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Whodunit?

Question

The Threedie brothers, Al, Bert, Chuck, Dick and Eddie, lived in a cabin 3 miles up the old mountain trail, and it was known they didn't get along all that well. This morning, Eddie was found dead behind the cabin, and his brothers, the only suspects in the case, were being questioned by Inspector Sherlock. It was known that, of the four, at least 3 were absolute truth-tellers, and none of them ever lied and told the truth in a single day. All four, of course, denied murdering their brother.

The Inspector started by asking each brother what he had done that morning:

1. Al: I was analyzing random groups of 3 numbers, and I found that if the numbers sum to zero then their product is the average of their cubes.
2. Bert: I was analyzing random polygons with 3 sides, and I found that if I trisected all their angles I could make an equilateral triangle.
3. Chuck: I planted a dozen apple trees out in the orchard, and I found a way to make eighteen rows of 3 trees, each row being dead-on straight.
4. Dick: I went out and ran 3 miles in the woods, and I've figured out that one of my 3 (living) brothers is lying.

The Inspector called in these clues to one of his friends at BrainDen, and in 3 shakes of a lamb's tail the case was solved. The sound you hear is your phone ringing. It's your chance to be famous!

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My guess:

1. Al's statement is provable algebraically.

2. Bert's is proven using Morley's trisector theorem

3. I see rocdocmac's drawing and I wonder if some of those three points are actually collinear.
@bonanova Can any 4 points be collinear?  If not, Chuck gets my accusation.  If they can be, I'll accuse Dick.

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Two guesses (both of which are more fun than logical).

"In the box" guess:

Spoiler

Chuck is the liar.

Each brother references 3 twice in their statements except Chuck.

1. Al: 3 numbers, cubes
2. Bert: 3 sides,  trisected
3. Chuck: 3 trees
4. Dick: 3 miles, 3 (living) brothers

Plus, Chuck is the third brother.  Kind of poetic.

"Out of the box" guess:

Spoiler

Eddie is the liar.

Dick ran 3 miles and saw Eddie.  When he got back to the cabin, he knew that the body had to be a fake and Eddie was still alive.

(I guess this also makes Dick a liar for saying he only has 3 living brothers, but that still leaves 3 absolute truth-tellers per the conditions of this problem.)

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Spoiler

Als' statement is invalid.    He therefore is the liar .

Berts' statement is true

Chucks statement is true

Dicks' statement must be true  , if there are 3 truth tellers

Edited by bonanova
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If I understand right, Bert's statement doesn't hold up starting with an equilateral triangle. But does making a false statement necessarily mean guilt or could it be a mistake on their part?

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Spoiler

The statements of the first two brothers appear to hold.

1. For any integer, if x+y+z=0, then x*y*z = (x³+y³+z³)/3

2. α + β + γ = 180; (α/3+β/3+γ/3) = (α + β + γ)/3 = 180/3 = 60

Chuck's statement will hold if it can be shown that it's possible to plant 12 trees in such a way that you get 18 straight rows with 3 trees each, the culprit would be Dick. If not proven, then Chuck is the liar and Dick is safe.

I think Chuck is the liar

Edited by rocdocmac
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Spoiler

It is possible to plant 12 trees in a maximum of 19 rows with each row containing 3 trees in a straight line. Thus, 18 rows with three trees in line seems feasible, but what would the configuration look like? Still a humdinger! C'mon Chuck, show us to prove you are guiltless or not!

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Spoiler

Dick is most probably the lier. Maybe that's why he ran into the woods?

One can almost certainly plant 12 trees in such a way that you have 18 rows with three trees per row ... meaning that Chuck's statement is true.

Edited by rocdocmac
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5 hours ago, rocdocmac said:
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The statements of the first two brothers appear to hold.

1. For any integer, if x+y+z=0, then x*y*z = (x³+y³+z³)/3

2. α + β + γ = 180; (α/3+β/3+γ/3) = (α + β + γ)/3 = 180/3 = 60

Chuck's statement will hold if it can be shown that it's possible to plant 12 trees in such a way that you get 18 straight rows with 3 trees each, the culprit would be Dick. If not proven, then Chuck is the liar and Dick is safe.

I think Chuck is the liar

Regarding (2), can you make a sketch for the Inspector?

Spoiler

He tried, and failed, finding none of the trisectors to be collinear

7 hours ago, Thalia said:

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If I understand right, Bert's statement doesn't hold up starting with an equilateral triangle. But does making a false statement necessarily mean guilt or could it be a mistake on their part?

"Absolute truth-teller" includes "without mistakes," and he did say random. Remember also that ...

Spoiler

A total of eight statements were made.

Edited by bonanova
Clarifying my statement to Thalia
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Can't prove it yet but the triangle still seems off. Equilateral was an attempt at checking. Not meant to cover all possibilities.

I'm not sure what rocdocmac proved other than adding up a third of the angles of a triangle gives you 60 degrees?

Edited by Thalia
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Spoiler

I misinterpreted Bert's statement! Doesn't work on paper the way I put it!

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Spoiler

Not the most elegant sketch, but here's one way to plant 12 trees in 18 rows of 3 each.

Thus, if Al and Chuck told the truth and Bert tried (but failed) to make a sketch for the inspector (none of the trisectors colinear), then Dick is the third truth teller. So far all four (living) brothers already got the blame, but it now appears that Bert is in trouble! Bert also mentioned that he "was analyzing random polygons with 3 sides" ... why didn't he say triangles?

Edited by rocdocmac
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1 hour ago, rocdocmac said:

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Not the most elegant sketch, but here's one way to plant 12 trees in 18 rows of 3 each.

Thus, if Al and Chuck told the truth and Bert tried (but failed) to make a sketch for the inspector (none of the trisectors colinear), then Dick is the third truth teller. So far all four (living) brothers already got the blame, but it now appears that Bert is in trouble! Bert also mentioned that he "was analyzing random polygons with 3 sides" ... why didn't he say triangles?

I thought that was weird at first too but I attributed it to the Threedie Bros obsession with 3's. I found a picture of a trisected equilateral triangle. None of the triangles with a corner at the trisected angles work because they're only 20 or 40 degrees at the corner. There's 6 smaller more or less identical triangles around the center. I haven't worked out the math but one of the angles is nearly 90 degrees so those are out too. So starting with an equilateral triangle seems to disprove Bert's statement unless he has unusual luck in picking random triangles or you count the original triangle.

I am curious about bonanova's 8 statements comment though. 2 statements per person?

Edited by Thalia
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10 hours ago, Thalia said:
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I am curious about bonanova's 8 statements comment though. 2 statements per person?

Spoiler

All four, of course, denied murdering their brother.

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On 1/4/2018 at 3:43 PM, bonanova said:

Regarding (2), can you make a sketch for the Inspector?

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He tried, and failed, finding none of the trisectors to be collinear

On 1/5/2018 at 7:38 AM, rocdocmac said:

Thus, if Al and Chuck told the truth and Bert tried (but failed) to make a sketch for the inspector

In my post I meant to assert the Inspector tried but failed to make the sketch following the method suggested by rocdocmac.

That is, the antecedent of "He" was meant to be "the Inspector." Bert, of course, had already succeded.

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Back to square 1!

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I don't think there's a way to be certain that any of the brothers killed Eddie.

On 1/4/2018 at 2:50 AM, bonanova said:

It was known that, of the four, at least 3 were absolute truth-tellers.

We know that all 4 aren't absolute truth-tellers.  But that doesn't mean that one of them is an absolute liar.

On 1/4/2018 at 2:50 AM, bonanova said:

All four, of course, denied murdering their brother.

All four of them might be telling the truth in this case.

On 1/4/2018 at 2:50 AM, bonanova said:

Dick: ... I figured out that one of my 3 (living) brothers is lying.

Whether Dick (or the brother to which he is referring) is lying or telling the truth, we still can't infer that he killed Eddie, since he may not be an absolute liar.

Edited by Molly Mae
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7 hours ago, Molly Mae said:
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I don't think there's a way to be certain that any of the brothers killed Eddie.

We know that all 4 aren't absolute truth-tellers.  But that doesn't mean that one of them is an absolute liar.

All four of them might be telling the truth in this case.

Whether Dick (or the brother to which he is referring) is lying or telling the truth, we still can't infer that he killed Eddie, since he may not be an absolute liar.

@Molly Mae Bravo. I should give you a solve (and a gold star) for this answer.

But the Inspector has a reputation to uphold -- he needs a conviction -- and he did appeal to us for help.

So, let me repair my flawed puzzle by adding this phrase about the brothers (and I'll add it to the OP as well.)

None of them lied and told the truth in a single day.

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Another go ...

On 1/4/2018 at 10:50 AM, bonanova said:
• Spoiler

Al: I was analyzing random groups of 3 numbers, and I found that if the numbers sum to zero then their product is the average of their cubes. (True)

Bert: I was analyzing random polygons with 3 sides, and I found that if I trisected all their angles I could make an equilateral triangle. (True - Morley's trisector theorem)

Chuck: I planted a dozen apple trees out in the orchard, and I found a way to make eighteen rows of 3 trees, each row being dead-on straight. (True - can be illustrated)

Dick: I went out and ran 3 miles in the woods, and I figured out that one of my 3 (living) brothers is lying. (False?)

How could DIck (whilst running in the woods) figure out beforehand what the later statements of the other three brothers (and his own) would be, i.e. "I did not kill Eddie"?

Edited by rocdocmac
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5 minutes ago, rocdocmac said:

Dick: I went out and ran 3 miles in the woods, and I figured out that one of my 3 (living) brothers is lying.

To clarify: (and I'll add it to the OP)

Dick: I went out and ran 3 miles in the woods, and I've figured out that one of my 3 (living) brothers is lying.

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One more try then!

Spoiler

Bert's statement referring to "make an equilateral triangle" (i.e the first Morley triangle) is only partially true. If all of the trisectors are intersected, one can obtain more than one equilateral triangle. Dick may then be telling the truth and refer to Bert as the lying brother. Probably not the answer yet!

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8 hours ago, rocdocmac said:

One more try then!

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Bert's statement referring to "make an equilateral triangle" (i.e the first Morley triangle) is only partially true. If all of the trisectors are intersected, one can obtain more than one equilateral triangle. Dick may then be telling the truth and refer to Bert as the lying brother. Probably not the answer yet!

Spoiler

If Bert could make more than one, he must also be able to make just one.

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Spoiler

Well, if all of them are telling the truth except for Dick, it must be Dick. Both of his statements could be false, while everyone else has given a truthful statement. Therefore, Dick must be lying, which means he is the only one who could be a liar.

Incidentally - "At least" three of the brothers tell the truth 100%. Could this mean that all of them are telling the truth, and Eddie just committed suicide?

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For Bert, can you draw lines between points of intersection (like the theorem rocdocmac mentioned) or does the triangle have to be formed from the trisecting lines as attached?

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3 hours ago, Thalia said:

For Bert, can you draw lines between points of intersection (like the theorem rocdocmac mentioned) or does the triangle have to be formed from the trisecting lines as attached?

Spoiler

Yes and No, respectively.
If you can now make an equilateral triangle, you've verified Bert's statement:

I found that if I trisected all their angles I could make an equilateral triangle.

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3 hours ago, flamebirde said:
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Incidentally - "At least" three of the brothers tell the truth 100%. Could this mean that all of them are telling the truth, and Eddie just committed suicide?

Spoiler

There's enough info to answer this question.

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