4. Miha has experimented with writing various numbers using only the digit 1 and the addition sign. For example, he found that there are only two natural numbers $n$ (13 and 4) for which the number 13 can be written using $n$ ones and the addition sign, since the number 13 can be written as the sum of thirteen ones or as $11+1+1$, using four ones.

How many different natural numbers $n$ are there for which the number 125 can be written using $n$ ones and the addition sign?

MATHEMATICAL COMPETITION

7. April 2001

## PROBLEMS FOR THE FOURTH GRADE
III/4. First, we have a sum of 125 units, then we replace eleven units from the sum with the number 11, thereby reducing the number of units by nine: we have 116 units, of which two are "bound" (in the number 11). In the next step, we again replace eleven units from the sum with 11, and we arrive at 107 units, of which four are "bound". We can continue the process as long as we have enough "unbound" units. At each step, we have $125-9k$ units (of which $2k$ are "bound"), where $k=0,1, \ldots, 11$. At $k=11$, we have 26 units (of which 22 are "bound"), so we cannot continue the process.

The number 125 can also be written using the number 111. There are two possibilities: $111+1+1+\cdots+1$ with 17 units and $111+11+1+1+1$ with 8 units. Thus, we have 14 different natural numbers $n$ for which it is true that using $n$ units and the addition sign, we can write the number 125 (these are $8,17,26$, 35 ... 116, 125).

Writing the number 125 with $\mathbf{125}$ units: $\mathbf{1}$ point. Replacing eleven units with the number 11: $\mathbf{1}$ point. Writing all 12 possibilities without the term 111: 4 points. Writing both possibilities with the term 111: 1 point.