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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 231

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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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,666

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.****No great discovery was ever made without a bold guess. **

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

Hi bobby

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

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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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,435

"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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,666

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.****No great discovery was ever made without a bold guess. **

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

Yes only one. Look at phro's answer.

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,666

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.****No great discovery was ever made without a bold guess. **

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

What?

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,666

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.****No great discovery was ever made without a bold guess. **

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

Yes I can.

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,666

When you have the time please post your proof.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,435

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)*

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

And I did it non-experimentally!

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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**wintersolstice****Real Member**- Registered: 2009-06-06
- Posts: 123

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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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,435

Hi anonimnystefy,

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

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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 231

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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**wintersolstice****Real Member**- Registered: 2009-06-06
- Posts: 123

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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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 *n*² and (*n*+*m*)² is 2*m**n*+*m*². Consider then all possible values of *n* and *m* such that 2*m**n*+*m*²=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−*m*², 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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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 231

Wow! Nehushtan, I like your solution!!

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**thickhead****Member**- Registered: 2016-04-16
- Posts: 873

anna_gg wrote:

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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