Treasure Revisited
The solution to the original post (yesterday) was to quick.
But we can hopefully find the numeric solution with some given data.
Let's say the Elm tree is at (30,35) and the Oak is at (80,44).
These are x,y coordinates.
Find the x,y of the treasure within a 1/2 unit.
But we can hopefully find the numeric solution with some given data.
Let's say the Elm tree is at (30,35) and the Oak is at (80,44).
These are x,y coordinates.
Find the x,y of the treasure within a 1/2 unit.
Labels: mathschallenge
posted by Ragknot at
11:42 AM
28 Comments:
This post has been removed by the author.
Ragknot, I guess you like finding treasure :)
I could cheat and use Miguel's answer from the other one.
Falcon! Eggplant! Rise & shine!!!
Chris,
I see the solution that seems the easiest to me. But it seems that it gives two solutions.
1. connect the trees with a line.
2. from the midpoint of the line, draw a circle that passes thru both trees.
3. The solution is the where a perpendicular thru the midpoint
intersects the circle.
4. But there's two intersections.
I can't see how the left, right from the instructions help find which one is the solution, without
using a point as the gallows.
You must define where the gallows is, otherwise the instructions can't be obeyed and then you haven't got a left and right.
...you only have to define the gallows mentally, then mentally obey the instructions, which you can do in a few seconds. It'll then be obvious which is the correct intersection to choose.
If the elm is at 9 o'clock, the oak at 3 o'clock, then the treasure is at 12 o'clock. Strangely enough, the clock hand does scale distances as well.
You would think that someone could work the math, with the numbers.
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Just to show how wonderfully useful complex numbers are for plane
geometry, I'll use Miguel's result from the other problem.
Let E, K and T denote the location of the Elm, the Oak and the
treasure respectively. Miguel proved that T = (K+E - i(K-E))/2.
In this case E=30+35i and K=80+44i.
So T=(110+79i-i(50+9i))/2 = (110+79i-50i+9)/2 = (119+29i)/2
So the treasure is located at (59.5, 14.5). Thank you Sqrt(-1).
Bingo! :)
Hi Miguel, I was impatient, I should have let you do it.
A small variant I just invented: if we know the position of the Gallows, the Elm and the Treasure, what is the procedure to find the Oak? Not mathematically (too obvious), but using instructions like in the original problem (counting paces, turning right or left, etc.).
No prob Chris... :)
Walk from E to T counting steps. Turn right and walk same number of steps :)
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This post has been removed by the author.
Miguel, Soz, that was too sneaky of me:) I'll try with G explicit.
You're right, Chris. In fact, it is easy to demonstrate that i(T-K)=T-E, and so:
a) T-K and T-E have the same length, and
b) T-K and T-E are normal (they make 90ยบ)
So, your answer is correct :)
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I hope Ragknot doesn't mind us doing this. I'll abbreviate it to save verbiage.
Walk G to E, turn right, walk same distance, now at P1.
Walk P1 to T, keep going (same distance) mark P2.
Walk to G, counting steps, then halfway back to P2, turn right, walk same again as you just did, to K.
Plant an acorn.
yep, that was my solution too. but when you posted your first I tried it and you were right: that's the best we can do :)
Chris,
Of course I don't mind. I wanted discussion and exploration.
Since I did not give a point for the gallows, I expected two solutions. Did any one give two solutions? I gotta check.
Thank you Ragknot. I just felt that Miguel and I, were perhaps a little too far off topic.
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This post has been removed by the author.
When I draw this points and select a gallows point, I get the treasure is at (50.5,64.5)
If I switch the left/right, I get the (59.5,14.5).
maybe I remembered the left / right backward?
My bad. I typed a sign wrong. Should have checked on a diagram.
Here's the corrected workthrough:
Let E, K and T denote the location of the Elm, the Oak and the
treasure respectively. Miguel proved that T = (K+E + i(K-E))/2.
In this case E=30+35i and K=80+44i.
So T=(110+79i+i(50+9i))/2 = (110+79i+50i-9)/2 = (101+129i)/2
So the treasure is located at (50.5, 64.5). Thank you Sqrt(-1).
Part 2
I hope I've got this right:
Let M be the mirror location.
Then the mid-point, C=(T+M)/2
But this is the same as the centre point of the Elm and the Oak
So C=(E+K)/2. Equating and rearranging => M=E+K-T
=>M=30+35i+80+44i-50.5-64.5i
Mirror at (59.5,14.5)
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