Problem 7.1. Each of the seven dwarfs thought of a natural number. They all know what the others have thought of. Snow White asked each of the dwarfs what number they thought of.

- The 1st dwarf remained silent.
- The 2nd dwarf said: “My number is the same as the number of the first dwarf.”
- The 3rd dwarf said: “My number is the sum of the numbers of the first and second dwarfs.”
- The 4th dwarf said: “My number is the sum of the numbers of the first, second, and third dwarfs.”

$-\ldots$

- The 7th dwarf said: “My number is the sum of the numbers of the first, second, third, fourth, fifth, and sixth dwarfs.”

It is known that the sum of the seven numbers thought of by the dwarfs is 46. It is also known that exactly one dwarf lied. What number could the lying dwarf have thought of? List all possible options.
Answer: 7 or 14.

Solution. Let the first dwarf have guessed the number $a_{1}$, the second $a_{2}$, the third $-a_{3}, \ldots$, the seventh $-a^{7}$.

Assume that the 7th dwarf told the truth. Then his number is exactly half of the sum of all numbers, that is, $a_{7}=\frac{46}{2}=23$. Reasoning similarly, we get that the sixth dwarf definitely lied, otherwise his number would have been $\frac{23}{2}$.

Therefore, the first five dwarfs did not lie, and they guessed the numbers $a_{1}=x, a_{2}=x$, $a_{3}=2 x, a_{4}=4 x, a_{5}=8 x$. Then

$$
23=a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}=16 x+a_{6}>16 x
$$

Since $x$ is a natural number, $x=1$ (if $x \geqslant 2$, then $16 x>23$). From this, it is not difficult to calculate $a_{6}=23-16 \cdot 1=7$ - this is the number guessed by the lying dwarf. It is clear that the numbers $1,1,2,4,8,7,23$ fit the condition.

## Assume that the 7th dwarf lied.

Therefore, the first six dwarfs did not lie, and they guessed the numbers $a_{1}=x, a_{2}=x$, $a_{3}=2 x, a_{4}=4 x, a_{5}=8 x, a_{6}=16 x$. Then

$$
46=a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}=32 x+a_{7}>32 x
$$

Since $x$ is a natural number, $x=1$ (if $x \geqslant 2$, then $32 x>46$). From this, it is not difficult to calculate $a_{7}=46-32 \cdot 1=14$ - this is the number guessed by the lying dwarf. It is clear that the numbers $1,1,2,4,8,16,14$ fit the condition.