[NatureNS] Christmas brain teazer

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From: Patrick Kelly <patrick.kelly@dal.ca>
Date: Wed, 20 Dec 2006 19:37:37 -0400
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That reminds me of something I remember reading once:

2 is not equal to 3 =96 not even for large values of 2.

Pat



On Dec 20, 2006, at 7:22 PM, David & Alison Webster wrote:

> Hi Chris & All,
>    But notice that  one doesn't quite get to divide by zero because =
the
> previous line (below) says that a*zero =3D b*zero and this condition =20=

> alone voids any conclusions that one might attempt to make about a and =
=20
> b.
>
> Yt, DW
>
>
> Factor :
> a ( a - b -c) =3D b ( a - b -c)
>
>
>
> c.majka@ns.sympatico.ca wrote:
>
>> Hi Dave,
>>
>> Nice riddle: made me scratch my head for a few minutes, However, as  =20=

>> Paul points out (a - b - c) =3D 0 and dividing by 0 (at the last =
step)  =20
>> allows one to prove whatever one wishes. ;->
>>
>> Cheers,
>>
>> Chris
>>
>> On 20-Dec-06, at 9:17 AM, David & Alison Webster wrote:
>>
>>> Dear All,     Dec 20, 2006
>>>    I recently came across an old article about Raymond Smullyan, a
>>> highschool dropout who e.g. satisfied the course requirements for a  =
=20
>>> PhD
>>> in Math and Philosophy by teaching the required courses. He also  =20=

>>> created
>>> many logic puzzles. One thing leads to another, so I blew the dust =20=

>>> off
>>> of an old copy of Martin Gardner's 1959 Mathematical Puzzles and
>>> Diversions.
>>>
>>>    One of his fallacies follows. I hope you have not seen this  =20
>>> previously.
>>> Proof that unequal numbers are equal
>>>
>>>    Given two numbers a & b such that b is smaller than a by an =
amount
>>> c; thus
>>>
>>> a =3D b + c
>>>
>>> prove that a =3D b
>>>
>>> multiply both sides by (a - b) to obtain--
>>>
>>> a^2 - ab =3D ab +ac - b^2  - bc
>>>
>>> subtract ac from both sides to obtain
>>>
>>> a^2 - ab - ac =3D ab - b^2  - bc
>>>
>>> Factor :
>>> a ( a - b -c) =3D b ( a - b -c)
>>>
>>> Divide both sides by ( a - b - c) to obtain
>>>
>>> a =3D b
>>>
>>> QED
>>>
>>>    "The road to correct conclusions is full of pitfalls" (DHW, Dec  =20=

>>> 20, 2006).
>>>
>>> Merry Christmas, Dave Webster, Kentville
>>>
>>>
>>
>>
>
>
>
>

=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20
=3D=3D
Patrick Kelly
Director of Computer Facilities
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20
=3D=3D
Faculty of Architecture and Planning
Dalhousie University
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20
=3D=3D
PO Box 1000 Stn Central                5410 Spring Garden Road
Halifax, Nova Scotia B3J 2X4           Halifax, Nova Scotia B3J 2X4
Canada                                 Canada
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20
=3D=3D
Phone:(902) 494-3294    FAX:(902) 423-6672   E-mail:patrick.kelly@dal.ca
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20
=3D=3D


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That reminds me of something I remember reading once:


2 is not equal to 3 =96 not even for large values of =
2.<fontfamily><param>Bookman</param>


Pat



</fontfamily>

On Dec 20, 2006, at 7:22 PM, David & Alison Webster wrote:


<excerpt>Hi Chris & All,

   But notice that  one doesn't quite get to divide by zero because the

previous line (below) says that a*zero =3D b*zero and this condition
alone voids any conclusions that one might attempt to make about a and
b.


Yt, DW



Factor :

a ( a - b -c) =3D b ( a - b -c)




c.majka@ns.sympatico.ca wrote:


<excerpt>Hi Dave,


Nice riddle: made me scratch my head for a few minutes, However, as=20
Paul points out (a - b - c) =3D 0 and dividing by 0 (at the last step)=20=

allows one to prove whatever one wishes. ;->


Cheers,


Chris


On 20-Dec-06, at 9:17 AM, David & Alison Webster wrote:


<excerpt>Dear All,     Dec 20, 2006

   I recently came across an old article about Raymond Smullyan, a

highschool dropout who e.g. satisfied the course requirements for a=20
PhD

in Math and Philosophy by teaching the required courses. He also=20
created

many logic puzzles. One thing leads to another, so I blew the dust off

of an old copy of Martin Gardner's 1959 Mathematical Puzzles and

Diversions.


   One of his fallacies follows. I hope you have not seen this=20
previously.

Proof that unequal numbers are equal


   Given two numbers a & b such that b is smaller than a by an amount

c; thus


a =3D b + c


prove that a =3D b


multiply both sides by (a - b) to obtain--


a^2 - ab =3D ab +ac - b^2  - bc


subtract ac from both sides to obtain


a^2 - ab - ac =3D ab - b^2  - bc


Factor :

a ( a - b -c) =3D b ( a - b -c)


Divide both sides by ( a - b - c) to obtain


a =3D b


QED


   "The road to correct conclusions is full of pitfalls" (DHW, Dec=20
20, 2006).


Merry Christmas, Dave Webster, Kentville



</excerpt>


</excerpt>




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=
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=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D

Patrick Kelly

Director of Computer Facilities

=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
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=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3