Math Is Fun Forum

NIH

Member

Registered: 2005-06-14

Posts: 33

Pick any five...

Take the first 10 natural numbers -- that's 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.  Pick any five of the numbers, and write them in a column, in increasing order.  Then take the five remaining numbers -- the ones you didn't pick -- and write them in a second column, in decreasing order.

In each row, subtract the smaller number from the larger one, giving you a third column.  (See the example below.)  Now add the numbers in the third column.

1    9    8   
4    7    3   
5    6    1   
8    3    5   
10    2    8   
Total:    25   

What is your total?  Everybody post their total...

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2 + 2 = 5, for large values of 2.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Pick any five...

3   10    7
4     9    5
6     5    1
7     2    5
8     1    7

Total:   25

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Roraborealis

Member

Registered: 2005-03-17

Posts: 1,594

Re: Pick any five...

1    8  = 7
3    6  = 3
7    5  = 2
9    4  = 5
10  2  = 8

Total: 25!

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School is practice for the future. Practice makes perfect. But - nobody's perfect, so why practice?

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Pick any five...

I have no imagination at the moment...

1  10  9
2  9    7
3  8    5
4  7    3
5  6    1
         --
         25

I'm annoyed at this because at a glance it seems easy to prove how it works, but on closer inspection it's not at all!
That or I'm being dumb...

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Why did the vector cross the road?
It wanted to be normal.

Zach

Member

Registered: 2005-03-23

Posts: 2,075

Re: Pick any five...

8 - 10 = 2
6 - 9 = 3
4 - 7 = 3
2 - 5 = 3
1 - 3 = 2
         ----
          13

I'm not sure if I got that correct, based on the point of the idea. Perhaps I got it wrong.

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Boy let me tell you what:
I bet you didn't know it, but I'm a fiddle player too.
And if you'd care to take a dare, I'll make a bet with you.

NIH

Member

Registered: 2005-06-14

Posts: 33

Re: Pick any five...

I'm not sure if I got that correct, based on the point of the idea. Perhaps I got it wrong.

The first column should be in ascending order.

I'm annoyed at this because at a glance it seems easy to prove how it works, but on closer inspection it's not at all!

I agree; it looks trivial, but it's not!

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2 + 2 = 5, for large values of 2.

Zach

Member

Registered: 2005-03-23

Posts: 2,075

Re: Pick any five...

Gah. Misread that then.

1 - 10 = 9
2 - 9 = 7
4 - 7 = 3
6 - 5 = 1
8 - 3 = 5
         -----
           25

---

Boy let me tell you what:
I bet you didn't know it, but I'm a fiddle player too.
And if you'd care to take a dare, I'll make a bet with you.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Pick any five...

Maths: It is still amazing to me how something that starts so simply (one pebble plus another pebble makes two pebbles) can become so rich and varied so quickly.

I mean, you can make a circle by standing on one foot and dragging the other one around on the sand. Who could possibly think that comparing the distance your feet are apart to the length your foot travels would be a number that has digits going on forever ...

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

NIH

Member

Registered: 2005-06-14

Posts: 33

Re: Pick any five...

Who could possibly think that comparing the distance your feet are apart to the length your foot travels would be a number that has digits going on forever ...

Indeed.  And who could imagine that that ratio is equal to 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ... ?  Or imagine that it is equal to sqrt(6/1^2 + 6/2^2 + 6/3^2 + ... ) ?

Last edited by NIH (2005-06-28 07:21:54)

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2 + 2 = 5, for large values of 2.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Pick any five...

Simply inspires awe.

I got into Excel and ran the first one for 50 terms (up to-4/99 ) and it got to 3.1216 (0.02 off)

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Pick any five...

You have to go incredibly far to get it accurate though.

I went to the 65536th term (filling an Excel column) and it comes to 3.141577..., meaning it's still only good to 4 decimal places!

I think NIH might have had a typo when telling us the second one...

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Why did the vector cross the road?
It wanted to be normal.

Mr T

Member

Registered: 2005-03-30

Posts: 1,012

Re: Pick any five...

i am well confused...

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I come back stronger than a powered-up Pac-Man
I bought a large popcorn @ the cinema the other day, it was pretty big...some might even say it was "large
Fatboy Slim is a Legend

NIH

Member

Registered: 2005-06-14

Posts: 33

Re: Pick any five...

I think NIH might have had a typo when telling us the second one...

Thanks -- it's fixed now!

Of course, this comes from the well known result, first proved by Euler, that pi^2/6 = 1/1^2 + 1/2^2 + 1/3^2 + ...

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2 + 2 = 5, for large values of 2.

Roraborealis

Member

Registered: 2005-03-17

Posts: 1,594

Re: Pick any five...

They were exchanging information about pi 'doccuments'..........

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School is practice for the future. Practice makes perfect. But - nobody's perfect, so why practice?

Mr T

Member

Registered: 2005-03-30

Posts: 1,012

Re: Pick any five...

exciting.. -_-

---

I come back stronger than a powered-up Pac-Man
I bought a large popcorn @ the cinema the other day, it was pretty big...some might even say it was "large
Fatboy Slim is a Legend

NIH

Member

Registered: 2005-06-14

Posts: 33

Re: Pick any five...

Here's a hint as to why the total is always 25.

In the examples above, color or shade (in the first two columns) the numbers 1, 2, 3, 4, and 5.  Do you see a pattern?

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2 + 2 = 5, for large values of 2.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Pick any five...

Always on unique lines ...

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

TomG

Member

Registered: 2005-07-18

Posts: 2

Re: Pick any five...

1 10 9
2 9 7
3 8 5
4 7 3
5 6 1

25

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Pick any five...

Yep! Still works!

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,413

Re: Pick any five...

1 20  19
2 19  17
3 18  15
4 17  13
5 16  11
6 15   9
7 14   7
8 13   5
9 12   3
10 11 1

That's 100!

Re-arranging,
1 17  16
2 20  18
3 13  10
4 19  15
5 16  11
6 18  12
7 15   8
8 14   6
9 11   2
10 12  2
100 yet again!

Should it be always (n/2)² for 0 to n???
(Just like the side total of a magic square containing numbers 1 to n² is
(n³+n)/2 ??? )

Last edited by Jai Ganesh (2005-07-18 17:03:46)

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Roraborealis

Member

Registered: 2005-03-17

Posts: 1,594

Re: Pick any five...

Er.....wasn't it numbers from 1 to 10?

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School is practice for the future. Practice makes perfect. But - nobody's perfect, so why practice?

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Pick any five...

Ganesh has extended the idea.

It seems logical. Let's try a 6 row one (1 to 12):

1   12   11
3   10    7
4     8    4
5     7    2
9     6    3
11   2    9

Total = 36

That is 6²

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Pick any five...

1  2  1

Total=1=(2/2)²

Last edited by mathsyperson (2005-07-19 02:17:49)

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Why did the vector cross the road?
It wanted to be normal.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Pick any five...

1 4 3
2 3 1
Total = 2²

1 4 3
3 2 1
Total = 2²

1 3 2
4 2 2
Total = 2²

2 4 2
3 1 2
Total = 2²

2 3 1
4 1 3
Total = 2²

3 2 1
4 1 3
Total = 2²

Is that all of them?

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Pick any five...

I think so.

--  --  --
Total=0=(0/2)²

Last edited by mathsyperson (2005-07-19 04:04:42)

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Why did the vector cross the road?
It wanted to be normal.