1. Erika and Klárka played a word logic game with the following rules: Player A thinks of a word composed of $z$ five different letters. Player $B$ says any word made up of five different letters, and player A reveals how many letters $B$ guessed in the correct position and how many in the wrong position. Letters are considered different even if they only differ by a diacritical mark (for example, the letters $A, \bar{A}$ are different). If player A, for example, thought of the word LOĎKA and $B$ said the word KOLÁC̆, player $A$ would respond that player $B$ guessed one letter in the correct position and two in the wrong position. This is abbreviated as "1+2" because both words only match in the letter $O$ including its position (second from the left) and in the letters $K$ and $L$, whose positions are different. Erika thought of a word with five different letters, and Klárka said the words KABÁT, STRUK, SKOBA, CESTA, and ZÁPAL. Erika responded to these words in the given order with $0+3, 0+2, 1+2, 2+0$, and $1+2$. Determine what word Erika could have been thinking of.
SOLUTION. The words ZÁPAL and STRUK do not have any letters in common. Therefore, as follows from the answers $1+2$ and $0+2$, among their letters, which together form the set $\mathrm{M}=$ $=\{\mathrm{Z}, \mathrm{A}, \mathrm{P}, \mathrm{A}, \mathrm{L}, \mathrm{S}, \mathrm{T}, \mathrm{R}, \mathrm{U}, \mathrm{K}\}$, there are all five letters of the sought word. In the word SKOBA, there should be exactly three of the sought letters. They are therefore the letters $\mathrm{S}, \mathrm{K}, \mathrm{A}$. (The remaining letters $\mathrm{B}$ and $\mathrm{O}$ do not belong to the set $\mathrm{M}$.) In the word CESTA, there should be only two of the sought letters, and both in the correct positions. They are the already found $S$ and $A$, which therefore belong to the third and fifth positions of the sought word (and the letter $T$ can be "excluded" from the set $M$). The letter $\mathrm{K}$ cannot be in the first or second position, as follows from the answers for the words KABÁT $(0+3)$ and SKOBA $(1+2)$. It is therefore in the fourth position, and it remains to determine the first two letters. In the word STRUK, there are only two of the sought letters (they must be $\mathrm{S}$ and $\mathrm{K}$), both in incorrect positions. Therefore, we "exclude" the letters $R, U$ from the set $M$ (and $T$, if we haven't done so yet). The remaining two sought letters therefore belong to the set $\{Z, A', P, L\}$. From the conditions for the word KABÁT, it follows that one of them is Á. In the word ZÁPAL, there is exactly one letter in the correct position. If it were $\mathrm{Z}$, we would have nowhere to place the letter Á. Therefore, Á is in the second position, and additionally, we can exclude the letter Z. The first letter of the sought word can be $\mathrm{L}$ or $\mathrm{P}$. It is easy to verify that the found words LÁSKA and PÁSKA satisfy all the conditions of the problem.

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