determine the number thought of

Question

Ten people are seated around a circular table. Each of the ten people thinks of a number and whispers it to his/her two neighbours. Then these ten people announce the average of the two numbers they heard each such that we have 1, 2, 3, 4, 5, 6,7, 8, 9, 10 written on a circle. What number did the person who said 6 think of?

Answer 1

Let $x_i$ be the number thought by the person who said $i$. Then we can write a system of equations: $$\begin{cases} x_2+x_{10} =2\\ x_1+x_3=4\\ x_2+x_4 = 6\\ x_3+x_5=8\\ x_4+x_6=10\\ x_5+x_7=12\\ x_6+x_8=14\\ x_7+x_9=16\\ x_8+x_{10}=18\\ x_1+x_9=20 \end{cases}$$

In matrix form we have : $$\left[ \begin{array}{cccccccccc|c} 1&0&0&0&0&0&0&0&1&0&20\\ 0&1&0&0&0&0&0&0&0&1&2\\ 1&0&1&0&0&0&0&0&0&0&4\\ 0&1&0&1&0&0&0&0&0&0&6\\ 0&0&1&0&1&0&0&0&0&0&8\\ 0&0&0&1&0&1&0&0&0&0&10\\ 0&0&0&0&1&0&1&0&0&0&12\\ 0&0&0&0&0&1&0&1&0&0&14\\ 0&0&0&0&0&0&1&0&1&0&16\\ 0&0&0&0&0&0&0&1&0&1&18 \end{array}\right]$$

Then rearrange the third row by subtracting the first row, and rearrange the fourth row by subtracting second row.

Then rearrange fifth row by subtracting new third row, sixth by subtracting new fourth, and so on. After all that steps we obtain:

$$\left[ \begin{array}{cccccccccc|c} 1&0&0&0&0&0&0&0&1&0&20\\ 0&1&0&0&0&0&0&0&0&1&2\\ 0&0&1&0&0&0&0&0&-1&0&-16\\ 0&0&0&1&0&0&0&0&0&-1&4\\ 0&0&0&0&1&0&0&0&1&0&24\\ 0&0&0&0&0&1&0&0&0&1&6\\ 0&0&0&0&0&0&1&0&-1&0&-12\\ 0&0&0&0&0&0&0&1&0&-1&8\\ 0&0&0&0&0&0&0&0&2&0&28\\ 0&0&0&0&0&0&0&0&0&2&10 \end{array}\right]$$

Divide two last rows by $2$ and get rid of every non-zero elements that are not on diagonal in ninth row by adding or subtracting ninth row, same in tenth column. We get:

$$\left[ \begin{array}{cccccccccc|c} 1&0&0&0&0&0&0&0&0&0&6\\ 0&1&0&0&0&0&0&0&0&0&-3\\ 0&0&1&0&0&0&0&0&0&0&-2\\ 0&0&0&1&0&0&0&0&0&0&9\\ 0&0&0&0&1&0&0&0&0&0&10\\ 0&0&0&0&0&1&0&0&0&0&1\\ 0&0&0&0&0&0&1&0&0&0&2\\ 0&0&0&0&0&0&0&1&0&0&13\\ 0&0&0&0&0&0&0&0&1&0&14\\ 0&0&0&0&0&0&0&0&0&1&5 \end{array}\right]$$

The result is then: $$\begin{cases} x_1=6\\x_2=-3\\x_3=-2\\x_4=9\\x_5=10\\x_6=1\\x_7=2\\x_8=13\\x_9=14\\x_{10}=5 \end{cases}$$ Person who said $6$ thought about number $1$