Who Killed Mr

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  • 1. Who killed Mr. X?The investigation of a crime by adetective mathematician!THODORIS ANDRIOPOULOS

2. The story takes place in 1900 in Paris 3. where one of the most significant mathematics conferences is being held. At the entrance of the hotel conferenceroom there is an inscription: 4. Let no one ignorant of Geometryenter these doors 5. From the podium the famousmathematician, Mr. X, states at the closing of his speech:Mathematics is the Absolute Truth. Sooner or later, it canprove whether atheory is right or wrong or it can characterize a sentence true or false. 6. The work of the first day of theconference is done. Mr. X is alone in the dining roomreading. 7. We mustA waiter enters know thetruth The truth is we have toclose the bar. Do youneedanything? 8. Mr. X asks for a glass of water and the waiter leaves to bring it. Themathematiciansare totally crazy 9. The waiter returns. 10. Mr. X is dead 11. The waiter is interrogated by detective Kurt. He describes the preceding scene and adds that one second before he entered the dining room for the secondtime, he heard the central hotel clock striking. 12. The detective confirms that the murderer had 20 seconds at his disposal to commit the crime, which is the time between the waiters exit and his re-entrance to the dining room. 13. Mr. Kurt talks with his assistant. Let us assume that themurderer committed the crime immediately after the waiterleft the dining room. Thatgave him 20 seconds to movearound until the clock struck. 14. I believe the murderer was walking as he left the dining room in order not tolook suspicious, so, estimating someone walks one meter per second,then the murderer could not have been more than 20 meters away from the scene of the crime when the clockstruck. 15. Detective Kurt questions the suspects 16. Here is the hotel ground plan 17. After the first suspects S1, S2, S3, S4, and S5 testify, their positions are placed on a diagram (a ground plan of the hotel), where M is the spot where the murder took place. The suspects could only move on the lines that represent the hotel corridors shown below.GS2 LJ F S3 S5S4 V D MC S1QW U 18. M w2 9 D w 2 2w 20KNOWN3w 1 EVIDENCE ELEMENTS:C w +8Mr. Karl Friedrich (S1) statesC // DM that he was at point A when C=w+8 S1the clock struck. C=3w-1CM=w2-2w-20 SOLUTION-ANSWER DM=w2-9C CThe triangles ABC and ADM are similar so = M DM w +8 3w 1w +83w 1 = 2 or = w 2 w 12 w 9 (w 4)(w + 3) (w 3)(w + 3) w + 8 3w 1or=or( w + 8) (w 3) = ( 3w 1) (w 4) w 4 w 3or 2w2 18w + 28 = 0 and the root is w=7 (We dont accept the root w=2 since it produces DM= -5) M=30m, so Mr. Karl Friedrich cannot be the murderer. 19. G 3x-y-3KNOWNS2EVIDENCE ELEMENTS: x+2y+10 2x+y+5Mr. Constantin (S2) states thatD= x+6y-8 he was at point H when the =x+2y+10 x+6y-8 clock struck. G=3x-y-3 MD M=2x+y+5 SOLUTION-ANSWER Z = H = 90o The triangles GM and D are equal so G=D and M= 3x-y-3= x+6y-8 and2x-7y= -5 and2x-7y= -5 and2x+y+5=x+2y+10x-y=5x=y+5 2(y+5)-7y= -5 and 2y+10-7y= -5 and -5y= -15 andx=y+5 x=y+5x=y+5 x=8 and y=3 M=24m , so Mr. Constantin cannot be the murderer. 20. bLb JKNOWNEVIDENCEELEMENTS: a S F12Mr. Isaac (S3) states that he was at=3 point K when the clock struck.L=LJ a 12 =J=12 MaNM, or a + 2b >(2a) 2 + (2b) 2 3We square both sides, (a + 2b)2 > (2a)2 + (2b)2 a 2 + 4ab + 4b 2 > 4a 2 + 4b 2 4ab > 3a 2 b > a , 4which is a valid operation, since is given b>a.The shortest distance that Mr. Isaac could have covered is KF+FM=24m, so he cannot be the murderer. 21. 40 G EVIDENCE KNOWNELEMENTS: Mr. Leonhard (S4) states thatDM=G=40 when the clock struck he wasM=24 24in corridor MD at a point such Y = 90 that the distance from corridor D plus the distance from MRD corridor M was equal to 24m.The detectives assistant cries out: Mr. Leonhard was at point M, so he is the murderer! SOLUTION-ANSWER We assume point R on the side MD. We take the heights RT to MH and RY to D. RY=TH (1), since RYHT is a rectangle. We take the height RX to MP. The triangles RM and RTM are equal, since they are both right, they have a common side RM and angles RMX = TRM , since angles RMX = RDY (the triangle MPD is isosceles) and angles TRM = RDY (corresponding angles on RT//DP). So, RX=TM (2) and from (1) and (2) we have: RX+ RY=M+=MH=24. This means that the point R can be any point on the side MD, therefore we cannot conclude whether Mr. Leonhard is guilty or innocent, since he could have been either to the right of the mid-point of MD or to the left of the mid-point of MD. 22. S5 V MEVIDENCEMr. Ren (S5) states that when the clock struck he was at point V and if the rectangle MVQW had an area equal tofour times its actual area and remained similar to the initial rectangle, then thedistance from point M would have been QW 60m.The detectives assistant cries out: One fourth of 60 is 15, so Mr.Ren was at a distance 15m from point M, so he is the murderer!SOLUTION-ANSWER Let E be the actual area of the rectangle and E the area of the similar rectangle2E d then its true that = E d since the ratio of the areas of two similar shapes is equal to the square of the ratio of their sides. 22E d 1 d 1 d=or= or =a n d fin a lly, d = 3 0 m 4E 60 4 60 2 60So, Mr. Ren cannot be the murderer. 23. EVIDENCE Mr. Pierre (S6) states that his distance from point M when the clock struckis given by the function d(x)=2x2-12x+43 for an appropriate value of x The detectives assistant cries out:We cant find the answer! We have to calculate the value of the function for aninfinite number of values of x. SOLUTION-ANSWER The function d(x) has a minimum value. Let us calculate this value: The discriminant is = -200 The minimum value of d(x) is dmin= = 254 So, Mr. Pierre cannot have been the murderer since his minimum distance from point M was 25m. 24. EVIDENCEMr. Blaise (S7) states: You can find my distance from point M when theclock struck, if you know that five times this distance increased by 10and the whole thing divided by twice this distance increased by 4, is equal to 2.5m. I know this statement is too long but I didnt have time to make it shorterThe detectives assistant cries out: The answer is 0 then Mr. Blaise was at point M, so he is the murderer!SOLUTION-ANSWER If d is the distance then it holds that: 5d + 10 = 2.5 or 5d + 10 = 2.5(2d + 4) or 5d + 10 = 5d + 10 2d + 4 or 0 d = 0. So, the equation is indeterminate. Then, the distance could be 0 or any positive number. Therefore, we cannot conclude whether Mr. Blaise is guilty or innocent. 25. The assistant informs the detective that an employee heard Mr. Pheidias (S8) saying to someone:Do as I tell you, and you will be rewarded with gold. The detective calls in Mr. Pheidias to give evidence. 26. EVIDENCE What Mr. Pheidias (S8) said was: Separate a line segment 10cm in length into two parts, one with length x and one with length10 10-x so that: x2=10.(10-x). Then calculate the ratio x . Do as I tell you, and you will be rewarded with gold.SOLUTION-ANSWER The equation becomes x2+10x-100=0, whose positive root is the number 5( 5 1) Then, 10 = 5 + 1 , which is the number of the Golden Ratio.x2 Any piece of artwork containing this number offers us the sense of harmony and beauty. Therefore, the gold that Mr. Pheidias promised was the golden number , so we cannot consider him guilty. 27. EVIDENCEMr. Evarist (S9) states that he knows who killed Mr. X.He knows, because they both come from France. Mr. Pierre (S6) lied to you. He killed Mr. X! I know the French very well, and they are all liars.SOLUTION-ANSWER Assistant: If Mr. Evarist is telling the truth, then Mr. Pierre lied to us and Detective: One moment. If Mr. Evarist is telling the truth that the French always lie, then since Mr. Evarist is also French, he is also lying that he knows the killer. Assistant: So Mr. Evarist is lying. Detective: If Mr. Evarist is lying about the French being liars, then the French tell the truth and so Mr. Evarist, as a Frenchman, is telling the truth. Assistant: If Mr. Evarist is telling the truth, then he is lying, on the other hand, if he is lying, then he is telling the truth. Detective Kurt confirms the conclusion with a grimace and, putting his hands to his temples, he thinks while staring out of the window. 28. Assistant: After all mathematics doesnt have all the answers. Detective: What did you say? Assistant: I said Mathematics cant solve all problems. Suddenly the detectives face lights up and he mumbles to himself as he leavesThat killed him!!! 29. Everyone is gathered in the conference room.Detective Court explains:In the last lines of his notes, Mr. X wrote:Sentence A: Mathematics cant provesentence A 30. If the above sentence is characterized as true, then its meaning is corroborated. So theconclusion is mathematics cant prove a truesentence. 31. However, if sentence A is characterized as false that means mathematics can prove a false sentence, which is not acceptable. 32. Conclusion: If sentence A is true mathematics cant prove it. 33. Mathematics is theAbsolute Truth. Sooner or later, it can prove whether a theory is right orwrong or it cancharacterize a sentence true orfalse. 34. It seems that Mr. X found out that Mathematics is not complete,in other words, that there will always be sentences or theories for which we cannotdetermine whether they aretrue or false. 35. Mr. X dedicated his life t the quest for truth and when it was revealed to him, it took hisown life. 36. The truth killed Mr. X!!! 37. The above story is thankfully imaginary!THE THEORYOF INCOMPLETENESS, as proven by Kurt Gdel in 1931,is unfortunately REAL!!! 38. The Protagonists who took part unintentionally are: Detective Kurt Kurt Gdel Austrian Mathematician (1906-1978) Mr. XDavid Hilbert German Mathematician (1862-1943) Suspect Karl Friedrich (S1) Karl Friedrich Gauss German Mathematician (1777-1855) 39. Suspect Constantin (S2)Co