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#51 2012-12-04 14:51:17

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,584

Re: A few questions

Hi Agnishom;

Will you please explain this line?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#52 2012-12-05 01:00:26

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
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Re: A few questions

Thanks

Question 5:
The 64 squares of an 8×8 chessboard are filled with positive integers in such a way that each
integer is the average of the integers on the neighbouring squares. (Two squares are neighbours
if they share a common edge or a common vertex. Thus a square can have 8, 5 or 3 neighbours
depending on its position).

Show that all the numbers are infact equal


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#53 2012-12-05 09:50:13

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 88,584

Re: A few questions

Hi Agnishom;


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#54 2012-12-05 14:41:31

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
Website

Re: A few questions

Hmm... I see

Question 6: Prove that the ten’s digit of any power of 3 is even. [e.g. the ten’s digit of 3^6 = 729 is 2].


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#55 2012-12-05 22:03:15

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,600

Re: A few questions

You can just start writing out the last two digits of all powers of 3. It should evetually get into a loop. Yiu can then note that all powers of three have an even ten's digit.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#56 2012-12-05 22:40:10

scientia
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Registered: 2009-11-13
Posts: 222

Re: A few questions

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#57 2012-12-06 01:06:10

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
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Re: A few questions

Hi anonimnystefy;
Would the loop be enough for a proof? neutral

Hi scientia;
Would you please explain a little? Especially, the last two lines


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#58 2012-12-06 01:29:41

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: A few questions

Agnishom wrote:

Hi scientia;
Would you please explain a little? Especially, the last two lines


The last two digits of
are the last two digits of
. Hence the ten's digit of
(call it N) is the last digit
plus any carry over from
. If
,
is a single-digit number so there is no carry-over: N is just the last digit of
. If
,
so the carry-over is 2: N is the last digit of
.

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#59 2012-12-06 02:00:30

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: A few questions

Here's another proof, by induction; perhaps you might find it easier to follow. smile


First note that the tens' digit of
is 0, which is even.

Suppose the tens' digit of

is even, say,
. If the its last digit is
, we can write

where

is a multiple of 100 and
(powers of 3 can only end with these digits).

Now multiply through by 3.

The tens' digit of

is thus the last digit of
plus the carry-over from
.

(i) If s = 1 or 3, there is no carry-over: the tens' digit is the last digit of

.

(ii) If s = 7 or 9, the carry-over is 2 because

or
, so the tens' digit is the last digit of
.

In either case, the tens' digit is even. Hence, by induction, the tens' digit of

is even for all
.

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#60 2012-12-06 03:49:28

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
Website

Re: A few questions

I think I like this proof more! Thank You smile


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#61 2012-12-06 13:45:14

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
Website

Re: A few questions

Question 7: Determine the set of integers n for which

is a square of an integer.

Last edited by Agnishom (2012-12-06 14:56:39)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#62 2012-12-06 15:21:01

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,584

Re: A few questions

Hi Agnishom;


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#63 2012-12-07 00:53:47

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
Website

Re: A few questions

In the first place how do I know that we have to multiply the sides by 4?
I mean how do we get that idea?

By the way, what do you mean by 'y'?

Last edited by Agnishom (2012-12-07 00:55:36)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#64 2012-12-07 04:00:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,584

Re: A few questions

In the first place how do I know that we have to multiply the sides by 4?
I mean how do we get that idea?

Whether that is a stroke of genius or just a load of hooey, I am not sure. Perhaps when you try to complete the square or collocate for the RHS it becomes easier to see.

That y is a k, sorry. I have changed it.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#65 2012-12-07 04:27:41

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
Website

Re: A few questions

And how did you manage to find the unique solution in the last few lines?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#66 2012-12-07 04:35:18

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,584

Re: A few questions

The solution comes from this line here.

The squares only have a few numbers that they can end in. This is another way of saying mod 10. The subtraction of two squares has only one choice that can end in a 7.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#67 2012-12-07 05:20:57

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,600

Re: A few questions

Another thing you could do is apply the difference of squares formula and set the factors you get equal to factors of 7.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#68 2012-12-07 15:51:42

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
Website

Re: A few questions

bobbym wrote:

The solution comes from this line here.

The squares only have a few numbers that they can end in. This is another way of saying mod 10. The subtraction of two squares has only one choice that can end in a 7.

I understand upto that much but what should we be doing after that?

anonymnistefy wrote:

Another thing you could do is apply the difference of squares formula and set the factors you get equal to factors of 7.

Will you please explain a little bit?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#69 2012-12-07 15:58:41

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,584

Re: A few questions


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#70 2012-12-07 16:07:15

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
Website

Re: A few questions

Hmm...I see. Thanks.

How do you know that those are the only solutions? There could be other numbers which ends with 6.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#71 2012-12-07 16:09:19

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,584

Re: A few questions

Hi;

That is what we get when we solve the two equations. The mods only told us that one term had to end in 6 and the other in 9, it doesn't tell us how many. Solving the equations give the only solutions.

I can show you another way that maybe is easier for you.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#72 2012-12-07 16:23:27

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,936
Website

Re: A few questions

Oh! Please do so smile


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#73 2012-12-07 16:49:17

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,584

Re: A few questions

Hi;

We can use the difference of two squares rule to factor it like this:

The factors of 7 are 7 * 1 and -7 * -1

You now have to solve 4 sets of 2 equations.

Same answer as before.


This way is easier to understand at first but I wanted to show you how it was done with mods first. For these MO problems a very good understanding of the properties of mods is required.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#74 2012-12-07 17:13:54

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: A few questions

Here's a thought:

In general the difference of two squares can be written as

                                    2    2       2            2    2             2
                             (x+i)  - x  =  x + 2xi + i  - x   =  2xi+i   =  i(2x+i).

Setting this equal to 7, the only solution for i and x both greater than zero is x=3 and i=1.

                  2   2      2     2
So this (x+i) - x   = 4  - 3   = 16-9 = 7.

So the (2k)^2 = 16 making k=2 or -2.
And the (2n+19)^2 must equal 9 so 2n+19=3 or -3
Hence 2n+19 = 3  or  2n+19=-3
              2n = -16  or     2n = -22
                n = -8   or       n = -11
smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#75 2012-12-07 18:08:59

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 3,853

Re: A few questions

Hi noelevans,

Have you seen the 'useful symbols' above the menu headings near the top of the page?

With the superscript '2', this

                                    2    2       2            2    2             2
                             (x+i)  - x  =  x + 2xi + i  - x   =  2xi+i   =  i(2x+i).

can be turned into that

                             (x+i)² - x²  =  x² + 2xi + i² - x²  =  2xi+i²  =  i(2x+i).


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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