Treasure hunt
A man has a treasure map. It shows an island with a gallows, an elm tree and an oak tree. The map has instructions. They say go to the gallows and walk to the elm tree, counting your paces. Then turn 90 degrees right and walk the same number of paces as you had just counted. Mark that spot. Go back to the gallows. Now walk to the oak tree, counting your paces. Turn 90 degrees left and walk the same number of paces as you had just counted. Mark that spot. The treasure is buried halfway between the two marked spots.
When he got to the island, he could find the trees, but not the gallows. How can he find the treasure?
There are quite a few ways to answer this problem. One is partly outside the box. It is a famous problem.
When he got to the island, he could find the trees, but not the gallows. How can he find the treasure?
There are quite a few ways to answer this problem. One is partly outside the box. It is a famous problem.
Labels: lateral thinking, mathemagic, mathschallenge, outside-the-box, puzzle
posted by Chris at
12:50 PM
22 Comments:
I drew this up with the gallows at a random point. I put the Elm and Oak at random points.
I then consructed circles at each tree with the radius being the distance to the gallows.
I constructed perpendiculars to the right an left (elm, then oak)
that extend to the circles for each tree.
I then found the midpoint of the line connecting the two intersections.
I then moved the gallows and did the same steps a couple of times.
Each time the midpoint was the same point.
So it does not matter where the gallows are located. Pick a point for the gallows and follow the steps.
you can view my maps in my blog.
http://ragknot.blogspot.com/
Ragknot, that was unbelievably quick. It was also very clearly explained. Your CAD system sure comes in handy. Needless to say, you got it right.
In view of your answer, I might as well reveal that the outside the box answer was to recognise that IF the puzzle is solvable, it cannot matter where the gallows was.
The original problem was designed to be solved using complex numbers. Anyone care to try?
About my CAD system...
I was the Engineering Cad proffessor for about 5 years at our local college.
Teaching AutoCad 3D was my favorite class.
AutoCad 3D rules
If, for any given positions of the oak and the elm, the treasure is always in the same spot regardless of where the gallows were, then one would think that there should be a geometric way of solving the problem, i.e. one which doesn't involve compass and ruler or, indeed, complex numbers.
well this is not a complete answer...if the gallow is C and trees are A and B then if C is a point on the line of simetry of AB then it does matter where the gallow is because the path from A and from B will finish in the same spot.and then u cannot know where the treasure is...but then u can say there is no halfway...so we can asume that C isnt a point on the line of simetry...so we can then choose a random point and take the actions necessary but then we must choose another point which is simetrical with the first one and do the same actions.this requires a lot of geometry and it is really impossible to show it without a picture...because it is possible to prove that if C isnt on the line of simetry than for any C the treasure is on the same spot...and not only drawing the picture a few times with different C points...that is not a mathematical way of solving this just pure guessing...
The treasure just happens to be in a very special location in relation to the trees.
One way to find it would be by pretending the gallows was at the elm tree, say. So the first marked spot would then be the elm tree. Now simply follow the instuctions for the oak tree bit.
Ancient geometers only allowed the use of a compass and straightedge. Rulers and protractors weren't allowed.
Ragknot didn't cheat, he used a computerised equivalent of compass and straightedge - he didn't measure angles or distances - he only copied distances and constructed right angles.
The information is complete (surprisingly, but that's by design), if it weren't Ragknot wouldn't have been able to find the treasure.
milos, I've just done another update for the weight problem.
The man should walk from the oak tree to the elm tree counting his paces till the midpoint of the trees,then turn 90 degrees right at midpoint of the trees and walk the same number of paces as just counted and he will be standing right where the treasure is.
or if the man walks in the opposite direction,that is,from the elm to the oak tree,he should turn 90 degrees left.
milos, I've made yet another update to the weight problem - my last one had disappeared - I might have deleted it in error.
Let E be the complex number that represents the position of the elm, K the oag and G the gallow. After going from G to E, turning right and walking the same length we reach P1=E-i(E-G); doing the same with the oak but turning left we reach P2=K+i(K-G). the treasure is at the midpoint: T=(P1+P2)/2=E+K+i(K-E), which does not depend on G.
To find the treasure, start in E (for example), walk to K, turn left and walk half the number of steps you took from E to K.
Hi Ragknot, I had seen your personal site. I'm embarrassed to say
that, a few years ago, I was an (unqualified) "Teacher of Physics"
for five whole weeks. I only really enjoyed teaching the,
affectionately named, "drongo" class. They were great, we had a
lot of fun together. I couldn't stand much else about the realities
of the job. On leaving, one of them told me that most of them were
jealous that I was able to resign. I think that good teachers
should be paid enormous salaries.
Hi Miguel. Close enough for a cigar.
You made a small error, T = (E+K+i(K-E))/2. Your last instruction is wrong, but I don't care. Lovely answer. Thank you.
SBA, that's correct.
As the other methods are a bit boring, I'll round up.
This puzzle was invented by Prof. George Gamow. He is best known for writing physics books for the layman. "Mr. Tompkins adventures in Wonderland" and others in the series, have Mr. Tomkins experience world's where the different fundamental constants of nature are given large values, so that e.g. quantum mechanics and relativity becomes the norm in everyday life.
---
SBA I removed one of my posts as I had made a mistake in it when responding. I should have checked better.
Miguel. I didn't go as far as I should, the way you defined and introduced variables and made your clear and clean argument was a real treat to see. Thank you again.
SBA, I started to write the "best way" answer, then realised that is what you'd done. Thanks.
For those of you who aren't familiar with complex numbers, the reason Miguels's solution works is because multiplying a complex number by i is equivalent to rotating that number 90 degrees anti-clockwise (left in this problem). And multiplying by -i rotates 90 degrees clockwise (right in this problem).
For a little more understanding, Google with "argand diagram".
Yep Chris, I did the two mistakes... *blush*. My only escuse is that I did it when I was in my office, and had sooooo little time to think. Sorry all.
Bit late, but I just re-read milos' post. I don't think it's fair to say Ragknot guessed. He did some experiments and made a crucial observation. I'm sure that he'd have been slightly surprised and would have tried an extreme test or two to check. He'd cracked the most important fact - the location of the gallows is irrelevant.
Even if he had been cheating, that shouldn't stop you publishing your own solution. If you give a better solution, you'll get public due respect from me:)
George Gamow was also the chap who first recognised that the universe started with a big bang. But it was Fred Hoyle who first used the phrase "big bang". Fred Hoyle also explained the creation of all the elements (except hydrogen). The big-bang was needed to explain that.
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