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Carcity
Supreme Hero
Blind Sage
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posted April 28, 2010 04:59 PM |
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Not really, I just wanted to embarass myself more about how easy you guys solved it while it took me 20 minutes to do it. But I guess test stress and morning tiredness gave it's help too (Yeah I'm totally trying to save my but now).
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Why can't you save anybody?
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ohforfsake
Promising
Legendary Hero
Initiate
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posted April 28, 2010 05:13 PM |
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I don't see why, no matter how smart (talented) someone might be, unless they've a sense of the problem in the first hand, I doubt many could do it "just like that", if anyone.
It's a lot about knowing purpose and preset and then finding a method to go from preset to purpose, and finding this method in itself can be hard, and what one really could use is a method for finding such methods.
Many have a good understanding on single subjects, most of all through memory of methods that works on these, however this form of understanding may not help much in a completely unrelated problem.
-UNRELATED-
I think it's much to do with the problem of defining intelligence, because there may exist a set of tasks one can adapt to much easier than another, and vice versa, what really need is generality, but how can one make something general enough?
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Living time backwards
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Carcity
Supreme Hero
Blind Sage
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posted April 28, 2010 05:22 PM |
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That is not intelligence, that is thinking logical.
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Why can't you save anybody?
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Azagal
Honorable
Undefeatable Hero
Smooth Snake
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posted April 28, 2010 08:28 PM |
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Quote:
The CHiang family decided to take the train to Tokyo from their old farm, the train was new and shiny and the trip to Tokyo was 57 miles long. They paid a total of 1755 Yuan for their tickets. Grandma and Grandpa got a 50% discount because they were retired, and Ling got 25% discount because she was a student. How much did Ling's parents pay each for their tickets?
Your math teacher fails at geography...
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"All I can see is what's in front of me. And all I can do is keep moving forward" - The Heir Wielder of Names, Seeker of Thrones, King of Swords, Breaker of Infinities, Wheel Smashing Lord
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted April 28, 2010 09:23 PM |
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Edited by dimis at 21:26, 28 Apr 2010.
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Q1. Irregular but predictable
Good one Azagal. 
Corribus, thanks for the video. I enjoyed it a lot too!
Ok, I guess we want problems, so I will start drawing some witty ones from a time-drawer.
Q1. It's easy to show that the sum of the five acute angles of a regular star (like the ones in the American flag or the one in the Soviet flag) is 180º (180 degrees). Prove that the sum of the five angles of an irregular star is also 180º (180 degrees).
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The empty set
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Warmonger
Promising
Legendary Hero
fallen artist
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posted April 28, 2010 09:42 PM |
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All these angles are inscribed angles build on the same curve as central angles, which sum up to 360 even though may belong to circles of different radii. So their sum is equal to half, which is 180.
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The future of Heroes 3 is here!
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted April 29, 2010 04:57 AM |
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Witty does not necessarily mean one liner. I don't think I get your proof either. Do you imply that every object that is inscribed in a circle or (even more) different circles has sum of angles 180º ? I know at least one object for which it does not hold; e.g. square.
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The empty set
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Warmonger
Promising
Legendary Hero
fallen artist
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posted April 29, 2010 06:29 AM |
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted April 29, 2010 07:10 AM |
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It is my bad, because I didn't give a picture. So here is an example of a regular and an irregular star; the regular clearly on the left:
Now, we want to prove that the sum of their angles (on the tips) is equal to 180º (180 degrees).
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The empty set
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ihor
Supreme Hero
Accidental Hero
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posted April 29, 2010 07:33 AM |
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Nice geometry problem. Not very difficult but interesting.
I'll give a chance for others to solve it.
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted April 30, 2010 01:56 AM |
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Edited by dimis at 02:04, 30 Apr 2010.
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N is a Number: A Portrait of Paul Erdös
By the way, I think this is an interesting movie for historical reasons too (lasts about an hour or so).
It is available in youtube in different parts: 1, 2, 3, 4, 5, 6, 7, 8, 9.
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The empty set
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friendofgunnar
Honorable
Legendary Hero
able to speed up time
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posted April 30, 2010 10:08 AM |
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I thought of a way, however it references the sum of the interior angles of a pentagon, which is 540 degrees.
Draw a line connecting each tip and you have a pentagon totalling 540 degrees. This 540 degrees is equal to X (the sum of all the star point angles) plus 10 "side" angles.
540 = X + Side Angles
There's also a pentagon in the middle of the star. Each angle of that pentagon is congruent to the inside angle of a triangle who's outside face makes up the outer pentagon. Thus the sum of the 10 side angles is equal to 5 triangles (5 * 180 = 900 degrees) minus the inner pentagon (540 degrees), which equals 360.
Side Angles = (5 * 180) - 540 = 360
Combine the two equasions and you have
540 = X + 360
X = 180 degrees
Still, it would be nice if the solution could be reached without referencing the sum of the interior angles of a pentagon. Can anybody think of a way?
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ihor
Supreme Hero
Accidental Hero
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posted April 30, 2010 12:32 PM |
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Edited by ihor at 12:32, 30 Apr 2010.
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You can split pentagon into 3 triangles by drawing 2 diagonals from one arbitrary vertex. So the sum of interior angles of pentagon is 3 times the sum of interior angles of triangle.
3 * 180 = 540.
This references to sum of interior angles of triangle which is basic stuff and could be easily proven.
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Ecoris
Promising
Supreme Hero
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posted April 30, 2010 01:03 PM |
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Edited by Ecoris at 13:05, 30 Apr 2010.
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You can also do as follows:
For each pair of star tips that are not "neighbours" you form the unique triangle using these two points and the lines from them. Two such triangles are displayed below as examples.
The last vertex is a vertex of the inner pentagon and if we sum the angles of all 5 triangles we get that
900 = 2*[sum of angles at tips] + [sum of angles in a pentagon]
and the result follows once we know that the sum of the angles in a pentagon is 540.
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Corribus
Hero of Order
The Abyss Staring Back at You
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posted April 30, 2010 03:20 PM |
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I was/am a little confused by what an "irregular star" is.
For instance, Ecoris uses this as the basis of his solution:
Quote:
For each pair of star tips that are not "neighbours" you form the unique triangle using these two points and the lines from them.
That seems to me to be an assumption that doesn't necessarily have to be so. It's possible to form a 5 pointed star that isn't satisfied by this assumption. Or did dimis have this assumption in mind when he specified "irregular star" (certainly it appears this might be so by the way he drew it)? I'm not sure.
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted April 30, 2010 04:23 PM |
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Edited by dimis at 16:29, 30 Apr 2010.
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Great! I used the pentagon initially too. However, I think there is a simpler solution given below:
Connect two neighboring vertices with a line (say C and D). Now, look at the triangles CDM and BME. We have angle(M1) = angle(M2) by construction, so b+e = c2 + d2. But in the bigger triangle ACD we have a + c1 + c2 + d1 + d2 = 180º, where by re-arranging and substituting we get a + c1 + d1 + b + e = 180º which is precisely what we want.
As I said I opened kind of a "time-drawer", so these are puzzles, where not everything will be defined strictly. Perhaps at certain points we have to guess. So yes Corribus, I assumed that the star was drawn the usual way we draw stars on a piece of paper. We can create it with 5 straight lines (the dots being part of them). In any case, I didn't alter the way the problem it was given. It was an exact copy of the way the problem was given.
Problem by: A. Korshkov
Alright, soon comes the next one ...
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The empty set
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted April 30, 2010 04:41 PM |
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Edited by dimis at 16:42, 30 Apr 2010.
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Q2. Expanding Gap
Using each of the numbers 1, 2, 3, and 4 twice, I succeeded in writing out an eight-digit number, in which there is one digit between the ones, two digits between the twos, three digits between the threes, and four digits between the fours. What was the number ?
(A. Savin)
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The empty set
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Corribus
Hero of Order
The Abyss Staring Back at You
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posted April 30, 2010 05:15 PM |
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Unfortunately, I'm not sure if there's some logic to the problem since I found the answer on my second guess.
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted April 30, 2010 05:18 PM |
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Edited by dimis at 17:19, 30 Apr 2010.
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As I said, these are puzzles, not very deep problems. But are you 100% sure that if you write down a number you can find the number that I wrote ? 
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The empty set
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ihor
Supreme Hero
Accidental Hero
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posted April 30, 2010 07:16 PM |
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I spent almost 10 minutes to guess it .
I'm not sure what do you mean dimis. I think there are 2 such numbers. So your number is the same as Corribus's or reversed one.
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