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 Forum index » Diversions » Perplex City Puzzle Cards » PXC: Orange Puzzle Cards
[SOLVED] #046 - Orange - Sum Shortcut
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Joined: 18 Feb 2005
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 		[SOLVED] #046 - Orange - Sum Shortcut
Text:


Fredrich Gauss was a legendary German mathematician, astronomer
and physicist. His contributions to science have been so great that
he is sometimes reffered to as the 'Prince of Mathematics'.

Even from an early age his talent was evident. His father guessed he had a child prodigy on his hands when at the tender age of three his
son spotted an error while he was calculating his payroll.

Gauss junior was also making similar waves at school. His class was
particularly rowdy and one afternoon his exasperated teacher set
them all a problem to try and keep them quiet for as long as possible.
The task was to add up all the numbers between 1 and 100. Most of
the children had barely put chalk to slate when the five-year-old
Gauss announced he had the soloution. What answer did Guass give
to his starled teacher?



Now. before you plough headlong into this problem and spend ages
summing the numbers one by one, is there an easier way to tackle this problem? You never know when such a shortcut might come in handy.


(Signed Kurt)
046_Orange_SumShortcut.jpg
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PostPosted: Fri Jul 08, 2005 6:36 pm
Last edited by Reason on Mon Jul 11, 2005 9:30 am; edited 1 time in total
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JebJoya
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 		Bah, too easy
Spoiler (Rollover to View):
- the average of the numbers is 50.5, multiplied by 100 (for the number of numbers there are) is 5050

so 5050


yay

Jeb
Spoiler (Rollover to View):

edit:
more specifically, you pair up the numbers -
100 + 1 = 101 101 / 2 = 50.5
99 + 2 = 101 101 / 2 = 50.5
etc...
51 + 50 = 101 101 / 2 = 50.5
thus the average is 50.5, and there are 100 numbers, so 50.5 * 100 = 5050 which is the answer

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PostPosted: Fri Jul 08, 2005 6:41 pm
Last edited by JebJoya on Fri Jul 08, 2005 6:50 pm; edited 2 times in total
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tanner
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 		and that story is so famous
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PostPosted: Fri Jul 08, 2005 6:43 pm
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jamesi
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 		So, is this one solved then?

/me is trying desperately to keep track
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PostPosted: Sat Jul 09, 2005 12:32 am
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JebJoya
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 		Yes. Simple answer is
Spoiler (Rollover to View):
5050


Niiiice and simple Smile

Jeb
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PostPosted: Sat Jul 09, 2005 12:32 pm
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Atrophied
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 		easy as pi Wink :

Spoiler (Rollover to View):
The following works for any summation of the type (1+2+3+...+(n-1)+n)

½n(n+1)
n=100

50(101) = 5050


I know it's already solved, but thought I'd post the equation anyway.
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PostPosted: Sun Jul 10, 2005 6:46 pm
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IQue
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Joined: 26 Jul 2005
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 		Thought i'd mention the matemathical term because it seems noone else has.

Spoiler (Rollover to View):

In swedish it's called an "aritmetisk talföljd" which would translate to something like "Arithmetic integerseries".
And the official formula for solving those is:

Sn = n*((a1 + a1 + (n-1)d)½)

Where Sn is the sum and n = number of integers, a1 = first number, d = difference between the numbers
And this could be simplified in our case as the previous post did.

Sn = 100*(101/2) = 5050


PostPosted: Tue Jul 26, 2005 2:44 pm
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JebJoya
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 		Just to add a bit to this, the point of this little story is that Gauss wouldn't have known the formula for the sum to n of an arithmetic progression. The story is normally used to show that it is often productive to not look at the obvious ways of solving a problem, and that Gauss, who was to grow up to be a kick-ass mathmo (if such a thing exists), had a mind that could see the non-obvious route to a solution... Roll on proof of the Riemann Hypothesis!

Jeb
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PostPosted: Tue Jul 26, 2005 3:33 pm
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doublecross
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 		As I remember the story, he noticed that instead of adding 1 to 2, then 3 and so on, he could add 1 to 100 and get 101, 2 to 99 and get 101, 3 to 98 and get 101 and so on up to 50 added to 51, meaning that the answer is 50 101s, or 5050.
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PostPosted: Fri Jul 29, 2005 7:50 am
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Bahamoth
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 		This all seems a bit pointless now, but, not being a mathematician, I did it a different way.

I figured that I should add 100 to 0, and 99 to 1 and so on, and this would leave 50 on its own so I would add that. It didn't occur to me to do that averaging stuff. I thought it was neater adding two numbers to make 100.

PostPosted: Sun Aug 07, 2005 4:38 pm
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ParityBit
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Bahamoth wrote:
This all seems a bit pointless now, but, not being a mathematician, I did it a different way.


Me too. Pondered this for a while, and then just used Excel, as the answer is the sum, not the method. But reading the above makes a lot of sense. Just wish I could get my brain to work that way. Confused

PostPosted: Fri Aug 19, 2005 10:11 pm
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Da Tee In TeeS of doom
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Joined: 12 Sep 2005
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 		Too easy
I can't believe we're done here! The #015 Milo was just red, but way harder taht this one! Are you guys sure the answer is just
Spoiler (Rollover to View):
5050
?
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PostPosted: Mon Sep 12, 2005 6:24 pm
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oliverkeers13
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 		Yes. And will you stop questioning long agreed solves? It's getting kind of annoying. If you have something to contribute, fine, but the repeated "are we sure" comments are just wasting space. Unless there is something on the card that we haven't used, these comments don't help.
This isn't meant to sound angry or anything, it's just a polite suggestion.
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PostPosted: Mon Sep 12, 2005 6:31 pm
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Da Tee In TeeS of doom
Boot

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 		Grey hazes
I shall stop that from now on...


BTW what are those grey/green hazes on the picture? There is some on the door, and some on some og the tables.
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PostPosted: Mon Sep 12, 2005 6:50 pm
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joshF
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 		hang on. the card says numbers between 1-100, but doesn't specify wether they are inclusive. Therefore we could be looking at the number range of 2-99 which makes the answer 4949. What do you think?

(PS. my first post here. just ordered my first cards Very Happy )

PostPosted: Mon Oct 03, 2005 5:24 pm
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