwp+ccrp) && (s==ccrp);
}
function run(guess, ccwp, ccrp)
{
var col = ["B", "C", "G", "R", "Y", "M"]; /* ? */
var cnt=0;
for(a=0; a<6; a++)
for(b=0; b<6; b++)
for(c=0; c<6; c++)
for(d=0; d<6; d++)
{
if(match([a,b,c,d], guess, ccwp, ccrp))
{
document.write(col[a] + col[b] + col[c] + col[d] + "<br>");
cnt++;
}
}
document.write(cnt+"<br>");
}
function start(guess_string, result_string)
{
var cols={};
cols["B"] = 0; cols["C"] = 1; cols["G"] = 2; cols["R"] = 3; cols["Y"] = 4; cols["M"] = 5;
var guess;
guess=[cols[guess_string[0]],cols[guess_string[1]],cols[guess_string[2]],cols[guess_string[3]]];
var ccwp=0, ccrp=0;
for(i=0; i < result_string.length; i++)
{
if (result_string[i] == "B") ccrp++;
if (result_string[i] == "W") ccwp++;
}
run(guess, ccwp, ccrp);
}
start("BBCC", "WWW");
start("BBCG", "WWBB");
start("BCGR", "WWWB");
start("BCGR", "WWBB");
start("BBBC", "B");
start("BBCG", "WWW");
document.close();
</script> |
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It gives 44 for the last one.
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Wikipedia, Google, Mathworld, Integer sequence DB
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daemonturk
Newbie
Posts: 46
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Re: Mastermind Puzzles (6 Colours)
« Reply #4 on: Feb 22nd, 2009, 8:27am » |
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44 is correct, but I'm not familiar with this language, is it C#? It would be good if I had a logical solution too 
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towr
wu::riddles Moderator
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Re: Mastermind Puzzles (6 Colours)
« Reply #5 on: Feb 22nd, 2009, 8:43am » |
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on Feb 22nd, 2009, 8:27am, daemonturk wrote:| 44 is correct, but I'm not familiar with this language, is it C#? It would be good if I had a logical solution too |
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It's javascript. (So it can just run in your browser; which I find convenient since I'm using it anyway).
I'll have to think a bit more about a logical solution.
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tohuvabohu
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Re: Mastermind Puzzles (6 Colours)
« Reply #6 on: Feb 22nd, 2009, 12:09pm » |
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If there are two B’s, and a fourth color then it’s either
C?BB 3
?CBB 3
G?BB 3
?GBB 3
If there are two B’s and G or C is repeated then it’s
CCBB
GGBB
If there is one B and a fourth color, then it’s
CGB? 3
CG?B 3
C?GB 3
?CGB 3
GCB? 3
GC?B 3
G?BC 3
?GBC 3
If there is one B, and C or G is repeated, then it’s
CGBC
GCBC
GGBC
CGGB
CCGB
GCGB
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daemonturk
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Posts: 46
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Re: Mastermind Puzzles (6 Colours)
« Reply #7 on: Feb 22nd, 2009, 11:14pm » |
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Thanks, tohuvabohu 
@towr: Your code is impressive, but I'm not familiar with javascript to appreciate it =(
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towr
wu::riddles Moderator
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Re: Mastermind Puzzles (6 Colours)
« Reply #8 on: Feb 23rd, 2009, 12:25am » |
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on Feb 22nd, 2009, 11:14pm, daemonturk wrote:| @towr: Your code is impressive, but I'm not familiar with javascript to appreciate it =( |
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It's really not that impressive. It just loops through all 64 possible colour sequences and picks the one that fit the guess+result. If you were familiar with javascript, you probably wouldn't appreciate it any better 
Most pieces would work the same in java, C, C++, and similar languages though.
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Aryabhatta
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Re: Mastermind Puzzles (6 Colours)
« Reply #9 on: Mar 14th, 2009, 11:34am » |
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Can we move this out of putnam section?
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daemonturk
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Posts: 46
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Re: Mastermind Puzzles (6 Colours)
« Reply #10 on: Mar 14th, 2009, 9:03pm » |
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It belongs here, its a conditional probability game.
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Eigenray
wu::riddles Moderator
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Re: Mastermind Puzzles (6 Colours)
« Reply #11 on: Mar 15th, 2009, 6:09am » |
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Well, it was in Putnam, until somebody moved it. Maybe I should have said something, but I thought it would be obvious.
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