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Nov2214
  heuristic: <where is the father?>
careful, this is a familyfriendly site! 

Nov2314
  al wazir: <Sneaky>: Good! 

Nov2314
  kellmano: hmmmmm c +21 = m and 5c + 30 = m + 6. Just do a bit of algebra and .... Hang on a minute! you're not suggesting an embryo is a child are you? A mother is 4 times older than her child. In 4 years time she will be 2.5 times older than her child. Where is the father? Bit dark for these pages perhaps. 

Nov2314
  kellmano: On a lighter note, and not too challenging, how many 3 digit numbers cannot be written as the sum of a string of consecutive integers? E.g. 120 does not meet the criteria since 18 + 19 + 20 + 21 + 22 = 120 

Nov2314
  kellmano: If you're the kind of person that likes a challenge try this: https://www.youtube.com/watch?v=Yaj... What's wrong with this argument? I'm big enough to admit I didn't get it, but then neither did most of the people in the comments section. Will tag in <micartouse>, <sneaky> and <al wazir> as I think this is a nice puzzle that you'll enjoy if you haven't seen it before. 

Nov2314   micartouse: <kellmano: What's wrong with this argument?> I'm almost sure I figured out what's wrong with the argument, but when I scrolled down, I didn't see my answer in the comments! I was very surprised as it only took a few minutes and I don't even remember much geometry. I actually think it's a lot simpler in nature than most of the commenters are making it, but I can wait to post my solution so I don't spoil the fun. 

Nov2314
  kellmano: <micartouse> I agree it doesn't seem to be in the comments. I could see that the objections to congruence were incorrect. 

Nov2314
  Sneaky: I've seen that proof about equilateral triangles a long time ago, and I don't remember anything about it, other than the fact it's a bogus proof. I think an easy way to reveal the flaw without much brainwork would be to draw a triangle that is clearly not equilateral and then follow his proof along with that drawing at hand. It's no coincidence he drew a triangle that was very nearly equilateral to prove his point. If you drew a triangle with three very different sides, I think the drawing itself should betray the tomfoolery. 

Nov2314   micartouse: <It's no coincidence he drew a triangle that was very nearly equilateral to prove his point. If you drew a triangle with three very different sides, I think the drawing itself should betray the tomfoolery.> Agree, it's a nice optical illusion. I almost have the sense someone discovered the fake proof when trying to work out a real one in the context of a different problem and then tried to retrace their steps to see what happened. 

Nov2314
  Sneaky: <On a lighter note, and not too challenging, how many 3 digit numbers cannot be written as the sum of a string of consecutive integers?> That's very interesting. I never thought about it until now. I believe it's more interesting to ask the generalized question: "Which integers cannot be written as a sum of a string of consecutive positive integers?" At first I was tempted to say that all integers can be expressed as such a sum but then I realized that the number 8 can't be done. (BTW, it's important to put the word "positive" in there, because otherwise you can sum to any number by starting in the negatives.) 

Nov2314   micartouse: <On a lighter note, and not too challenging, how many 3 digit numbers cannot be written as the sum of a string of consecutive integers?> I found this difficult to chew on. I believe the answer is 897. I had to think of the answer visually: All numbers with an odd factor can be expressed as a rectangle with an odd number of rows. Then you can make one edge diagonal just by pulling blocks off the top and putting them on the bottom. Example: Your example of 120 can be expressed as 5 rows of 24 blocks. Then rearrange the rows so they have 22, 23, 24, 25, and 26 blocks (it appears your example had a typo). This procedure can't be followed with any numbers without odd factors. The only 3 digit numbers with this property are powers of 2: 128, 256, and 512. This leaves 897. 

Nov2414
  PhilFeeley: < kellmano: hmmmmm c +21 = m and 5c + 30 = m + 6. Just do a bit of algebra and ....> I didn't think my algebra was that rusty, but solving your equations gives 4c = 3, which seems quite meaningless. I'm not sure where I went wrong. My system for the solution was:
c+21=m and 5c+6=m+6, which gives
c=5.25 years, which fits. 

Nov2414
  al wazir: <PhilFeeley: I didn't think my algebra was that rusty, but solving your equations gives 4c = 3, which seems quite meaningless.> Think again. 

Nov2414
  kellmano: yes that is right micartouse. All odd numbers or numbers with an odd factor can be written in this way. The only numbers that are neither odd nor have an odd factor are powers of 2 (which can be shown using prime factorisation rather easily). 

Nov2414   micartouse: I noticed my visual solution to the consecutive integer problem has flaws in it so it's inadequate as a proof, but it at least gets one in the ballpark of how one could go about proving it algebraically. For the equilateral triangle problem, I'm pretty sure the flaw is that the angle bisector crosses through the midpoint of the opposite segment. Therefore M and X are the same point, and all his subsequent shapes would collapse. 

Nov2414   micartouse: <Burning Ropes> You are given two ropes and a lighter. Both ropes take exactly one hour to burn. Some parts of the ropes are thicker than other parts of the rope, therefore not all parts of the rope will burn at the same speed (burning half the rope will not necessarily take 30 minutes). Find a way to measure 45 minutes of time. 

Nov2414   micartouse: I couldn't figure this one out, liked the answer. 

Nov2414
  Sneaky: <I'm pretty sure the flaw is that the angle bisector crosses through the midpoint of the opposite segment.> if the triangle is in fact equilateral, yes. If it's not, then his proof seems to hold upwhich is disturbing. That's like saying you can't prove that an equilateral triangle is equilateral but you can prove that any other type of triangle is. I think that the real problem lies in this point "X". He never demonstrated that this point "X" exists, and it's not obvious that the angle bisector will intersect with the perpendicular. Why can't it be parallel, or just veer off to the right or left? But he just drew it that way and everything stems from that. 

Nov2414   micartouse: <Sneaky> Dang you're right  it has to be isosceles to bisect the opposite side. :( I'm back to square one then  no wonder I was surprised at how quickly I got the answer; it was crap. 

Nov2514
  kellmano: <sneaky> I think it would be fairly easy to prove that unless the triangle is isosceles the angle bisector and perpendicular bisector will meet at one point outside the triangle. You are right to be suspicious about the diagram but for the wrong reason. 

Nov2514   Schwartz: Nice <Burning Ropes> puzzle, micartouse. Here's a quick riddle.. not in the usual spirit of this page though: A man is pushing his car along the road when he comes to a hotel. He shouts, "I'm bankrupt!" Why? 

Nov2514
  OhioChessFan: Lovely, <Schwarz> and I am sure I'm not the only person who likes the nonmath questions asked on this page(I think "road" is a bit of an unfair red herring, although it would be kind of hard to ask the question any other way). The math questions might as well be like trying to do Sunday puzzles blindfold and I don't even bother. 

Nov2514
  kellmano: <micartouse> doesn't involve making a figure of 8 from one of them and lighting the middle does it? That's what I thought first but it doesn't work and I can't think of another way to do it. 

Nov2514
  al wazir: <micartouse>: Light one end of of one rope and both ends of the other. When the second rope is completely consumed, that is, when the flames from the two ends meet, exactly half an hour has elapsed. Now light the intact end of the first rope. When the flame from that end meets the one from the end originally lit, 45 minutes have elapsed. 

Nov2514   micartouse: <al wazir> Yes, well done. I was picturing things like kellmano's post and setting up various contraptions with weights or water but just couldn't make progress. <OCF> Agree, I enjoy reading a diverse puzzle set from many posters, even if I only dip my toes in the water with the math ones. 


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