## Task 4 - 260514

Pioneer Klaus Knobler performs as a magician in front of his pioneer group. After his eyes are blindfolded, he asks a spectator to take out an odd number of matches, but at least 13, from a matchbox.

These matches should be placed on the table in two parallel rows, with the upper row containing exactly one more match than the lower row. After this is done, Klaus Knobler

(1) removes any number $a$ (at least 1, but less than 7) of matches from the upper row, then

(2) removes as many matches from the lower row as there are still left in the upper row, and then

(3) removes all remaining matches from the upper row.

Afterwards, Klaus Knobler names the number of matches left on the table to the amazed audience. How many are there?

By what reasoning can Klaus Knobler find this number without knowing the number of matches initially placed on the table?
Klaus Knobler can reason as follows:

Assume that at the beginning, in addition to the a matchsticks, there were b more sticks in the upper row. Thus, initially, there were $\begin{array}{r}a+b \\ a+b-1\end{array}$ sticks in the upper and lower rows, respectively. After executing (1), there are $\begin{array}{r}b \\ a+b-1\end{array}$ sticks in the upper and lower rows, respectively. After executing (2), there are $\begin{array}{r}r a-1\end{array}$ sticks in the upper and lower rows, respectively. Finally, after executing (3), $a-1$ sticks remain on the table. Therefore, Klaus Knobler only needs to form the predecessor of the number $a$ he mentioned in (1) and state it as the number of matchsticks remaining on the table.

Solutions of the I. Round 1986 adopted from [5]

### 2.27.2 II. Round 1986, Class 5