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Ten girls are sitting around a table. Each of them picks a real number and whispers it to the two neighbors immediately to the left and to the right. (Hence: each girl communicates one number, and receives two numbers.) Each girl then loudly announces the average of the two numbers she received. The announced numbers in order around the circle are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Question: What was the number picked by the girl who announced the average number 6?

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3  
by the way, why girls? :) – Oray Feb 20 at 10:12
17  
@Oray because, click bait. ;) – NVZ Feb 20 at 12:34
3  
@NVZ 10 girls, one table, one real conversation? – Yakk Feb 20 at 17:51

11 Answers 11

up vote 23 down vote accepted

1

Consider 10 real numbers $a$ through $j$ and have each girl's whispered number be $2a, 2b, 2c$ etc (these do not have to be even numbers; $a$ through $j$ are not necessarily integers.)

It follows that

$$\dfrac{2a + 2c}2 = 1 \quad\text{or}\quad a + c = 1\\[10pt]\dfrac{2b + 2d}2 = 2\quad\text{or}\quad b + d = 2$$

etc

The girl who says 6 is sitting at g. Rearranging the

$$e + g = 5$$

equation into

$$g = 5-e$$

and then substituting around the system of equations, we get:

$$\begin{align}g &= 5-e \\&= 5 - (3-c) \\&= 2 + c \\&= 2 + (1-a) \\&= 3 - a \\&= 3 - (9-i) \\&= -6 + i \\&= -6 + (7-g) \\&= 1 - g\end{align}$$

therefore

$$2g = 1$$

which is what she says.

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EDIT: After the fact, realize this is the same proof as Kate's. I will leave it here because I find it might be a bit clearer, but hers was first and also complete, so upvote that one.

~~~

Lets the number that the girls say be $S_i$ and the number the girls think be $T_i$. We then know $S_i=i$ and:

$$S_1=\frac{T_{10}+T_2}{2}=1 \implies T_{10}+T_2=2$$

We can repeat this for the rest and get:

$$S_2=T_1+T_3=4, S_3=T_2+T_4=6, S_4=T_3+T_5=8$$ $$S_5=T_4+T_6=10,S_6=T_5+T_7=12, S_7=T_6+T_8=14$$ $$S_8=T_7+T_9=16, S_9=T_8+T_{10}=18,S_{10}=T_9+T_1=20$$

We have 10 unknowns and 10 equations. In fact, we have 2 sets of 5 equations with 5 unknowns since the even subscripts are never in an equation with odd subscripts.

Lets solve the even subscript set first since $S_6=6$ and we are trying to solve $T_6$. $$T_{10}+T_2=2 \implies T_2=2-T_{10}$$ $$T_2+T_4=6 \implies 2-T_{10}+T_4=6 \implies T_4=4+T_{10}$$ $$T_4+T_6=10 \implies 4+T_{10}+T_6=10 \implies T_6=6-T_{10}$$ $$T_6+T_8=14 \implies 6-T_{10}+T_8=14 \implies T_8=8+T_{10}$$ $$T_8+T_{10}=18 \implies 8+T_{10}+T_{10}=18 \implies T_{10}=5$$

Thus, $T_6=6-T_{10}=6-5=1$ and $S_6=6$, so the girl who said 6 thought 1.

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2  
You've a typo in the last cluster of equations... T2+T4=6 ==> should be 2-T10+T4=6. (Too minor a change for a suggested edit.) – Brock Adams Feb 18 at 20:35
    
This was essentially the method I used. Glad to see I was doing it right, even if a bit more verbose than necessary. – corsiKa Feb 18 at 20:57
    
Seeing as you wanted $T_6$, I'm a little surprised that you solved for $T_{10}$ first. – Neil Feb 19 at 15:20
    
@Neil Meh... I just went around the circle. Seemed more natural to start at the beginning, but yes, that would have saved a step. – Trenin Feb 19 at 15:25

Less rigorous and mathematical, but perhaps more intuitive approach:

(a) note that for each girl,

there is no relationship at all between the number whispered and the number said; they are parts of different cycles.

(b)

in the group of girls we care about, the thought-of numbers averaged to $[1, 3, 5, 7, 9]$.

(c)

Note that this is symmetrical, so the number that was a component of both 1 and 9 must be 5. From there, the rest follows:
$5 + a = 9\times2 \therefore a = 13$
$13 + b = 7\times2 \therefore b = 1$

(d) to check the work, go the other way around the circle:

$5 + d = 1\times2 \therefore d = -3$
$-3 + c = 3\times2 \therefore c = 9$
$9 + b = 5\times2 \therefore b = 1.$
(The other 5 girls each thought of the number 1 higher than the girl sitting to one side of her).

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4  
"Note that this is symmetrical": I don't find your argument clear here. – cfh Feb 19 at 12:43
    
@cfh The average of the five numbers is 5, you can see that by adding them all together twice and pairing them to get each of the averages in (1, 3, 5, 7, 9) counted once. The average of the five numbers plus the x that participates in both the 1 and the 9 is also 5: you can add all of those together and pair them to get each of the averages in (1, 5, 9) counted once. If adding x to the list of numbers has no effect on the average, then x must equal the average. – hvd Feb 20 at 18:13
    
What I meant by symmetrical was that they are symmetrically distributed about the average; one number is the average, another is two higher, another is two lower, another is four higher, another is four lower -- and they are arranged such that the two-above and two-below are equidistant -- physically, around the circle -- from the average. The unknown numbers to the left and right of the average response must be equidifferent, and since the next reponses are equidifferent (as each other, from the average), the next unknowns must also be equidifferent, etc. – Vynce Feb 24 at 20:07
    
@cfh ... which means that the number between 1 and nine (a) must be equidifferent from 1 and 9, ergo 5 and (b) must be exactly as much above 5 as it is below 5, ergo 5. As I said, intuitive, but I'm afraid not mathematically rigorously explained. – Vynce Feb 24 at 20:15

Let's write down the sums:

$a_1 +a_3 = 2$
$a_2 +a_4 = 4$
...
$a_9 +a_1 = 18$
$a_{10} +a_2 = 20$

Then we can obtain two arithmetic series where the terms increase by 4:

$a_3,a_7,a_1,a_5,a_9$
$a_4,a_8,a_2,a_6,a_{10}$

The one who said 6 must have picked $a_7$, so we can find it using this and $a_7+a_9=14$ (or $a_7+a_5=10$), which means $a_7=1$.

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2  
I was with you until right after "next term increases by 4". Could you clarify what happens after that point? – user1717828 Feb 18 at 17:54
    
$a_7$ can be inferred from $a_7+12=a_9$ and $a_7+a_9=14$ or $a_7+8=a_5$ and $a_5+a_7=10$. – Nautilus Feb 18 at 19:21
1  
Uhhmm... erhmmm alrite thanks. – user1717828 Feb 18 at 19:50
    
This answer is very straightforward and makes sense. Nice job. – Timtech Feb 18 at 22:06

The answer is

1

I have used simple matrix multiplication while solving the problem;

Let say who said 1 is A, 2 is B, 3 is C, 4 is B ... J said 10;

$\frac{J+B}{2}=1 \ for \ A$

an so on;

\begin{bmatrix} & & 1& 2& & & \\ & & A& B& & & \\ 10& J& & & C& 3& \\ 9& I& & & D& 4& \\ 8& H& & & E& 5& \\ & & G& F& & & \\ & & 7& 6& & & \end{bmatrix}

So If we put every equation as a matrix form;

$\begin{bmatrix} 1& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 1\\ 1& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 1 \end{bmatrix}\cdot \begin{bmatrix} J\\ A\\ B\\ C\\ D\\ E\\ F\\ G\\ H\\ I \end{bmatrix}=\begin{bmatrix} 2\\ 4\\ 6\\ 8\\ 10\\ 12\\ 14\\ 16\\ 18\\ 20 \end{bmatrix}$

Then we need to take inverse of the first matrix and multiply with the right hand side;

$\begin{bmatrix} J (10)\\ A (1)\\ B (2)\\ C (3)\\ D (4)\\ E (5)\\ F (6)\\ G (7)\\ H (8)\\ I (9) \end{bmatrix}=\begin{bmatrix} 5\\ 6\\ -3\\ -2\\ 9\\ 10\\ 1\\ 2\\ 13\\ 14 \end{bmatrix}$

The one who announced 6 was F then the answer is 1. You may find the other numbers as above.

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2  
This method is overkill for this question. – f'' Feb 19 at 21:51
    
@f'' :) i agree – Oray Feb 19 at 22:36
1  
I didnt like the qustion, but I like this answer. thumb up – Woeitg Feb 20 at 20:55
    
@Woeistg thanks :) – Oray Feb 20 at 20:55

Let Girl<#> be the girl who announced #. For example, Girl6 is the girl who announced 6.

Similarly, let Value<#> be the number chosen by, and whispered by, Girl<#>. For example, Girl6 chose Value6, and whispered Value6 to Girl5 and Girl7.

Let x = Value6.

As shown below, x = Value6 = 1. The girl who announced 6 chose 1.

Value4 = 10 - x. This allows (Value4 + Value6)/2 = ((10 - x) + x)/2 = 10/2 = 5.
Value2 = x - 4. This allows (Value2 + Value4)/2 = ((x - 4) + (10 - x))/2 = 6/2 = 3.
Value8 = 14 - x. This allows (Value6 + Value8)/2 = (x + (14 - x))/2 = 14/2 = 7.
Value10 = x + 4. This allows (Value8 + Value10)/2 = ((14 - x) + (x + 4))/2 = 18/2 = 9.
(Value2 + Value10)/2 = 1. Thus, ((x - 4) + (x + 4))/2 = 1, or 2x/2 = 1, or x = 1.

x = Value6 = 1. The girl who announced 6 chose 1.

Check by substitution.
Girl.. Choice Announced
Girl1... __...... 1
Girl2.... -3...... 2
Girl3... __...... 3
Girl4..... 9...... 4
Girl5... __...... 5
Girl6.... 1...... 6
Girl7... __...... 7
Girl8... 13...... 8
Girl9... __...... 9
Girl10... 5.... 10

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@Neil -- Explanations of math problems should be pronounceable. Variable names can be short phrases, not just one letter with a subscript. The set-up of a problem should clearly describe all of the relevant variables. – Jasper Feb 19 at 14:55
1  
@f'' -- Thank you for rejecting edits that deviate from the original intent of the post. – Jasper Feb 19 at 14:56
1  
@Daedric -- Please reject edits that make posts worse. – Jasper Feb 19 at 14:57
1  
@Gamow -- Please reject edits that make posts worse. – Jasper Feb 19 at 14:57
    
Apart from Neil, these pings don't notify anyone. Also, I thought this post was improved with Neil's edit. – grgarside Feb 20 at 14:35

Let $g_n$ be the number the $n$th girl picked, so that $(g_{n-1}+g_{n+1})/2=n$ for $1\le n\le 10$ (where indices wrap around the circle, so $g_0=g_{10})$. Then $$ \begin{align} g_6 &= (g_6-g_4)/2 + (g_6+g_4)/2\\ &= (g_6+(g_8-g_8)+(-g_{10}+g_{10})+(g_2-g_2)-g_4)/2 + (g_6+g_4)/2\\ &= (g_6+g_8)/2 -(g_8+g_{10})/2+(g_{10}+g_2)/2-(g_2+g_4)/2+(g_6+g_4)/2\\ &= 7-9+1-3+5=\boxed{1} \end{align} $$

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  "2" whispers v to .
                      ` .
                       . "3"  =  (v+w)/2
                     .'   .        .     `.
  "4" whispers w to '.    .        .       :
                      `.                    :
                       . "5"  =  (w+X)/2     :
                     .'   .        .          :
  "6" whispers X to '.    .        .           + -->  3+9  =  (v+w+y+z)/2  =  12
                      `.                      :                            .
                       . "7"  =  (X+y)/2     :                             .
                     .'   .        .        :                             .
  "8" whispers y to '.    .        .       :                             .
                      `.                 .'                             .
                       . "9"  =  (y+z)/2                               .
                     .'   .        .                                 .
 "10" whispers z to '.    .        .                                .
                      `.                                          .
                       . "1"  =  (z+v)/2                        .
                     .'   .        .                          .
  "2" whispers v to '     .        .                        .
                          .        .                     .
                   3+5+7+9+1  =  v+w+X+y+z  =  25     .
                                            :      .
                                  v+w+y+z   =  24
                                            :

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\:$ :
$\qquad\qquad\qquad\qquad\quad~~\!\;$ "6" picked X = 25 - 24 = 1

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We are given that the number $b_i$ announced by girl $i$ is the average of the numbers $x_{i^+}$ and $x_{i^-}$ chosen by the neighboring girls $i^+$ and $i^-$:

$$2 b_i = x_{i^+} + x_{i^-},$$

where

$$\begin{array}{rcl} i^+ & = & i + 1 - \left\lfloor \frac{i}{N} \right\rfloor N, \\ i^- & = & i - 1 + \left\lfloor \frac{N + 1 - i}{N} \right\rfloor N \end{array}$$

and $N$ is the number of girls (that is, $i^+$ and $i^-$ loop around the interval $[1, N]$). This forms a linear equation system, which can be expressed on matrix form as $A\mathbf{x} = \mathbf{b}$ where the number at row $i$, column $j$ of the matrix $A$ is

$$a_{i,j} = \begin{cases} \frac{1}{2}, \ i-j \equiv_N 1,\\ \frac{1}{2}, \ j-i \equiv_N 1,\\ 0 \ \ \mathrm{otherwise.} \end{cases}$$

For $N = 5$, $2A$ looks like:

$$\begin{array}{ccccc} &0 &1 &0 &0 &1 \\ &1 &0 &1 &0 &0 \\ &0 &1 &0 &1 &0 \\ &0 &0 &1 &0 &1 \\ &1 &0 &0 &1 &0 \end{array}$$

Solving this equation system for $\mathbf{x}$ with $N = 10$ and $\mathbf{b} = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]$ gives

$x_6 = 1$.

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6. For her to announce an average of 6, the two numbers she receives must add up to 12. This happens to be the average of the two averages announced to either side of her. This pattern holds around the circle, which makes sense if the announced average is also the number picked.

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2  
Wait, I forgot to take into account that this is a circle. :( will edit answer once I reconsider – Irishpanda Feb 18 at 13:08
1  
haha, I had this same thought process initially – question_asker Feb 18 at 16:03

1

Let $g_n$ be the number picked by the girl who said $n$. The average of $g_6$ and $g_4$ is two more than the average of $g_2$ and $g_4$, so $g_6$ must be four more than $g_2$. Symmetrically $g_6$ four less than $g_{10}$. $g_6$ is therefore the average of $g_2$ and $g_{10}$, but we already have already been told what this is.

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