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 2005-07-15, 07:35 #23 Wacky     Jun 2003 The Texas Hill Country 21018 Posts So, Mally, Your reasoning is faulty... If you consider the orientation of the tetrahedron to be critical, and differentiating with respect to distinct cases, why do you stop with up vs down? Is an upright tetrahedron with one side of the base oriented in a North-South direction that same as one where that side is oriented in an East-West direction? What about my example of a ravine? There the tetrahedron is oriented neither up nor down. Since you claim that there are exactly two distinct solutions, which one is mine?
 2005-07-15, 16:36 #24 Fusion_power     Aug 2003 Snicker, AL 11110000002 Posts I have to agree with Wacky and Xilman on this one. A tetrahedron is the only viable form that fits the requirements, but the tetrahedron can be oriented vertex up, vertex down, or vertex at an angle as per the example of the ravine with two trees in the bottom and two trees up top. By extension, the "ravine" could be the dip between two mountains of varying heights such that the tetrahedron could assume just about any angle. I vote one solution, the tetrahedron. Orient it any way you choose.
 2005-07-17, 07:32 #25 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22·33·19 Posts Lateral thinking puzzle. [QUOTE=Wacky]So, Mally, Your reasoning is faulty...]UnQuote/ :surprised Lateral thinking puzzle. After reviewing the post above I am retracting my statement that there are only two solutions. Lets see how the idea developed. Paul hit the nail on the head by giving the general solution which can be represented by a tetrahedron. [tetra(h)] Numbers explained that its a hypothetical frame and not a mechanical one. Wacky went a step further when he noted a depression would also serve the purpose implying an inverted tetra (h). Subsequently he pointed out that on a flat surface itself there are infinite positions. Fusion Power on wackys tip introduced the ravine where the tetra (h) could be accommodated according to virtually any configuration and went off to sleep !.. It s not the case of who is right or wrong or allying oneself to any party. This could lead to a flaming war. Just the day before Prof. ewmayer locked out a post for this very same reason. Please read his reasons and guidelines in ‘Exponential digits’ Each post contributes to the preceding one. I would dare say even wrong posts help in the final analysis. Napoleon Hill says “When two or more minds get together the third mind is formed”. Very true!. Since I introduced the problem I learnt very much more than when I posted it It will be interesting to know how Edward de Bono himself tackled this problem. I quote from his book ‘Lateral thinking’. Quote [ .. but one tree is planted at the top of a hill and the other trees are planted on the side of the hill. This makes them all equidistant from one another (in fact they are at the angles of a tetra(h). One can also solve the problem by placing one tree at the bottom of a hole and the others around the edges of the hole] Unquote/ Hope this clears the air and my statement is acceptable. Mally .
2005-08-05, 07:14   #26
Citrix

Jun 2003

160510 Posts

Quote:
 Originally Posted by mfgoode Try this one for size. It has only two possible answers! A landscape gardner is given instructions to plant 4 special trees so that each one is exactly the same distance from each other. How would you arrange the trees? Ref. 'Lateral Thinking' by Edward de Bono. Mally

Actually there are infinite answers to this question. The 4 corners of any rohmbus will work and there are infinite of them.

Citrix

2005-08-05, 08:30   #27
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

23×3×5×97 Posts

Quote:
 Originally Posted by Citrix Actually there are infinite answers to this question. The 4 corners of any rohmbus will work and there are infinite of them. Citrix
Wrong answer, even though there are an infinite number of solutions.

A rhombus is defined as a plane quadrilateral with each pair of opposite sides parallel. One special case of a rhombus is a square. Opposite corners of a square are separated by a distance sqrt(2) times adjacent corners, so the four corners are not all the same distance from each other.

In Euclidean space, the four points must be at the vertices of a regular tetrahedron. There are an infinite number of orientations of a tetrahedron, so an infinite number of solutions.

Paul

 2005-11-24, 15:27 #28 lycorn     "GIMFS" Sep 2002 Oeiras, Portugal 62216 Posts That leads us to the original topic of this thread: the hapless chap that killed himself. He was actually quite desperate about solving the gardener´s puzzle... Thought about several types of frames (mechanical and non-mechanical ) and eventually fell asleep on a rhombus-shaped bed. Woke up with the loud noise and ran to the front door to find a huge (regular) tetrahedron in the middle of the road. And he sighed: "Silly me! This is the one and only solution to the problem" . And he ran up the stairs in shame, went to a different room to avoid being misled once again by the shape of his bed and shot himself using a conic bullet. His last thought was: "Perhaps a cone-shaped frame, with the appropriate height, will also be a solution... Oh well, forget about it, this problem is killing me!"
 2005-11-24, 17:16 #29 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22·33·19 Posts Lateral thinking puzzle. Lycorn that's a fitting ending to an unusual story. If you include the background to it, it may become a best seller and earn you millions of euros if you only publish it. Thank you I enjoyed the humour you have inculcated to a boring problem which has so many erroneous solutions! It was as is said in Latin 'Pons arsinnorum' and this translated literally is the 'Bridge of Asses' as my geometry teacher would say of the 6th Theorem of Euclid if I remember it well. Mally
 2005-11-25, 22:06 #30 lycorn     "GIMFS" Sep 2002 Oeiras, Portugal 2×5×157 Posts
2005-11-26, 10:33   #31
fatphil

May 2003

3×5×17 Posts

Quote:
 Originally Posted by xilman In Euclidean space, the four points must be at the vertices of a regular tetrahedron. There are an infinite number of orientations of a tetrahedron, so an infinite number of solutions.
Holes and hills are out of fashion in landscape gardening nowadays, I've been told. My alternative would be:
Put 2 trees at opposite sides of the top of the grand canyon, and two at the base of the valley itself.

However, you're solution of just placing them in 4 different continents is the best one, as it's not very easy. Do you know where works? (I can google for the/a solution, as ISTR the newsgroup and the poster that solved it.)

And now solve in Lobachevskian space!

Great thread hijack, BTW, to whomever performed it!

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