6. How many numbers from 1 to 999 without the digit "0" are written in the Roman numeral system exactly one symbol longer than in the decimal system?

(P. D. Mulyenko)

Reference. To write a number in Roman numerals, you need to break it down into place value addends, write each place value addend according to the table, and then write them sequentially from the largest to the smallest. For example, let's write the number 899. According to the table, $800 = \text{DCCC}$, $90 = \text{XC}$, $9 = \text{IX}$, we get DCCCXCIX.

| 1 I | $10 \text{X}$ | $100 \text{C}$ | $1000 \text{M}$ |
| :--- | :--- | :--- | :--- |
| $2 \text{II}$ | $20 \text{XX}$ | $200 \text{CC}$ | $2000 \text{MM}$ |
| 3 III | $30 \text{XXX}$ | $300 \text{CCC}$ | $3000 \text{MMM}$ |
| $4 \text{IV}$ | $40 \text{XL}$ | $400 \text{CD}$ |  |
| 5 V | $50 \text{L}$ | $500 \text{D}$ |  |
| $6 \text{VI}$ | $60 \text{LX}$ | $600 \text{DC}$ |  |
| $7 \text{VII}$ | $70 \text{LXX}$ | $700 \text{DCC}$ |  |
| $8 \text{VIII}$ | $80 \text{LXXX}$ | $800 \text{DCCC}$ |  |
| 9 IX | $90 \text{XC}$ | $900 \text{CM}$ |  |
Solution. Note that, regardless of the digit place and other digits of the number, a decimal digit $a$ is written as:

- one Roman numeral when $a=1$ and $a=5$,
- two Roman numerals when $a$ is $2,4,6,9$,
- three Roman numerals when $a=3$ and $a=7$,
- four Roman numerals when $a=8$.

Thus, in suitable numbers, only the digits 1, 5, and exactly one of the digits $2,4,6,9$ are used (otherwise, the total length of the record will be longer by more than one character). This means that 4 single-digit numbers, $4 \cdot 2 + 2 \cdot 4 = 16$ two-digit numbers, and $4 \cdot 2 \cdot 2 + 2 \cdot 4 \cdot 2 + 2 \cdot 2 \cdot 4 = 48$ three-digit numbers are suitable. In total, 68 numbers.

Criteria. Correct answer without justification - 2 points.

One of the above cases is missing or extra cases are considered - 0 points.

The condition about the absence of zero is incorrectly interpreted (for example, as "excluding zero," meaning that the length of the Roman record of the number should be one more than the number of non-zero digits - no more than 3 points.