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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Hi Agnishom;

Will you please explain this line?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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Thanks

Question 5:**The 64 squares of an 8×8 chessboard are filled with positive integers in such a way that eachinteger is the average of the integers on the neighbouring squares. (Two squares are neighboursif they share a common edge or a common vertex. Thus a square can have 8, 5 or 3 neighboursdepending 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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Hi Agnishom;

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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Hmm... I see

Question 6: ** Prove that the tens digit of any power of 3 is even.** [e.g. the tens 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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

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

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Hi anonimnystefy;

Would the loop be enough for a proof?

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

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

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

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

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 writewhere

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

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

is even for all .Offline

I think I like this proof more! Thank You

'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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Question 7: **Determine the set of integers n for which **

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

'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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Hi Agnishom;

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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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)*

'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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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

'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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

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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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?

'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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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Hmm...I see. Thanks.

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

'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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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Oh! Please do so

'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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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

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

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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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

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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