# The treasure on a tropical island

My great-great-great-grandpa left this note for my family when he passed away, but no one dared try it out.

On the tropical island located at 90.888°N, 123.456°E, there's a gallows where we used to hang criminals and betrayers. There are also a palm tree and a banana tree. Stand under the gallows and walk straight to the palm tree. Take note of how many steps you've walked. Turn right perpendicularly and walk the same number of steps. Take note of this spot and return to the gallows. Now walk straight to the banana tree and take note of how many steps you've walked. Turn left perpendicularly and walk the same number of steps. Walk to the spot you've noted previously and stop halfway. Dig down straight and the treasure will be there.

I've arrived at the island and found both trees. But there's no gallows visible, most likely because it has decayed completely in the centuries that have passed. Unfortunately, this means I can't follow the guide closely. The island is large enough that digging randomly will only be a waste of stamina, so I have no idea what to do next.

Can I find the treasure anymore or should I give up?

• I’m not sure if this is a trick question or not, but the max degrees North is 90...... 90.888 degrees North latitude doesn’t exist on a sphere that defines 0 degrees latitude at the sphere’s equator. (I assume because of the geometry tag and lack of lateral-thinking tag that this doesn’t matter, hence comment and not answer.) Sep 10 '20 at 18:22
• @El-Guest ...and even if it did exist, it certainly wouldn't be tropical. Sep 10 '20 at 18:25
• Good spot. It is intended that where the island is is irrelevant to the puzzle, so I composed those nonsensical descriptions on purpose.
– iBug
Sep 10 '20 at 18:30
• Wasn't this in Gamow's 1, 2, 3 ... infinity? Sep 11 '20 at 13:57

Put the whole configuration into...

...a complex plane

Let G,P,B,T be positions of gallows, both trees and a treasure.

Then the first point $$P_1$$ is a postion of the palm tree plus a distance from gallows rotated by a right angle and added to P:

$$P_1 = P + (P-G)\cdot(-i)$$

The second point $$B_1$$ is similarly related to B, but with a left turn:

$$B_1 = B + (B-G)\cdot i$$

Then the treasure is halfway between them: $$T=\frac{P_1+B_1}2$$ which is

$$T = \big((P + (P-G)\cdot(-i)) + (B + (B-G)\cdot i)\big)/2 \\ = \big((P+B) + (B-P)i + (G-G)i)\big)/2 \\ = \frac{P+B}2 + \frac{B-P}2i$$

As we can see, the position of gallows...

disappears from formula – the treasure's position can be found from positions of both trees only:

walk half the way from the palm tree to the banana tree, then turn left and walk another half way. That's the point to dig.

Simply

Walk from the palm tree to the banana tree, noting how many steps you take. Then walk halfway back, turn right, walk the other half, and dig.

Proof:

Let the palm tree, banana tree, and gallows be $$P,B,G = (0,1),(0,-1),(x,y)$$, respectively, and the two noted points be $$F,S$$.
The right turn at the palm means that $$PF$$ is $$PG$$ rotated 90 degrees counterclockwise, and the left turn at the banana means that $$BS$$ is $$BG$$ rotated 90 degrees clockwise.
Since $$PG=\langle x,y-1 \rangle, PF=\langle 1-y,x \rangle$$ and $$F = (1-y,1+x)$$; likewise, since $$BG = \langle x,1+y\rangle, BS=\langle 1+y, -x\rangle$$ and $$S=(1+y,-1-x)$$. Averaging the two yields the location of the treasure, $$(1,0)$$.
Interactive Diagram!

• Consider the case where the gallows is at one of the trees. Sep 11 '20 at 13:15
• I think it can be assumed that each path is at least one step long. Sep 14 '20 at 1:53
• It doesn't matter whether the gallow is on one of the trees, the result is the same. Sep 14 '20 at 2:36
• "walk the full distance in that direction". I think you mean half the distance here. Though it is a bit unclear what you consider the full distance, I assumed the distance between the trees. Sep 14 '20 at 8:09
• I said that in my original answer, then forgot my coordinate system and thought it was wrong, oops. Sep 14 '20 at 13:47

Well... let's math it out a little. FIrst, variables

Ax, Ay: the gallows
P1x, P1y: the Palm tree
P2x, P2y: the point we reach after walking to the Palm tree
B1x, B1y: the Banana tree
B2x, B2y: the point we reach after walking to the banana tree
Tx, Ty: the target point.

Objective:

Trying to find Tx/Ty purely from B1x/B1y and C1x/C1y

Assumptions:

We'll assume that the island is effectively flat.

...and now let's see what we can derive from the text.

the target location is in between the two post-tree spots, so...
Tx = (P2x+B2x)/2
Ty = (P2y+B2y)/2

Gallows to palm, turn right. Gallows to banana, turn left.

turn right: eventual delta in x is orig delta in y eventual delta in y is negative orig delta in x
turn left: eventual delta in y is orig delta in x eventual delta in x is negative orig delta in y

P2x = P1x + P1y - Ay
P2y = P1y + Ax - P1x

B2x = B1x + Ay - B1y
B2y = B1y + B1x - Ax

Tx = (P1x + P1y - Ay + B1x + Ay - B1y)/2
Ty = (P1y + Ax - P1x + B1y + B1x - Ax)/2

Tx = (P1x + B1x + P1y - B1y)/2
Ty = (P1y + B1y + B1x - P1x)/2

And thus, in conclusion...

The treasure is somewhere in the circle defined such that the line between the two trees serve as are its diameter - radius of half the distance, centered on the midpoint. Further, returning to the originals, and knowing that you can distinguish between banana and palm lets you reduce it to a half-circle. If the palm is to your right, the dig point will be on your side of the trees. If it is to your left, the dig point will be on the opposite side of the trees

So... you still have some digging to do, but hopefully not too much.

• Note that there's no coordinate system defined in the question, so you can define your own and assign any coordinate to the entities described, as long as they don't contradict themselves. This should make describing locations easier and clearer.
– iBug
Sep 10 '20 at 18:29
• @iBug that's true, and also unnecessary. The math pretty much checks out regardless. Technically, I probably could have simplified things a bit for myself if I'd started with the gallows as 0,0, though. Sep 10 '20 at 18:30
• @iBug Am I correct? Sep 10 '20 at 18:34
• I think you are missing the point. Tx and Ty end up to be independent of Ax and Ay. So it doesn't matter where the gallows were. There is only one point to dig. Sep 13 '20 at 13:15
• @FlorianF, yea, I feel this answer is incomplete. The other answer is better. It has instruction on how to find the treasure too. Sep 14 '20 at 2:39