Larissa and Jorge are playing with numbered cards from 1 to 6 that must be placed in the spaces on the board below to form a six-digit number.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-008.jpg?height=68&width=797&top_left_y=477&top_left_x=664)

Jorge places the first card, and then the turns alternate between the two. Larissa's goal is to get the largest number possible, and Jorge's goal is to get the smallest number possible. Larissa has the cards with the digits 1, 3, and 5, and Jorge has the cards with the digits 2, 4, and 6. If both players are smart, what number will appear at the end of the game?
(a) 254361
(b) 253416
(c) 251634
(d) 256134
(e) 251346
The correct option is (b).

The formation of a 6-digit number is illustrated below.

| hundred  thousand | ten  thousand | unit  thousand | hundred | ten | unit |
| :---: | :---: | :---: | :---: | :---: | :---: |

To obtain the smallest number possible, the smallest digits should be placed as far to the left as possible (in the thousand place), and to obtain the largest number possible, the largest digits should also be placed as far to the left as possible (in the thousand place).

Jorge plays first: To obtain the smallest number possible, he places the smallest digit he has, which is 2, in the hundred thousand place. If he did not do this, Larissa would place her 5 in this place on her next turn and thus obtain a larger number.

| 2 | ten  thousand | unit  thousand | hundred | ten | unit |
| :---: | :---: | :---: | :---: | :---: | :---: |

Now it's Larissa's turn: To obtain the largest number possible, she places the largest digit she has, which is 5, in the ten thousand place, as the hundred thousand place is already occupied.

| 2 | 5 | unit  thousand | hundred | ten | unit |
| :---: | :---: | :---: | :---: | :---: | :---: |

Now, Jorge has the digits 4 and 6, and Larissa has 1 and 3. Therefore, Larissa's digits are smaller than Jorge's, which determines Jorge's strategy: he should try to place his digits as far to the right as possible, with the 6 to the right of the 4. In turn, Larissa should try to place her digits as far to the left as possible, with the 3 to the left of the 1.

Jorge plays: He places the digit 6 in the unit place.

| 2 | 5 | unit  thousand | hundred | ten | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: |

Larissa plays: She places her 1 in the ten place.

| 2 | 5 | unit  thousand | hundred | 1 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: |

Now, Jorge has only the digit 4 and Larissa has the 3. He then places the 4 in the hundred place and Larissa places the 3 in the unit thousand place, ending the game.

| 2 | 5 | 3 | 4 | 1 | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- |

Thus, the final number, obtained if both players are smart, is 253416.