Enthusiastic Forum !

[tillson]

let a =b
therefore a*a = b*a....................................multiply by a
therefore a*a-b*b= b*a-b*b.........................subtract b*b
therefore (a+b)(a-b) = b(a-b)......................factorise
Therefore a+b = b......................................divide by (a-b)
therefore 2b =b..........................................because a=b
therefore 2=1...................................divide by b
QED

This falls down on line 5, divide by (a-b). If a=b then (a-b)=0. This leads on to a situation where 0 is divided by 0. (a-b)/(a-b). The common but, incorrect assumption is then made that 0/0=1. This is what the trick relies on. At the stage where you divide through the equation by (a-b) to leave a+b = b, the trickster has lead you to believe that (a-b)/(a-b) = 1, otherwise the equation would collapse to 0 on both sides. It is generally accepted that 0/0 can't be determined, or is an indeterminate number. Whatever its value, it certainly isnt equal to 1, so the whole equation is invalid and a nonsense.


It reminds me of this old chestnut from my days of studying mathematics. (slightly modified)

You are on a game show and you stand to win the electric bicycle of your choice. There are three inverted cups numbered 1,2 and 3 standing on a table. Under one of the cups are the keys to the bike of your dreams. All you have to do is lift the cup which covers the keys. You have two chances to select the correct cup.

You lift the first cup, number 2...............there is nothing beneath it. So now it is a choice between the two remaining cups, Cup 1 and Cup 3. You put your hand on Cup 3 and are just about to lift it. The host shouts stop! He offers you the chance to change your mind and pick the other cup.

The question is, if you swap, are your chances of success increased?

 

[flecc]

Your line 5 conclusion matches that of the wikipedia article I referred to earlier on page 1 of this thread here. Though I observed the way the fault arose from line 4, the result is the same, the conclusion is invalid.

 

[Deleted member 4366]

This falls down on line 5, divide by (a-b). If a=b then (a-b)=0. This leads on to a situation where 0 is divided by 0. (a-b)/(a-b). The common but, incorrect assumption is then made that 0/0=1. This is what the trick relies on. At the stage where you divide through the equation by (a-b) to leave a+b = b, the trickster has lead you to believe that (a-b)/(a-b) = 1, otherwise the equation would collapse to 0 on both sides. It is generally accepted that 0/0 can't be determined, or is an indeterminate number. Whatever its value, it certainly isnt equal to 1, so the whole equation is invalid and a nonsense.


It reminds me of this old chestnut from my days of studying mathematics. (slightly modified)

You are on a game show and you stand to win the electric bicycle of your choice. There are three inverted cups numbered 1,2 and 3 standing on a table. Under one of the cups are the keys to the bike of your dreams. All you have to do is lift the cup which covers the keys. You have two chances to select the correct cup.

You lift the first cup, number 2...............there is nothing beneath it. So now it is a choice between the two remaining cups, Cup 1 and Cup 3. You put your hand on Cup 3 and are just about to lift it. The host shouts stop! He offers you the chance to change your mind and pick the other cup.

The question is, if you swap, are your chances of success increased?

You have a better chance of winning if you change your mind. Unknown to some, I do in fact have a degree in maths, so it's easy to figure out.

 

[tillson]

You have a better chance of winning if you change your mind.

Correct. Go to the top of the class.

 

[flecc]

You have a better chance of winning if you change your mind. Unknown to some, I do in fact have a degree in maths, so it's easy to figure out.

I can only answer that by asking why it is that professional mathematicians disagree and state categorically that this is a mathematical fallacy, indeed a well known one amongst many such, as I'd previously remarked.

 

[tillson]

I can only answer that by asking why it is that professional mathematicians disagree and state categorically that this is a mathematical fallacy, indeed a well known one amongst many such, as I'd previously remarked.

I think this one has stimulated quite a bit of debate amongst mathematicians. When I first saw it, I worked through the problem myself before reading many of the proposed solutions. My conclusion is that the chances of winning are enhanced by a change of mind, but I know that this is not a view shared by everyone and I could be wrong. However, I can't see where the error is in my workings, if indeed there is an error.

 

[indalo]

why it is that professional mathematicians disagree and state categorically that this is a mathematical fallacy, indeed a well known one amongst many such

It's simply because mathematicians invented and resolved conundrums such as this many years ago and observe rules to preserve the purity of maths. If the gentleman here is not fibbing and indeed possesses a degree in mathematics, I don't understand why he is insistent in his view of this.

Indalo

 

[indalo]

I think this one has stimulated quite a bit of debate amongst mathematicians. When I first saw it, I worked through the problem myself before reading many of the proposed solutions. My conclusion is that the chances of winning are enhanced by a change of mind, but I know that this is not a view shared by everyone and I could be wrong. However, I can't see where the error is in my workings, if indeed there is an error.

This was dealt with not long ago on TV and I couldn't see how there was any advantage to be gained but the program clearly demonstrated otherwise. Can't remember the show though.

Indalo

 

[flecc]

This was dealt with not long ago on TV and I couldn't see how there was any advantage to be gained but the program clearly demonstrated otherwise. Can't remember the show though.

Indalo

Ditto, I also saw it and cannot recall.

 

[flecc]

I think this one has stimulated quite a bit of debate amongst mathematicians. When I first saw it, I worked through the problem myself before reading many of the proposed solutions. My conclusion is that the chances of winning are enhanced by a change of mind, but I know that this is not a view shared by everyone and I could be wrong. However, I can't see where the error is in my workings, if indeed there is an error.

The chance can be mathematically shown to be better if one changes one's mind, but perhaps the calculation premise is false? If the two cup stage is viewed as a new stage in isolation, it has certain simlarities to that other old one about spun coins. i.e. If a coin is spun 9 times and each times lands as heads, what is the chance of it landing tails on the tenth spin?

 

[Orraman]

You lift the first cup, number 2...............
The question is, if you swap, are your chances of success increased?

Is this the same game?

The contestant points to one door of three.
The host opens a different door which he knows to be empty and asks if the contestant would like to change.

The Magical Maze by Ian Stewart says change, referencing a 100,000 run of a computer simulation and invites the reader to create their own simulation.

Being slow I struggled but found ~ 2/3 if change ~ 1/3 if not.

Dave

 

[tillson]

Is this the same game?

The contestant points to one door of three.
The host opens a different door which he knows to be empty and asks if the contestant would like to change.

The Magical Maze by Ian Stewart says change, referencing a 100,000 run of a computer simulation and invites the reader to create their own simulation.

Being slow I struggled but found ~ 2/3 if change ~ 1/3 if not.

Dave

Yes, that's the same one. It comes in various guises, but essentially they are all mathematically the same.