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**gmn****Member**- Registered: 2020-01-21
- Posts: 6

I recently found this site, and binge-solved some that looked interesting. In the process I found that few of official answers are either wrong or not entirely correct.

#1) Marble Mix Up puzzle

Official answer is that "Sally knew what she had because she knew what was left" but, since she did not look at her marbles or her label, Sally did not have any more information than person before her (Harry), so Harry should have been able to deduce exactly same answer.

For example, we can mark B/L as "number of blue marbles in someones box" / "number of blue marbles on label" ( where 0<= B,L <=3 ), and it will uniquely identify situation at any box. Also, we can mark as M ( 0..2 ) number of blue marbles out of those two marbles they took out of their box.

It is easy to see that only way to "know" what is in their box, after taking M blues, is if label has either same number of blues L==M ( in which case B=M+1) , or if label has one more blue L==M+1, in which case B=M . This is because box can never have LESS blues than what we took as M, so box is either B==M+0 or B==M+1. And we can be sure which one only if label is also one of those two values (in which case box content is other value ). As an example, if we take two blues ('bb', M=2), we know box must be either 'bbb' (B=3) or 'bby' (B=2). If label shows 'bby' (L=2) , we know box is B=3 ; conversely if label shows 'bbb' L=3, we know that box must be 'bby' B=2. But more importantly, if label shows any other number , we can not be sure about box content. So if label is 'byy' (L=1) - we can not eliminate neither of two box possibilities (B==2 or B==3)

Assuming others can see what marbles those before them draw, and hear when others tell their last color, they will be able to deduce both B and L value of those players who "knew" before them ( if Tom took two blue marbles and said 'blue', Duck knows Toms B=3, but also knows his L=2 )

So, lets look at the puzzle.

1) Tom takes 'bb' M=2 , see his label L and "knows" B, meaning label must have been either 2 or 3 (L=2|3), and box is consequently either 3 or 2 (B= 3|2) .. in other words, Tom has 2/3 if he said 'yellow', or 3/2 if he said 'blue'

2) Duck takes 'by' M=1, see his label and "knows". He can know if his L=1|2 ( implying B=2|1), but hearing what color Tom announced can also help him:

2a) if Tom said 'yellow', then Duck certainly have one blue, regardless of his label: B_tom=2 -> B_duck <> 2 , and due to M_duck=1 -> B_duck=1 ( Duck has 1/? and say 'yellow', Tom had 2/3 )

2b) if Tom said "blue", Duck can not ignore his label ( must have L=1|2 to know), but then it would limit his possible box to B=2 and his label to L=1 ( since if B_tom=3 -> L_tom=2 -> L_duck <>2 -> L_duck=1 (he knows only if L=1|2 ) -> B_duck=2 ( Duck has 2/1 and say 'blue', Tom had 3/2 )

3) Harry takes 'yy' M=0 ( so he can have B=0|1 ) , see his label and does NOT know right away- that means his label is neither 0 nor 1 ( ie L_harry=2|3) , or he would know right away. BUT ... depending of what Tom and Duck said before him:

3a) if Tom said 'yellow', Harry knows that Duck has 1/? , Tom had 2/3 -> Harry can't have B==1 (Duck has it) , nor L==3 (Tom has it) -> B_harry=0 and L_harry=2 (0/2). Further, only B=3 remains -> B_sally=3 ; only L=0|1 remain and B_duck==1 -> L_duck=0 -> L_sally=1 [ Duck 1/0, Harry 0/2 , Sally 3/1 ]

3b) if Tom said 'blue', Harry knows Duck has 2/1, Tom had 3/2 -> L_harry=3 (since 2 is taken) -> he knows all labels so far (1,2,3) -> L_sally=0 -> he also knows Sally's box has options B_sally=1|0, just like his ( since B=2&3 are taken with Tom and Harry) -> B_sally=1 ( since her label is L_sally==0) -> only remaining box value is 0 -> B_harry=0 [ Harry 0/3 , Sally 1/0 ]

Note that above allows more than one color distribution (B/L):

a) Tom 2/3, Duck 1/0, Harry 0/2, Sally 3/1

b) Tom 3/2, Duck 2/1, Harry 0/3, Sally 1/0

Above means not specifically that official answer is incorrect, but rather that :

- **answer is incomplete**, since it mention only "b" version from above where Tom said 'blue', but Tom could have said 'yellow' as well and both Harry/Sally would know

- **problem itself is invalid**, since it is not possible that Harry would not know ( Hurry must be able to deduce his last marble, regardless of what he see on his label )

Above mention about invalid problem would be true regardless of wording of this puzzle, or even regardless of validity of my solutions above. If we assume that all of them are capable of logical conclusions, and if Harry and Sally have same information, then Harry must be able to get to same conclusion as Sally. If Harry have MORE information (and he did have more, ie he saw his label which Sally did not), this holds even more. On the other hand, if we assume that Harry is incapable of logical thinking, then Sally could not rely on his "don't know" answer (he may had label 'yyy', marbles 'yy' .. and still fail to deduce that his box is 'yyb') - in which case Sally would make mistake if she announce that she has 'yyb', as official solution suggests. So, it is NOT possible for Harry not to know, and Sally (without any additional information that Harry did not have ) to know.

**POTENTIAL FIX , so that Marbles problem is not invalid:** *... all parts at start remain same... and then Harry takes two yellow marbles and announce without even looking at his label " I now know what marbles everyone have".*

This would force 'a' version of result from above, since in 'b' version both Harry and Sally could have B=0|1, but in 'a' version B=1 is taken by Duck, so Harry and Sally have 0|3, and 3 is impossible with M=0 ('yy') - so Harry would know he has B=0 and complete solution is as one in 'a' ( Tom 2 blues, Duck 1 blue, Harry 0 blues, Sally 3 blues ), while problem remains valid and we avoid illogical situation where Sally knows better than Harry (who had more info btw).

#2) Merry Go Round Puzzle

For this one, it is correct result, but **incorrect multiple results** are implied : "There must have been a multiple of 3 as well as 4 plus one children. So, the minimum number of children riding on this Merry-Go-Round could be thirteen."

In fact, only single result with 13 kids is possible: x/3+3/4*x= number of kids = x+1 -> (*12) -> 4x+9x=12x+12 -> x=12 -> number of kids = x+1 = 13 ( single possible solution )

Other implied results that are "multiple of 3 as well as 4 plus one children", like 25 kids (x=24) are NOT valid answer, since 24/3+3/4*24=8+18=26 <> 25 !

*EDIT* This forum should not be so picky about name D*I*C*K (which they changed everywhere with 'math') - after all, they choose to use that name in the puzzle. I tried now to change back all 'math' to 'D-u-c-k', but if forum change that also ... just consider that wherever you see 'Duck' or out-of-place 'math', it actually means D*i*c*k ( as name of second person from Marble puzzle ) .

*Last edited by gmn (2020-01-21 10:49:24)*

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 8,913

hi gmn

Welcome to the forum.

I joined the forum after trying some of the puzzles too. But not those two so I've had a look now.

I tried the Merry Go Round using algebra and got a linear equation with that one solution. So you're right. I suspect that the wording of the solution may be Sam Loyd's own. He liked to set puzzles that could be done just by thinking about them, rather than using 'heavy' maths like algebra. The 'solution' could be improved by adding something like "In fact, this is the only solution." I cannot edit those pages so we'll just have to hope that MIF himself does it for us.

As regards the Nanny algorithm to change certain words, it's a problem faced by all forums. Do you moderate every post before it is published or remove offensive ones after the event. The 'Nanny' let's us do the latter which is easier for the moderators and improves communications generally. I've 'fallen foul' of it myself and I just smiled to myself and found another way to say what I wanted. It's not perfect but it does allow us to function.

I've had a quick go at the marbles puzzle and I couldn't even get a solution. I'm wondering if some crucial words are missing. I'm helping out at my local school today so I'll have a longer look later when I get some time.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 8,913

Marble Mix Up Puzzle.

I agree; there's something wrong with this puzzle.

I'm using B and Y to indicate marbles; b and y to indicate labels.

TOM draws BB and looks at his label. He says he knows the third marble. This can only be done if his label is bbb or bby.

RICHARD (huh!) draws BY and looks at his label. He says he knows the third marble. This can only be done if his label is bby or byy.

HARRY draws YY and his label doesn't enable him to deduce his marble **. So his label must be bbb or bby.

At this point the three labels bbb, bby and byy have all been seen by these three so

SALLY must have yyy as her label. As she cannot have YYY HARRY must have this.

So why didn't he work this out at ** ? Perhaps his logical ability isn't as good as SALLY's. Maybe TOM, RICHARD and HARRY make their draws, look at their labels and make their decisions but don't announce them until all three have completed all three steps and are then not allowed to change their decision. In that case, HARRY doesn't yet know what the other two have concluded so he cannot deduce his third marble. This is a complicated explanation so a simpler way is to have HARRY going first without knowing the other two's decisions; then TOM, then RICHARD. That way when SALLY comes to make her decision she has more information than HARRY.

Bob

*Last edited by Bob (2020-01-22 22:31:53)*

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**gmn****Member**- Registered: 2020-01-21
- Posts: 6

I mostly agree, but have some comments:

1) "Why did not Harry work out his last color?" is a good question, but I believe that different interpretations of problem do not leave room for easy answer. Listing your suggestions :

1a) Perhaps his logical ability isn't as good as SALLY's ? I already explained why this is not valid - if his logical ability was low, he could have stated "I don't know" after drawing YY, even if he had yyy/yyb on his label - which would then result in wrong answer by Sally anyway.

1b) Maybe they do not announce their last marble color until all have finished draw/look/reason ? Also hardly valid - problem strongly imply that Tom announced his color before 'Richard' etc. But even if we accept that they all announce at same time ... then Sally would have to announce at same time with them, and would not know what to say ( she could not use info from their announcements to make her decision)

1c) Order of announces is different, with Harry - Tom - Richard - Sally , instead of Tom-Richard-Harry-Sally ? This is also hardly valid - problem text very strongly imply order ( it mention them in Tom-Richard-Harry-Sally order ). Although, if Harry was first, then maybe it could work, providing that single deductible solution exists. But in order to really fix 'invalidity' of current problem, change of text would be needed anyway ( to rearrange Harry as first mentioned )

2) "SALLY must have yyy as her label" : this is only partially correct - it correspond to 'b' version of possible answer ( Tom 3/2, Duck 2/1, Harry 0/3, Sally 1/0) from my previous post. But another possibility also exists ( 'a' version from my post with Tom 2/3, Duck 1/0, Harry 0/2, Sally 3/1 )

2a) "RICHARD (huh!) draws BY and looks at his label. He says he knows the third marble. This can only be done if his label is bby or byy." - this is also partially correct, and reason for above issue : RICHARD can know his third marble **regardless of his label**, if he hears Tom saying "yellow" for 3rd marble. In that case Tom has BBY, and Richard can only have BYY. Thus option that you marked as 'would not work' (in your finished answer ) could, in fact, work - I explained that line of reasoning in my 2a/3a parts of previous post.

3) one potential different interpretation of the problem that I did not deeply consider ( and that could *maybe* explain differences) is if they are not able to either see what marbles others draw, or hear what others announced.

3a) It can not be both (since then Sally would not have any information to base her " I know without looking" reasoning .

3b) if they do not see what two marbles others draw, but hear what others announce as color of last marble?

3c) if they see what two marbles others draw, but do not hear what others announce ?

3d) if they see what two marbles others draw, do not hear color that others announce but do hear if they announce (meaning they hear if other "deduced" , but not what color they deduced ?

While it is possible that one of above 3a..d interpretations of problem could fix smaller issue about multiple solutions ( current answer only mentions one possible solution, but another one also exists ), my key point and main argument in initial post was that nothing short of different problem text can fix major issue : **If Sally is able to deduce solution without any additional information , then whoever was just before her must have been able to do the same.**

That statement is true for ANY possible problem of this type - regardless if our specific solutions above are correct or not, if there is only single or multiple solutions, what exact colors were in boxes, etc ... point is that whatever Sally is able to deduce, Harry must be able to deduce also.

*Last edited by gmn (2020-01-23 08:46:06)*

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 8,913

hi gmn,

Thanks for your comments. Here's the situation. MathsIsFun made both sites. Some years ago he made me an administrator on the forum. I can do some things here but I cannot alter the puzzle itself.

I have had a go at a rewording of the puzzle. I've tried to make it clear, unambiguous and with a single solution. Please would you have a go at it and see if you agree. If we both think it's ok then I'll email MIF and suggest he changes it.

**Marble Mix Up Puzzle**

Years ago, to puzzle his friends, a scientist gave one of four containers containing blue and/or yellow marbles to each of the friends; Alan, Brian, Colin and Diana.

There were 3 marbles in each container, and the number of blue marbles was different in each one. There was a piece of paper in each container telling which color marbles were in that container, but the papers had been mixed up and were ALL in the wrong containers.

He then told his friends to take 2 marbles out of their container, show the others, read the label without the others seeing, and then tell him, but not each other, the color of the third marble.

Alan took two yellow marbles out of his container and looked at the label. He announced he could not tell the color of his remaining marble.

Brian took 2 blues marbles from his container. After looking at his label he was able to whisper the color of his remaining marble to the scientist.

Colin took 1 blue and 1 yellow marble from his container. He looked at the label in his container and he was also able to whisper the color of his remaining marble.

Diana, without even looking at her marbles or her label, was able to tell the scientist what color her marbles were.

Can you tell what color marbles Diana had?

Can you also tell what color marbles the others had, and what label was in each of their containers?

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**gmn****Member**- Registered: 2020-01-21
- Posts: 6

hi bob,

I think that your modified version would work, and that it now has single solution:

It still holds true that Colin would be also able to tell what marbles and labels everyone else has, even before Diana - but now it does not invalidate problem itself, since it is not in collision with what Colin actually did.

Interesting version of this problem would be one that starts the same, but when it comes to Colin it does not tell us what marbles he draw:*Colin took two marbles from his container and, even without looking at his label, he was able to whisper the color of his remaining marble.*

Note that above line makes it different problem, with different solution and even different answer to "Can you also tell what color marbles the others had?" ... so I just mentioned it as interesting 'different' problem, but your version should be one to replace original problem.

*Last edited by gmn (2020-01-26 06:43:58)*

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 8,913

gmn wrote:

( Bd=1='byy', Ld=0='yyy' ); ( Bc=2='bby', Lc=1='byy' ); ( Bb=3='bbb', Lb=2='bby' ); (Ba=1='byy', La=3='bbb')

You've got Bd = byy and Ba = byy. I'm hoping you meant Ba = yyy.

I'll message MIF.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**gmn****Member**- Registered: 2020-01-21
- Posts: 6

bob bundy wrote:

You've got Bd = byy and Ba = byy. I'm hoping you meant Ba = yyy.

Yes, you are correct, that was typo for Ba - I updated my post to fix that.

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