459. The first player is best off naming 2 or 3, since in these cases only one outcome of the dice throw will lead to his defeat. If he names 1, then an unfavorable outcome would be rolling a 3 or 6. If he names 2, then only rolling a 5 would be unfavorable. If he names 3, then only a 4 would be unfavorable. If he names 4, then a 3 or 4 would be unfavorable. If he names 5, then a 2 or 3 would be unfavorable. If he names 6, then a 1 or 5 would be an unfavorable outcome. A complete analysis of this game is not possible here, but I will say that if you score 5, 6, 9, 10, 14, 15, 18, 19, or 23 points at any position of the die, you will definitely lose. If you score 7 or 16 at any position of the die, you will win. The chances of success with other point totals depend on how the die lies.
459. Reversing the Die. For this game, one die is needed. The first player calls out any number from 1 to 6, and the second player rolls the die. Then they take turns flipping the die in any direction, but no more than a quarter turn at a time. The number of points, \( \mathrm{K} \), called out by the first player, is added to the number of points that appear on the top face after rolling the die and each subsequent turn. The player who manages to reach a total of 25 points on their turn or forces the opponent to exceed 25 points on their next turn wins.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-162.jpg?height=197&width=260&top_left_y=1198&top_left_x=904)

Let me provide an example game. Player \(A\) calls out 6, and player \(B\), after rolling the die, gets 3 points (as shown in the image), making the total 9 points. Then \(A\) turns the die to show 1 point on top, making the total 10 points. Player \(B\) turns the die to show 3 points on top (total 13 points). Player \(A\) turns the die to show 6 points on top (total 19 points). Player \(B\) turns the die to show 3 points on top (total 22 points). Player \(A\) turns the die to show 1 point on top (total 23 points). Finally, player \(B\) turns the die to show 2 points on top, reaching a total of 25 points and winning the game.

What number should \(A\) call out to have the best chance of winning? Remember that the numbers on opposite faces of the die always add up to 7, i.e., they are paired as \(1-6, 2-5, 3-4\).