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## #1 2012-04-10 20:21:03

anna_gg
Member
Registered: 2012-01-10
Posts: 232

### Perfect Squares

Consider an integer x. If we add 30, then the result is a perfect square. If we subtract 30, the result is also a perfect square. How many such integers are there?"

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## #2 2012-04-10 20:54:19

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Perfect Squares

Hi anna

Did you try setting up the equations? Let us know what you have tried.

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
The knowledge of some things as a function of age is a delta function.

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## #3 2012-04-10 22:50:16

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Perfect Squares

Hi all;

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #4 2012-04-11 12:10:25

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Perfect Squares

Hi bobby

That is not the only one.There is one more solution.

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
The knowledge of some things as a function of age is a delta function.

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## #5 2012-04-11 13:45:13

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,594

### Re: Perfect Squares

"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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## #6 2012-04-11 13:45:52

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Perfect Squares

That is not the only one.There is one more solution.

That is true, but only one more?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #7 2012-04-11 13:50:06

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Perfect Squares

Yes only one. Look at phro's answer.

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
The knowledge of some things as a function of age is a delta function.

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## #8 2012-04-11 14:01:48

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Perfect Squares

How about one past where he looked?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #9 2012-04-11 14:04:36

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Perfect Squares

What?

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
The knowledge of some things as a function of age is a delta function.

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## #10 2012-04-11 19:43:10

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Perfect Squares

He must have searched up to some number. That was the limit of his search. Can you provide a reason why there is not a number passed his search?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #11 2012-04-11 20:18:48

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Perfect Squares

Yes I can.

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
The knowledge of some things as a function of age is a delta function.

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## #12 2012-04-11 20:29:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Perfect Squares

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #13 2012-04-11 21:07:49

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,594

### Re: Perfect Squares

Hi Bobby,

I only searched as far as shown in my image, because at that point:

The numbers that are squared (I don't know what they're called) to produce the perfect squares must differ by at least 1.

EDIT: The column E heading should be "If Col D = integer, print B + 30".

Last edited by phrontister (2012-04-11 21:20:13)

"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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## #14 2012-04-11 21:36:54

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Perfect Squares

And I did it non-experimentally!

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
The knowledge of some things as a function of age is a delta function.

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## #15 2012-04-11 21:39:11

wintersolstice
Real Member
Registered: 2009-06-06
Posts: 125

### Re: Perfect Squares

anonimnystefy wrote:

And I did it non-experimentally!

I did it diferent again:D

Last edited by wintersolstice (2012-04-11 21:44:11)

Why did the chicken cross the Mobius Band?
To get to the other ...um...!!!

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## #16 2012-04-11 22:21:47

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,594

### Re: Perfect Squares

Hi anonimnystefy,

Yes, I like that proof. Good reasoning, and with a process of elimination at the end that leaves only those two possibilities.

"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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## #17 2012-04-16 03:48:38

anna_gg
Member
Registered: 2012-01-10
Posts: 232

### Re: Perfect Squares

Here is my solution: It is based to the fact that each perfect square N^2  is the sum of the first  N odd numbers (5^2 = 25 = 1+3+5+7+9).
Thus the difference of any 2 perfect squares should equal to the sum of consecutive odd numbers (and this should equal 60).
Starting from 1, we write down the sums of the odd numbers:
1+3+5+7+9+11+13 = 49 while 1+3+5+7+9+11+13+15 = 64. Thus we cannot make 60 starting from 1.
We do the same with 3: 3+5+7+9+11+13=48 while 3+5+7+9+11+13+15 = 63. Not possible.
Starting from 5:
5+7+9+11+13+15=60 Here we are.
So, one perfect square is 4 and the next is 64, their difference being 60, so the first number we are looking for is 34 (34-30 = 4 and 34+30 = 64, both of them perfect squares).
Similarly, we find that the only other sum of consecutive odd numbers equaling 60 is 29+31.
Therefore one perfect square is 1+3+5+...+27=196 (14^2) and the next is 1+3+5+...+27+29+31=256 (16^2) and the second number we are asking for is 226 (226-30 = 196, 226+30 = 256).
There is no other series of successive odd numbers equaling 60, so these two are the only numbers with this property.
Obviously this is not a proper "proof"; it is more based on a "guess and try" method, but it works!

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## #18 2012-04-16 03:59:35

wintersolstice
Real Member
Registered: 2009-06-06
Posts: 125

### Re: Perfect Squares

anna_gg wrote:

Here is my solution: It is based to the fact that each perfect square N^2  is the sum of the first  N odd numbers (5^2 = 25 = 1+3+5+7+9).
Thus the difference of any 2 perfect squares should equal to the sum of consecutive odd numbers (and this should equal 60).
Starting from 1, we write down the sums of the odd numbers:
1+3+5+7+9+11+13 = 49 while 1+3+5+7+9+11+13+15 = 64. Thus we cannot make 60 starting from 1.
We do the same with 3: 3+5+7+9+11+13=48 while 3+5+7+9+11+13+15 = 63. Not possible.
Starting from 5:
5+7+9+11+13+15=60 Here we are.
So, one perfect square is 4 and the next is 64, their difference being 60, so the first number we are looking for is 34 (34-30 = 4 and 34+30 = 64, both of them perfect squares).
Similarly, we find that the only other sum of consecutive odd numbers equaling 60 is 29+31.
Therefore one perfect square is 1+3+5+...+27=196 (14^2) and the next is 1+3+5+...+27+29+31=256 (16^2) and the second number we are asking for is 226 (226-30 = 196, 226+30 = 256).
There is no other series of successive odd numbers equaling 60, so these two are the only numbers with this property.
Obviously this is not a proper "proof"; it is more based on a "guess and try" method, but it works!

Have you seen my proof? It's very similar in that it's based on consecutive odd numbers.

Why did the chicken cross the Mobius Band?
To get to the other ...um...!!!

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## #19 2016-02-17 03:21:53

Nehushtan
Member
Registered: 2013-03-09
Posts: 912
Website

### Re: Perfect Squares

I was bored so I searched Anna's old posts for some more mentally challenging puzzles, and I found this. Allow me to present my solution.

anna_gg wrote:

Consider an integer x. If we add 30, then the result is a perfect square. If we subtract 30, the result is also a perfect square. How many such integers are there?"

So two perfect squares differ by 60. The difference between two perfect squares n² and (n+m)² is  2mn+m². Consider then all possible values of n and m such that 2mn+m²=60. NB: (i) Obviously m has to be even. (ii) We can also assume WLOG than both m and n are positive.

From here on 60−m², and hence n, will always be negative. Hence two are only three such integers, which are

Last edited by Nehushtan (2016-02-17 03:39:35)

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## #20 2016-04-04 21:27:55

anna_gg
Member
Registered: 2012-01-10
Posts: 232

### Re: Perfect Squares

Wow! Nehushtan, I like your solution!!

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## #21 2016-04-16 03:48:14

Member
Registered: 2016-04-16
Posts: 1,086

### Re: Perfect Squares

anna_gg wrote:

Consider an integer x. If we add 30, then the result is a perfect square. If we subtract 30, the result is also a perfect square. How many such integers are there?"

Let x be the number.

Last edited by thickhead (2016-04-21 22:16:15)

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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