# What are a stone on a cone on a bone and a stone on a bone?

The following cryptic message is supposed to tell me the secret code I have to give the guard that he lets me in.

A bone and a bone are a cone on a bone.

A bone on a cone is the opposite of a cone on a bone.

The opposite of a stone is a stone.

A stone on a bone and the opposite of a cone on a bone are a bone.

A stone on a bone for each bone on a cone is a stone on a bone on a cone.

A cone and a bone on a cone are the opposite of a stone on a bone.

A cone for each stone on a bone is a stone on a cone.

A cone on a bone for each cone on a bone is a bone on a bone.

What are a stone on a cone on a bone and a stone on a bone?

Apparently I need to tell the guard a specific stack of objects. But what is it?

Here's a hint I've found in a fortune cookie:

Remember that every thing has value.

A manager gave me this valuable hint:

The following insight from a self-help book might also be helpful:

We also need the negative things in life.

I've also got a hint from a TV series fan:

You really should know the power of three.

• I'm not sure whether to get stoned or a boner now – Avigrail Aug 19 '16 at 5:51
• Seeing as we're talking about stacks, should "the opposite of A on B" mean, "B on A"? – RoadieRich Aug 19 '16 at 15:59
• @RoadieRich: No. – celtschk Aug 19 '16 at 16:41

OK, we have a lot of hints now and I think they're enough.

We are working in

balanced ternary

and "X on Y on Z" means

the balanced-ternary number ZYX; that is, X + 3Y + 9Z

where

a bone is +1, a cone is -1, and a stone is 0.

The other operations are

"and" = addition, "for each" = multiplication, "opposite" = negation.

Accordingly, a stone on a cone on a bone and a stone on a bone equal

[0 + 3*(-1) + 9] + [0 + 3] = 9

which we describe as

a stone on a stone on a bone.

Let's go through the statements we've been given and check them.

A bone and a bone are a cone on a bone.

1 + 1 = (-1) + 3*1. 2 on both sides, check.

A bone on a cone is the opposite of a cone on a bone.

1 + 3*(-1) = -[-1 + 3*(1)]. -2 on both sides, check.

The opposite of a stone is a stone.

-0 = 0, check.

A stone on a bone and the opposite of a cone on a bone are a bone.

0 + 3*1 - (-1 + 3*1) = 1.

A stone on a bone for each bone on a cone is a stone on a bone on a cone.

(0 + 3*1) * (1 + 3*-1) = 0 + 3*1 + 9*-1. -6 on both sides, check.

A cone and a bone on a cone are the opposite of a stone on a bone.

[-1] + [1 + 3*-1] = -[0 + 3*1]. -3 on both sides, check.

A cone for each stone on a bone is a stone on a cone.

-1 * (0 + 3*1) = (0 + 3*-1). -3 on both sides, check.

A cone on a bone for each cone on a bone is a bone on a bone.

(-1 + 3*1) * (-1 + 3*1) = (1 + 3*1), or 2*2=4, check.

• You got it! Bone. – celtschk Aug 22 '16 at 14:34
• In fairness, by the time I looked at it today the hints were more or less a solution :-). Then again, there's a geometry-ish puzzle still open where the hints are basically a very detailed solution outline and everyone, including me, has been avoiding the work of actually doing it... – Gareth McCaughan Aug 22 '16 at 14:38

This can only be a partial solution, because I don't understand any rule which contains the word "each". Anyway, Let $s$ = a stone, $c$ = a cone, $b$ = a bone, $1$ = a stack with nothing on it. Say $x+y$ for $x$ and $y$, and $xy$ for $x$ on $y$. Say $x^{-1}$ for the opposite of $x$. Say $xy=1$ or $yx=1$ if it's stated that $x$ is the opposite to $y$. (Actually, seeing as we have two operators, "and" and "on", I am not sure which operator "opposite" refers to.) Then

$$\begin{matrix} & b+b & = & cb & [1]\\ & bc & = & (cb)^{-1} & [2]\\ & ss & = & 1 & [3] \\ & sb+(cb)^{-1} & = & b & [4] \\ & c+bc & = & (sb)^{-1} & [6] \\ \Rightarrow & sb & = & b+cb & (4) \\ &&=&b+b+b & (1)\\ & sb+c+bc & = & 1 & (6)\\ \Rightarrow & b+c & = & 1\\ & scb & = & s(b+b) & (1) \\ && = & sb+sb\\ \Rightarrow & scb+sb & = & sb+sb+sb \\ && = & b+b+b+b+b+b+b+b+b \end{matrix}$$

At least, that's one interpretation. Of course, I might've been attacking quite the wrong problem.

• You're partly on the right way, but not completely. Interestingly, your first conclusion (the line labelled $(4)$) doesn't seem to follow from the preceding lines (at least I cannot see how), but translates to a correct statement. – celtschk Aug 19 '16 at 7:26
• Since we deal with stacks of at least 2 objects probably the opposite of y/x would be x/y. For a stack made of 1 object the extra condition opposite of x = x is given in line 3 since 1/x wouldn't make sense. – Avigrail Aug 19 '16 at 9:12
• Surely 4 is just what you get by combining "A stone on a bone and the opposite of a cone on a bone are a bone" and "A bone on a cone is the opposite of a cone on a bone", so it's not surprising it's correct. – Gareth McCaughan Aug 19 '16 at 16:50
• In my interpretation of the question I too came up with a system of equations, but I didn't interpret stacks to be multiplication, or opposites to be inverses. – Avik Mohan Aug 19 '16 at 17:17

Is it:

A stone?

Based on:

stone = 1; bone = 2; cone = 4;

Where:

'on' is to divide, 'and' is to add, and 'for each' is to multiply.

A bone and a bone are a cone on a bone.

b+b = c/b, thus c=2b.

A bone on a cone is the opposite of a cone on a bone.

I assume this to mean, b/c is the opposite of c/b

The opposite of a stone is a stone.

This only holds for 1/1, so s=1

A stone on a bone and the opposite of a cone on a bone are a bone.

Math happened. B=2 and C=4

So I skipped ahead and tried my theory so far.

What are a stone on a cone on a bone and a stone on a bone?

(1/4)/2 + 1/2 = 1

There might be a valid statement in there, but I doubt it....

• No. And your interpretation already violates the first statement: "a bone and a bone" would be 4, while "a cone on a bone" would be 2. – celtschk Aug 19 '16 at 21:02