6. How many numbers from 1 to 999 are written in the Roman numeral system with the same number of symbols as 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 VI | $60 \text{LX}$ | $600 \text{DC}$ |  |
| $7 \text{VII}$ | $70 \text{LXX}$ | $700 \text{DCC}$ |  |
| 8 VIII | $80 \text{LXXX}$ | $800 \text{DCCC}$ |  |
| $9 \text{IX}$ | $90 \text{XC}$ | $900 \text{CM}$ |  |
|  |  |  |  |
Solution. Note that, regardless of the digit place and other digits of the number, a decimal digit a is written as:

- zero Roman numerals when $a=0$,
- 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, for a number to meet the condition, it must:

1) either consist only of the digits 1 and 5,
2) or contain a pair of digits 0 and $x$, where $x$ is one of the digits $2,4,6$ or 9, and possibly one digit 1 or 5,
3) or consist of one digit 3 or 7 and two zeros.

There are a total of $2+2 \cdot 2+2 \cdot 2 \cdot 2=14$ numbers of the first type and 2 numbers of the third type (DCC and CCC). There are exactly 4 two-digit numbers of the second type, and 32 three-digit numbers $-2 \cdot 2 \cdot 4 \cdot 2=32$ (2 ways to place the digit "0", two ways to choose the position for the single-character digit, 4 ways to write the two-character digit, and 2 ways to write the single-character digit). In total, $14+$ $2+4+32=52$ numbers.

Criteria. Correct answer without justification - 2 points.

Each of the missing cases described above --2 points.

Arithmetic error in each of the cases --1 point.