Lucka wrote the number 12345 on a piece of paper and cut it twice between the digits. She thus obtained three smaller cards with three numbers. She rearranged these cards in two ways, thereby obtaining two different five-digit numbers. The difference between these two numbers was 28926.

Between which digits did Lucka cut the paper?

(M. Petrová)
For clarity, let's write the problem as a written subtraction, whose result is 28926, and where the asterisks on each line represent digits from 1 to 5:

![](https://cdn.mathpix.com/cropped/2024_04_17_96587d6cc949a3c812fag-5.jpg?height=157&width=263&top_left_y=1726&top_left_x=902)

The digit in the units place can only be obtained in this way (with a carry-over from the tens place):

$$
\begin{array}{r}
* * * * 1 \\
-* * * * 5 \\
\hline 28926
\end{array}
$$

With the remaining available digits, the digit in the tens place can only be obtained in one of the following ways:

$$
\begin{aligned}
& * * * 41 \quad * * * 51 \\
& \begin{array}{r}
-* * * 15 \\
\hline 28926
\end{array} \frac{-* * * 25}{28926}
\end{aligned}
$$

a) In the first case, the sheet must be cut as follows: $1|234| 5$ and the cards rearranged as $5|234| 1$ and $234|1| 5$. In this case, the difference meets the requirements:

$$
\begin{array}{r}
52341 \\
-23415 \\
\hline 28926
\end{array}
$$

b) In the second case, the sheet must be cut as follows: $1|2| 34 \mid 5$, which are three (not two) cuts. This case does not meet the requirements.

Lucka cut the sheet between the digits 1 and 2 and between the digits 4 and 5.

Note. The necessity of the cut between the digits 1 and 2 already follows from the first observation. The necessity of the cut between the digits 4 and 5 (and subsequent rearrangement) follows from this reasoning: if the digit 5 were preceded by 4, then we would need to be able to complete

![](https://cdn.mathpix.com/cropped/2024_04_17_96587d6cc949a3c812fag-6.jpg?height=157&width=257&top_left_y=932&top_left_x=905)

In such a case, the digit in the tens place of the subtrahend would have to be 7, which we do not have available.

Another solution. The sheet can be cut in six ways:

$$
1|2| 345, \quad 1|23| 45, \quad 1|234| 5, \quad 12|3| 45, \quad 12|34| 5, \quad 123|4| 5
$$

The three resulting cards can be rearranged in six ways, which we schematically (in ascending order) write as follows:

$$
A|B| C, \quad A|C| B, \quad B|A| C, \quad B|C| A, \quad C|A| B, \quad C|B| A .
$$

By examining all possibilities, we can find the cutting and rearrangement that meets the requirements.

Instead of examining possible differences (of which there are 15 for each cut), we can proceed by adding the required difference of 28926 to the relevant six numbers and checking if any of the results are among the given six numbers. The largest number that can be formed with the digits from 1 to 5 is 54321. Thus, the largest number to which it makes sense to add the required difference is 25395. This significantly reduces the examination and looks as follows (repeating or previously rejected possibilities are written in parentheses):

a) After cutting $1|2| 345$, the numbers that can be formed are

$$
12345, \quad 13452, \quad 21345, \quad 23451, \quad 34512, \quad 34521
$$

After adding 28926, we get

$$
41271, \quad 42378, \quad 50271, \quad 52377, \quad(>54321, \quad \ldots)
$$

No option meets the requirements.
b) After cutting $1|23| 45$, the numbers that can be formed are

$$
12345, \quad 14523, \quad 23145, \quad 23451, \quad 45123, \quad 45231
$$

After adding 28926, we get

$$
(41271), \quad 43449, \quad 52071, \quad(52377, \quad>54321, \quad \ldots)
$$

No option meets the requirements.

c) After cutting $1|234| 5$, the numbers that can be formed are

$$
12345, \quad 15234, \quad 23415, \quad 23451, \quad 51234, \quad 52341
$$

After adding 28926, we get

$$
(41271), \quad 44160, \quad 52341, \quad(52377, \quad>54321, \ldots)
$$

The option $23451+28926=52341$ meets the requirements.

d) After cutting $12|3| 45$, the numbers that can be formed are

$$
12345, \quad 12453, \quad 31245, \quad 34512, \quad 45123, \quad 45312
$$

After adding 28926, we get

$$
(41271), \quad 41379, \quad(>54321, \quad \ldots)
$$

No option meets the requirements.

e) After cutting $12|34| 5$, the numbers that can be formed are

$$
12345, \quad 12534, \quad 34125, \quad 34512, \quad 51234, \quad 53412
$$

After adding 28926, we get

$$
(41271), \quad 41460, \quad(>54321, \quad \ldots)
$$

No option meets the requirements.

f) After cutting $123|4| 5$, the numbers that can be formed are

$$
12345, \quad 12354, \quad 41235, \quad 45123, \quad 51234, \quad 54123
$$

After adding 28926, we get

$$
(41271), \quad 41280, \quad(>54321, \ldots)
$$

No option meets the requirements.

The only option that meets the requirements is in case c): Lucka cut the sheet between the digits 1 and 2 and between the digits 4 and 5.

Note. In case a), the difference between the largest and smallest number in the given six is 22176. All other differences are smaller, so the required difference of 28926 cannot be obtained, and it was not necessary to try anything further.