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PrintXXIX Rioplatense Mathematical Olympiad
Argentina counting and probability
Problem
Consider a table with its columns and rows labelled to from left to right and bottom to top respectively. Lucía writes the numbers from to in increasing order by steps, in the following way:
Number is in the cell of row and column .
In the first step, she fills every cell which is a neighbor of , anti-clockwise. In the second step, she fills every cell which is a neighbor of any cell filled in the previous step, clockwise. She continues this way in each step, filling the cells that are neighbors of the cells filled in the previous step and changing the orientation.
The figure shows the table after Lucía completed the first three steps. In which step will Lucía write the number ? In which row and which column is written?
| Row 4 | 25 | 24 | 23 | 22 | 21 | 20 | 19 | ||
| Row 3 | 26 | 9 | 10 | 11 | 12 | 13 | 18 | ||
| Row 2 | 27 | 8 | 5 | 4 | 3 | 14 | 17 | ||
| Row 1 | 28 | 7 | 6 | 1 | 2 | 15 | 16 | ||
| 47 | 48 | 49 | 50 | 51 | 52 | 53 |
In the first step, she fills every cell which is a neighbor of , anti-clockwise. In the second step, she fills every cell which is a neighbor of any cell filled in the previous step, clockwise. She continues this way in each step, filling the cells that are neighbors of the cells filled in the previous step and changing the orientation.
The figure shows the table after Lucía completed the first three steps. In which step will Lucía write the number ? In which row and which column is written?
Solution
First, let us observe that every step contains four more numbers than the previous one. In fact, the number of cells filled in step is . After steps, there are numbers in the table. We can compute that sum by multiplying by the sum of the numbers from to and then adding .
On the other hand, after steps there are numbers in the table. Hence, is written in step .
Now, notice that the last number written in step is and, since is odd, in that step the numbers were filled anti-clockwise, so is in column and row . Finally, since the numbers of step are filled clockwise, is in column and row .
On the other hand, after steps there are numbers in the table. Hence, is written in step .
Now, notice that the last number written in step is and, since is odd, in that step the numbers were filled anti-clockwise, so is in column and row . Finally, since the numbers of step are filled clockwise, is in column and row .
Final answer
Step 32; row 6; column 18
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