Just as in the Roman numeral system, for example, $9=\mathrm{IX}=\mathrm{X}-\mathrm{I}$, meaning the I numeral is considered negative because it stands before a larger value numeral, in the Arabic numeral system, certain digits can also be assigned a negative sign. This is typically expressed by placing a line over the digit, for example, $19=20-1=2 \overline{1}$ (twenty minus one), $263=3 \overline{4} 3=3 \overline{3} \overline{7}=1 \overline{8} 7 \overline{7}$.

How many ways can 1969 be written using this method with at most 6 digits?
We present two observations. Allowing negative digits as well, the highest place value digit of a positive number can only be positive, as this notation is similar to paying a sum of money with change, where change can only be given from a larger amount, and we receive less back than we gave. Furthermore, it would be redundant to assign a negative sign to the digit 0.

A two-digit number, if it contains a one, i.e., does not end in 0, can also be written in a new way by rounding up to the nearest ten and specifying the difference from it: $69=70-1=7 \overline{1}, 8=08=1 \overline{2}, 94=10 \overline{6}$; a round (two-digit) ten, however, cannot be (two-digit) written in the new way.

Furthermore, if we allow negative digits from right to left in more and more places, then in the found notations of a number, the tens, hundreds, ... digits can only be replaced by a negative digit in cases where the digit in question is not 0. For example, for the tens place with a negative digit:

$$
969=9 \overline{71}
$$

both notations can lead to a new form:

$$
10 \overline{49}=10 \overline{31}
$$

Similarly,

$$
1969=197 \overline{1}=20 \overline{4} 9=20 \overline{31}
$$

from here, allowing negative digits in the hundreds place as well, only the first two notations can lead to a new form:

$$
2 \overline{1} 69=2 \overline{1} 7 \overline{1}
$$

because (1) the rounded tens value of the last two notations is a round hundred in both cases. According to this, 1969 can be written in 6 different ways with 4 digits, 5 of which are new.

Switching to 5- and 6-digit notations - where the 5th and 6th digits are not needed in the usual notation of our number, we agree that an extended notation (writing) with one or two leading 0s is not considered new. Thus, from each notation of (1) and (2), we get only 1 five-digit and 1 six-digit notation, by writing the following in place of the first digit 1 xyz and 2 uvw (where the letters represent the last three digits):

$$
1 \overline{9} x y z, \quad 1 \overline{8} u v w, \quad \text { or } \quad 1 \overline{9} \overline{9} x y z, \quad 1 \overline{9} \overline{8} u v w
$$

From the above $6+6+6=18$ notations, 17 contain at least one negative digit.

Maria Buruzs (Tata, Eötvös J. Gymnasium, 1st year) Katalin Párkány (Budapest, Berzsenyi D. Gymnasium, 1st year)