[bonanova]

Andy's great-grandfather Bud was a '49er who rode west in the California gold rush.

He never struck it rich, but lived and died in poverty in Coloma, near Sutter's original

mill - a place Andy had visited many times. All this had been handed down as rich

family tradition. But today when Andy opened the shoebox of old heirlooms, all of

that changed. There lay a letter written by Bud to his wife Clara. And it told a different

story. Andy's heart began to race as he read ...

Quickly Andy's eyes darted to great-grandpa Bud's verbal treasure map.

The distances to the cairns had been erased! Perhaps Bud feared his letter would

fall into the hands of thieves. But what good, then, to mail it? Andy knew approximately

where the old cabin had stood, near the South Fork of the American River, and he

decided, blurred information or not, it was worth another trip.

The place had changed. Only a worn trail was left, and a young forest where once

had laid an open plain. The stakes placed by Bud are surely gone, thought Andy,

but possibly the cairns still stand. And after a short morning search, Andy found

the cairns. But what next?

Sipping coffee in the Wagon Wheel Cafe that noon, Andy sketched out what he knew.

From his cabin, now gone, Bud had walked unknown, possibly different, distances and

built cairns. Turning left at one, and right at the other, Bud duplicated the original

distances and driven stakes into the ground. Midway between the stakes, also now

gone, and 3 feet below the surface, the gold nuggets had been buried.

Andy spent 30 minutes placing dots and drawing circles on a napkin, but made no

progress. The coffee flowed, but no answer came.

On a hunch Andy called back to Morty's and as luck would have it, Alex answered the

phone. Andy explained his situation and asked if Alex could help. That afternoon,

Andy took a shovel to Bud's secret cache and dug up the treasure.

How did Alex direct Andy to the treasure?

[Guest]

Exactly between the 1st and scond cairns

[bonanova]

Exactly between the 1st and scond cairns

I called Alex, and he said that's not it ...

[Guest]

For everyone's benefit, I've retyped the handwritten note (sorry, bn, but it's kinda hard to read...almost looks as though it was created with a mouse in Photoshop? ).

"For my own safety, I have not disclosed, even to those closest to me, the whereabouts of the strike. Instead, I have collected the largest of the nuggets and placed them where no map should be needed for retrieval. I write this to you, dearest Clara, so that in the event of my demise, you will recover my find and escape the poverty I have for so long now been accustomed.

I proceeded from my porch steps a distance of X paces and there constructed with some effort a cairn - large enough, I (?), to survive flood or fire. Thereupon, I turned right, exactly as I could discern, and marked an equal number of paces. There I drove a large stake in the ground. Returning to the cabin, I walked Y paces, this time in a different direction, a 2nd cairn I there constructed. But now I turned left, more or less exactly, proceeded another Z paces, and planted another stake. At the midpoint between the stakes, exactly, I buried the nuggets. Supposing the depth to be 3 feet, more or less."

Who are Morty and Alex? The story doesn't mention them until the very end, but doesn't say what connection they have to Andy.

[Prof. Templeton]

Who are Morty and Alex? The story doesn't mention them until the very end, but doesn't say what connection they have to Andy.

They are bn's "goto" characters and make appearances in several of his puzzles.

[Guest]

For everyone's benefit, I've retyped the handwritten note (sorry, bn, but it's kinda hard to read...almost looks as though it was created with a mouse in Photoshop? ).

"For my own safety, I have not disclosed, even to those closest to me, the whereabouts of the strike. Instead, I have collected the largest of the nuggets and placed them where no map should be needed for retrieval. I write this to you, dearest Clara, so that in the event of my demise, you will recover my find and escape the poverty I have for so long now been accustomed.

I proceeded from my porch steps a distance of X paces and there constructed with some effort a cairn - large enough, I (?), to survive flood or fire. Thereupon, I turned right, exactly as I could discern, and marked an equal number of paces. There I drove a large stake in the ground. Returning to the cabin, I walked Y paces, this time in a different direction, a 2nd cairn I there constructed. But now I turned left, more or less exactly, proceeded another Z paces, and planted another stake. At the midpoint between the stakes, exactly, I buried the nuggets. Supposing the depth to be 3 feet, more or less."

Who are Morty and Alex? The story doesn't mention them until the very end, but doesn't say what connection they have to Andy.

Clarification: From ANdy's summary he implies that the distance from cairn to stake in each case is the same as the distance from cairn to cabin, but in the letter there are three erased numbers - are the second to blanks meant to be the same distance?

[Guest]

I think I have a solution that would pinpoint the location of the gold without even needing to know how many paces were stepped each time, if Bud had purposely buried the gold in such a location that the number of steps was irrelevant.

the locations of the two cairns and the approximate location of the cabin. Not knowing which of the two cairns was the first one, he can start by assuming one of them is the first cairn mentioned in the note. He can measure the distance to that one from the place where the cabin once stood, turning right, measure that same distance, and plant a new stake (let's call it Stake 1). He can then tie a piece of string (String A) to Stake 1, and stretch the string back across the cairn and on further in the opposite direction (as if he had turned left at the cairn instead of right, in case it were the second cairn and not the first). He would then repeat the process with the second cairn (calling that Stake 2 and String B). So he has two stakes and two strings laid out. If he measures the shortest distance from String A to Stake 2 and divides it by two, he can construct a parallel string to String A at that offset distance (call it String C), along which the treasure could be buried, as any point on String A where the second stake could have been located will have a point on String C that is halfway between String A and Stake 2. Again, he can repeat the process with the String B and Stake 1, creating String D that is parallel to String B and halfway between it and Stake 1. Where Strings C and D cross must be where the gold is buried, as this is the only spot that does not depend on which cairn was first or second. While the number of paces could throw this solution off if they were known, the fact that they are unknown allows Andy the freedom to assign them to be anything to leads him to the point where the "X" formed by Strings C and D marks the spot.

I hope that makes sense, it seems to make sense in my head anyways. I hope I'm not assuming too much to make this solution work.

[Guest]

Clarification: From ANdy's summary he implies that the distance from cairn to stake in each case is the same as the distance from cairn to cabin, but in the letter there are three erased numbers - are the second to blanks meant to be the same distance?

Assuming the above...

At a point equidistant from the two cairns, such that triangle hole-cairn-cairn is an equalateral isocelese triangle with Cairn-cairn the hypotenuse

Proof is an exercise for the reader, cos its home time.

[bonanova]

Clarification: From ANdy's summary he implies that the distance from cairn to stake in each case is the same as the distance from cairn to cabin, but in the letter there are three erased numbers - are the second to blanks meant to be the same distance?

Yes, Y and Z are the same number, as implied by "another" in the text.

[bonanova]

I think I have a solution that would pinpoint the location of the gold without even needing to know how many paces were stepped each time, if Bud had purposely buried the gold in such a location that the number of steps was irrelevant.

the locations of the two cairns and the approximate location of the cabin. Not knowing which of the two cairns was the first one, he can start by assuming one of them is the first cairn mentioned in the note. He can measure the distance to that one from the place where the cabin once stood, turning right, measure that same distance, and plant a new stake (let's call it Stake 1). He can then tie a piece of string (String A) to Stake 1, and stretch the string back across the cairn and on further in the opposite direction (as if he had turned left at the cairn instead of right, in case it were the second cairn and not the first). He would then repeat the process with the second cairn (calling that Stake 2 and String B). So he has two stakes and two strings laid out. If he measures the shortest distance from String A to Stake 2 and divides it by two, he can construct a parallel string to String A at that offset distance (call it String C), along which the treasure could be buried, as any point on String A where the second stake could have been located will have a point on String C that is halfway between String A and Stake 2. Again, he can repeat the process with the String B and Stake 1, creating String D that is parallel to String B and halfway between it and Stake 1. Where Strings C and D cross must be where the gold is buried, as this is the only spot that does not depend on which cairn was first or second. While the number of paces could throw this solution off if they were known, the fact that they are unknown allows Andy the freedom to assign them to be anything to leads him to the point where the "X" formed by Strings C and D marks the spot.

I hope that makes sense, it seems to make sense in my head anyways. I hope I'm not assuming too much to make this solution work.

The cabin is long gone; and its former whereabouts are not known with enough precision, now, to be helpful.

Bud obviously intended for Clara to retrace his steps starting from the porch.

Andy's problem is that the starting point is no longer available to him, and that the distances were erased.

We can assume Bud erased them for security reasons at some point, not because they were irrelevant.

Alex had another idea.

[Guest]

I don't know the math (I'm not so great at math), so I'll leave that for someone else to prove, but

Walking to the midpoint of the two cairns, and then turning 90° and walking the same distance (half the distance between the two cairns) and then digging down 3 feet. If no gold nuggets, he returned to the midpoint, and walked the other direction the same distance, and dug down 3 feet, and found the gold.

I did some iterations in CAD, and given a constant distance between the cairns, no matter the location of the cabin, the gold always ends up being in the same two possible locations (given that you don't know which cairn is which), creating a square between the two cairn points and the two gold points.

Very interesting... I'd like to see the math (so maybe someday I won't be so bad at math)

[Guest]

I don't know the math (I'm not so great at math), so I'll leave that for someone else to prove, but

Walking to the midpoint of the two cairns, and then turning 90° and walking the same distance (half the distance between the two cairns) and then digging down 3 feet. If no gold nuggets, he returned to the midpoint, and walked the other direction the same distance, and dug down 3 feet, and found the gold.

I did some iterations in CAD, and given a constant distance between the cairns, no matter the location of the cabin, the gold always ends up being in the same two possible locations (given that you don't know which cairn is which), creating a square between the two cairn points and the two gold points.

Very interesting... I'd like to see the math (so maybe someday I won't be so bad at math)

Haha, I've been doing the same thing, and I get the same results. Still working on the proof though. I guess I did assume too much in my first attempt, and didn't realize that Y=Z.

[bonanova]

I don't know the math (I'm not so great at math), so I'll leave that for someone else to prove, but

Walking to the midpoint of the two cairns, and then turning 90° and walking the same distance (half the distance between the two cairns) and then digging down 3 feet. If no gold nuggets, he returned to the midpoint, and walked the other direction the same distance, and dug down 3 feet, and found the gold.

I did some iterations in CAD, and given a constant distance between the cairns, no matter the location of the cabin, the gold always ends up being in the same two possible locations (given that you don't know which cairn is which), creating a square between the two cairn points and the two gold points.

Very interesting... I'd like to see the math (so maybe someday I won't be so bad at math)

Yes that's the answer also given by armcie.

There is a beautiful proof using complex numbers

and a messier one just using [x, y]'s and angles and distances.

I'll leave these as an extension to the puzzle.

For now, here's an image that shows how, unexpectedly, Bud's "verbal map" was a great one.

No map, no cabin, no distances, just the cairns - and the algorithm for using them.

The dots show locations of the assumed cabin and the stakes Bud would have placed.

Use the red cabin, the black cabin, or any starting place at all.

Or start at Cairn 1 for that matter [almost the red cabin case] they all lead to the treasure.

Seeing where the instructions lead to, we see we could also

walk from Cairn 1 halfway to Cairn 2, turn left and walk an equal distance to the treasure.

The only uncertainty is which cairn is which: you may end up digging a bogus hole.

[bonanova]

Without loss of generality, let the cairns be at {-1,0} and {1, 0}

Let the cabin be at {x, y}, and the stakes at {x₁, y₁} and {x₂, y₂}

Move a distance {dx, dy}, then turn and go the same distance.

If you turn right, you'll move {dy, -dx}; if you turn left, you'll move {-dy, dx}.

From cabin to cairn1:

{dx, dy} = {-1, 0} - {x, y} = {-x-1, -y}

then turning right:

{x₁, y₁} = {-1, 0} + {-y, x+1} = {-y-1, x+1} [stake 1]

From cabin to cairn2:

{dx, dy} = {1, 0} - {x, y} = {-x+1, -y}

then turning left:

{x₂, y₂} = {1, 0} + {y, 1-x} = {y+1, -x+1} [stake 2]

A the midpoint of the stakes is the treasure:

{(x₁+x₂)/2, (y₁+y₂)/2} = {(-y-1+y+1)/2, (x+1-x+1)/2} = {0,1} [Treasure]

If you turn left at cairn1 and right at cairn2 you get {0,-1}

The treasure location is independent of the cabin location, at a corner of a square

whose diagonally opposite corners are on the cairns.

[bonanova]

Here's the other proof:

Place Cairn1 and Carin2 at 0 and 1 in the complex plane.

Place Cabin at A.

Stake1 = Ae^ipi/2

Cabin is at B from Cairn2; thus at [A=1+B] from Cairn1

Stake2 = 1 + Be^-ipi/2

The treasure is at (Stake1 + Stake2)/2 = (Ae^ipi/2 + 1 + Be^-ipi/2)/2 = (1 + [A-B]e^ipi/2)/2 = (1+i)/2, independent of A