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No. 7 Ten Green Bottles

 

Four friends, the Triangle, the Square, the Pentagon and the Hexagon were attempting to sing the popular song 'Ten Green Bottles'. Unfortunately they could not agree how many bottles were still on the wall, the Hexagon sang 21, the Pentagon sang 23 and the square sang 31. For this brainteaser you must explain how the confusion arose and suggest how many bottles the triangle thought were on the wall.

 

hexagon	21
Pentagon	23
Square	31
Triangle	111

 

The solution was of course

 

Whilst they were all singing the same number, each person used a different number base to express themselves equivelent to the number of sides they had.

 

Congratulations go to "Jonathan Salmon" for sending in the first correct answer. The following section contains every correct answer in the order it was received. Thank you all for taking part.

 

From: Jonathan Salmon
Subject: #7 Ten Green Bottles

The problem here is which base the friends are singing in.
If they all actually think they're talking about the same number but in 
different bases then:

Number  | Base | Number  | Shape
in Base |      | Base 10 |
--------+------+---------+--------
 21     |    6 |      13 | Hexagon
 23     |    5 |      13 | Pentagon
 31     |    4 |      13 | Square
111     |    3 |      13 | Triangle

From: Arjan Bosma
Subject: Brainteaser #7

The triangle would sing 111 bottles were on the wall.

From: Collins, Victor
Subject: Brainteaser solution

The triangle thinks the number should be 111.  Each is using a different
numeric base with the number of points in their respective polygon as the
base.

Victor Collins

From: Iain Davies
Subject: Teaser #7

Each friend speaks in his own base.
There are 13 bottles left.  Each one is saying 13 in a different base.

Hexagon:	(21)base6	= 13
Pentagon	(23)base5	= 13
Square		(31)base4	= 13
Triangle		(111)base3	= 13

So Triangle says there are 111 bottles left on the wall.

From: Albert Wong
Subject: Proposed solution to Brain Teaser #7

The confusion is that the four friends were counting in different bases; =
hexagon in base 6, pentagon in base 5, square in base 4, and triangle in =
base 3.  If they were all counting in base 10, there would be 13 bottles =
left on the wall.  However, by counting in different bases, the hexagon =
counted 21, pentagon counted 23, square counted 31, and triangle counted =
111.

From: Inge Klobasa
Subject: No. 7

The numbers are not based on 10 but on the number of edges of the
friends, so they all sing about 13 bottles:

2*6 + 1 = 13
2*5 + 3 = 13
3*4 + 1 = 13

So the triangle sings there are 111 bottles on the wall:

1*9 + 1*3 + 1 = 13

From: Derrick Swinarsky
Subject: #7 Solution

Triangle thinks there are 111 bottles

Hexagon used base 6, Pentagon used base 5, Square uses base 4, therefore
Triangle uses base 3.

There are actually 13 bottles left.

Derrick Swinarsky

From: David B. Anderson
Subject: Re: Re: No. 7 Ten Green Bottles

Doh, my triangle must have had too many bottles already.  The real correct
answer would be 111 (i.e. 1*3^2 + 1*3^1 + 1*3^0).  The reasoning is in my
original message below.

From: Simon Grimes
Subject: RE: number 7

Ok, so now I've had my glaring error pointed out to me I'll give it another
go

13 base 10 in base 3 is

	9s	3s	1s
	1	1	1

there you go, 111 is the answer hopefully.

From: Jim Miller
Subject: Ten Green Bottles

All the friends are counting in their own number base system:
hexagon base 6
pentagon base 5
square base 4
triangle base 3  ====== the triangle would sing 111 (13 in base 10)

From: Brandon Fliflet
Subject: brainteaser #7

Each friend is counting in the base equal to the number of sides on each
primitive:

hexagon  --- base 6 ---  21(base6) = 13(base 10)
pentagon --- base 5 --- 23(base5) = 13(base 10)
square    --- base 4  --- 31(base4) = 13(base 10)
triangle   --- base 3 --- 111(base3) = 13(base 10)

From: Jason Bell
Subject: No. 7 Ten Green Bottles

The hexagon counts in base 6
21 in base 6 is 13 in base 10

The pentagon counts in base 5
23 in base 5 is 13 in base 10

The square counts in base 4
31 in base 4 is 13 in base 10

The triangle counts in base 3
111 in base 3 is 13 in base 10

The only thing I don't understand is why
they're all at 13 when the song is 10
green bottles...

From: dabj
Subject:

They are using different base depending on the number of angles they have.
Hexagon 21  Base6
Pentagon 23 Base5
Square 31   Base4
Triangle ?  Base3
So the triangle will say 111, since they all mean 13 (in base 10 that is)

From: John Buffington
Subject: #7 revisited again

The hexagon,pentagon,and square all use a number system
based on "their" number of sides. 

Hexagon  21 base 6 = 13 base 10
Pentagon 23 base 5 = 13 base 10
Square     31 base 4 = 13 base 10

Therefore if the triangle uses the same number system we get

Triangle    111 base 3 = 13 base 10.

From: Johannes Päivinen
Subject: Brainteaser #7 (again)

All of the shapes saw the same amount of bottles, but used a different
numeric system to express it.

The hexagon used a 6-base system (2*6+1=13). The pentagon used a 5-base
system (2*5+3=13) and the square a 4-base system (3*4+1=13). Respectively
the triangle would use a 3-base system (1*3^2+1*3+1=13).

So the correct answer is 111.

From: David M. Turner
Subject: No. 7 Ten Green Bottles

Solution:

The triangle said 111 (base 3)

Each shape used its number of sides as the base in which it said the
number (13).

From: Balazs Vasvari
Subject: No. 7 Ten Green Bottles

No. 7 Ten Green Bottles

The triangle would sing 111 because all songs equal to 13(decimal) in their
own bases. (hexagon base 6, pentagon base 5, square base 4), i just dont
understand why 13, maybe they had a bad day, or they're employed by Mr.
13gon, who would sing "10" in his own base :)  (yet another theory: they
learned the song from a guy 7gon, who sang "13" which equals to 10(decimal),
and hey this is teaser NO. SEVEN so it's obvious ;))

From: Rasheed Abdal-Aziz
Subject: RE: Teaser #7

Firstly Trianlge thought there was 111 bottles...
Not Forty one, but Four - One...

A human has ten digits on there hands, which is where Decimal arose...
Hexagon has Six sides, so his number base is Six.. Pentagon is five, square
i four, triangle is three.

So
Hexagon: 21 = (2 * 6) + 1   (which is thirteen)
Pentagon:   23 = (2 * 5) + 3   (which is thirteen)
Square: 31 = (3 * 4) + 1   (which is thirteen)
Triangle:   111 = (1* 3^2) + (1 * 3) + 1   (which is thirteen)

They agree on the number of bottles, they just could not illiterate..

From: Geraint Bundy
Subject: RE: Brainteaser # 7

Duh! Must be sleepy today!

111, darnit

Never used base 3 before, mind

Like the brainteasers - keep them up!

From: Rafael Kaufmann
Subject: (correction) Re: Solution to BT #7

As you pointed out...

> 41 is not base 3 for 13!

The correct answer follows:

The number of bottles on the wall, in radix-10, is 13. The Hexagon was
using radix-6 math (2 * 6^1 + 1 = 13); the Pentagon, radix-5 (2 * 5^1 +
3 = 13); and the Square, radix-4 (3 * 4^1 + 1 = 13). Accordingly, the
Triangle will use radix-3 math, wherein there will be (1 * 3^2 + 1 * 3^1
+ 1) 111 bottles on the wall.

From: Michael Snyder
Subject: Brain Teaser #7

This one isn't hard once you realize that each shape is counting by the
number base equivalent to the number of sides that they have, just
remember to keep your bases straight.

The hexagon is counting in base 6, the pentagon is counting in base 5, the
square is counting in base 4, so the triangle must be counting in base 3.

21 in base 6 = 13.
23 in base 5 = 13.
21 in base 4 = 13.
111 in base 3 = 13.

The triangle will say that there are 111 bottles on the wall.
There are actually 13 bottles on the wall.

Mike

From: Rick v. Bijnen
Subject: Brainteaser #7

The triangle thinks there are 111 bottles.
Because they all sing about 13 bottles:

Each n-gon has a n-based numerical system like the binary
numerical system has 2 possibility's per digit: 0 or 1 (2-based)

So the 21 the hexagon is singing about represents
13 in decimal counting because it has only 6 possible
values per digit: 0,1,2,3,4 or 5 <--> 2x6^1+1x6^0=13

Same goes for the pentagon--> 2x5^1+3x5^0=13
The square also sang 13 couplets: 3x4^1+1=13

13 in the 3based numerical system = 1x3^2+1x3^1+1x3^0 = 111 decimal

Greetings,

Rick van Bijnen

From: Eric Marcus
Subject: No. 7 Ten Green Bottles

Hexagon: 21 (base 6) = 13 (decimal)
Pentagon: 23 (base 5) = 13
Square: 31 (base 4) = 13

(1 x 3^2) + (1 x 3^1) + (1 x 3^0) = 13

Triange: 111 (base 3) = 13

From: Ppeck5
Subject: Answer to Puzzle #7 (ten green bottles) [corrected prev. answer]

The triangle sang 111. The confusion is that each can only count by its
number of sides hence each counts by base (it's sides). Although each totaled 
13
(eg. hexagon : 1 + 2(6) = 13) and not 10 but I used 13 to figure out the
triangles number: 111(1(1) +1(3) + 1(9) = 13)

From: David Faux
Subject: No. 7 Ten Green Bottles

The hexagon, pentagon, square, and triangle all count using different
bases corresponding to the number of points they possess (same as we use
base 10 because of the number of fingers we have). The total number of
bottles on the wall (in base 10) is 13 and the triangle sees it as 111
bottles.

The hexagon: (2 * 6^1) + (1 * 6^0) = 13 => 21
The pentagon: (2 * 5^1) + (3 * 5^0) = 13 => 23 
The square: (3 * 4^1) + (1 * 4^0) = 13 => 31 
The triangle: (1 * 3^2) + (1 * 3^1) + (1 * 3^0) = 13 => 111

From: Andrew Beers
Subject: No. 7 answer

The answer is 111.  Each friend expresses the number of bottles in its
own base.  Hexagon sings "21", which is the base 10 number 13 written
in base 6.  Pentagon and square also express the number 13 in bases 5
and 4, respectively.  Triangle must sing "13" in base 3, which is 111.

It's unclear why they are singing about bottle number 13 if the song is 
about 10 green bottles, but hey.

From: Ness
Subject: Brainteaser #7

Each shape uses a numerical base equal to the number of sides (or angles, if
you prefer) it has. Thus, the Hexagon is using base 6, the Pentagon is using
base 5 and the Square is using base 4. What they *really* mean is 13base10
(which we all know is the only True and Proper base, otherwise why would God
give us 10 fingers, eh?!). The Triangle would sing 111base3 (although it
wouldn't *actually* sing the 'base3' part, or the puzzle 'd be spoiled)

From: James M. Donovan
Subject: BT #7

Hi:

        Base 10 13 in base 3 is 111.

From: Technolord
Subject: Me idiot (ref: BT#7)

I idiotically sent triangle's answer "41" to be a base3 number... okay,
that was 111 :P (shame on me!)

From: BlueSky
Subject: Ten Green Bottles

Hey,

I guess they each counted in different bases.  Hexagon in Base6, Pentagon in
Base5, Square in Base4 and... Triangle in Base3.

So the Triangle must think there are 111 bottles [on the wall]. (13 in
decimal).

From: Tomer Brisker
Subject: Teaser #7

The triangle will say 111. The mixup is  because each polygon counts in it's
base- the hexagon in base 6, the pentagon in base 5, and so on.
(21)6=(23)5=(31)4=(111)3