Problem 6. Fox Alice and Cat Basil have grown 20 fake banknotes on a tree and are now filling in seven-digit numbers on them. Each banknote has 7 empty cells for digits. Basil calls out one digit at a time, either "1" or "2" (he doesn't know any others), and Alice writes the called digit in any free cell of any banknote and shows the result to Basil.

When all cells are filled, Basil takes as many banknotes with different numbers as possible (if there are several with the same number, he takes only one), and the rest go to Alice. What is the maximum number of banknotes Basil can get, no matter how Alice acts? [8 points] (A. V. Shapovalov)
Answer: 2.

Solution: Basil can always get two banknotes: he knows the place where the last digit should be written and names it so that it differs from the digit in the same place on some other banknote. Then the numbers on these two banknotes will be different, and the cat can take them.

We will show how Alice can ensure that there are no more than two different numbers. She arranges the banknotes one above the other so that the cells for the digits form a table.

When the cat names a one, Alice writes it in the leftmost column where there is a free cell (in any of the cells), and when the cat names a two - in the rightmost column.

If both ones and twos start to appear in some column, then all other columns are already filled: ones on the left, twos on the right. This means there will be at most one column where both ones and twos appear. Therefore, if the numbers do differ, they will differ only by the digit in this column. Since there are only two digits, there will be no more than two different numbers.

![](https://cdn.mathpix.com/cropped/2024_05_06_031fac0a0a7981eac22cg-11.jpg?height=362&width=301&top_left_y=917&top_left_x=576)

$\begin{array}{lllllllllllll}1 & 1 & 1 & 1 & 2 & 2 & 2\end{array}$

![](https://cdn.mathpix.com/cropped/2024_05_06_031fac0a0a7981eac22cg-11.jpg?height=56&width=287&top_left_y=971&top_left_x=583)

![](https://cdn.mathpix.com/cropped/2024_05_06_031fac0a0a7981eac22cg-11.jpg?height=50&width=289&top_left_y=1073&top_left_x=582)

$\begin{array}{lllllll}1 & 1 & 1 & 2 & 2 & 2 & 2\end{array}$

$\begin{array}{lllllll}1 & 1 & 1 & 2 & 2 & 2 & 2\end{array}$

The 11th Oral City Mathematical Olympiad for 6-7 grades will take place on March 17, 2013.

Students who have received a diploma or certificate as a prize winner or honorable mention in at least one of the following mathematical competitions are invited to participate:

- Mathematical Festival (19.02.12 or 17.02.13),
- 10th City Oral Olympiad (09.03.12),
- Winter Archimedes Tournament (22.01.12 or 20.01.13),
- Spring Archimedes Tournament for 5th grade (individual competition, 01.04.12).

To participate in the Olympiad, you need to register in advance by March 5, 2013. Details on the website olympiads.mccme.ru/ustn/

## Information on Enrollment in 5-8 Grades with Advanced Mathematics in 2013

| School | Phone, URL | Address | Grades | Periods |
| :---: | :---: | :---: | :---: | :---: |
| 2 | (499) $137-17-69$  (499) $137-69-31$  www.sch2.ru | Fotieva St., 18  (m. "Oktyabrskaya") | 7,8 | March-May |
| 25 | sch25.ru  nabor@mathbaby.ru | University Ave., 7  (m. "University") | $7 ;$  additional in 8 | from February |
| 54 | (499) 245-99-72  (499) 245-54-25  moscowschool54.ru | Dovator St., 5/9  (m. "Sportivnaya") | 8 | February-May |
| 57 | (495) $691-85-72$  (495) $691-54-58$  sch57.msk.ru | Mal. Znamensky Lane, 7/10,  bld. 5 (m. "Borovitskaya") | 8 | from March 13  Wednesdays |
| 179 | (495) $692-48-51$  www.179.ru | Bol. Dmitrovka St., 5/6,  bld. 7 (m. "Okhotny Ryad") | 7,8 | from March 22 |
| 192 | (499) $137-33-55$  (499) $137-72-85$  www.sch192.ru | Leninsky Prospekt 34-A  (m. "Leninsky Prospekt") | 5,$7 ;$  additional in 8 | April-May  Fridays  at $16^{00}$ |
| 218 | (499) 976-19-85  school218.ru | Dmitrovskoye Shosse, 5a  (m. "Dmitrovskaya") | 8 (IUP) | from March 25  to May 18 |
| 444 | (495) 465-23-52  $(495) 465-60-52$  schv444.mskobar.ru | Nizh. Pervomayskaya St., 14  (m. "Pervomayskaya") | 8 | March-May |
| 1329 | co1329.mskzapad.ru  nabor@mathbaby.ru | Nikulin St., 10  (m. "Yugo-Zapadnaya") | 5,8 | from February |
| 1543 | (495) $433-16-44$  $(495) 434-26-44$  www. $1543 . r u$ | 26 Baku Commissars St., 3, bld. 5  (m. "Yugo-Zapadnaya") | 8 | April |
| 2007 | (495) 716-29-35  fmsh2007.ru | Gorchakova St., 9, bld. 1  (m. "Gorchakova St.") | $5-7 ;$  additional in 8 | April |
| Intel-  lectual | (499) 445-52-10  sch-int.ru | Kremenchugskaya St., 13  (m. "Slavyanskiy Bulvar") | $5 ;$  additional  in 6,7 | March-June |

The information is provided by the schools to the MCCME. It is published free of charge.

Detailed information on enrollment in these and other classes is available on the website www.mccme.ru

Current information on olympiads is available on the website www.olimpiada.ru

The page of the Mathematical Festival (problems, solutions, lists of winners) www.mccme.ru/matprazdnik /