Henry Ernest Dudeney/Modern Puzzles/204 - Turning the Die/Solution

by : $204$

 * Turning the Die
 * This is played with a single die.
 * The first player calls any number he chooses, from $1$ to $6$, and the second player throws the die at hazard.
 * Then they take it in turns to roll over the die in any direction they choose, but never giving it more than a quarter turn.
 * The score increases as they proceed, and the player wins who manages to score $25$ or forces his opponent to score beyond $25$.


 * I will give an example game.
 * Player $A$ calls $6$, and $B$ happens to throw $3$, making the score $9$.
 * Now $A$ decides to turn up $1$, scoring $10$;
 * $B$ turns up $3$, scoring $13$;
 * $A$ turns up $6$, scoring $19$;
 * $B$ turns up $3$, scoring $22$;
 * $A$ turns up $1$, scoring $23$;
 * and $B$ turns up $2$, scoring $25$ and winning.


 * What call should $A$ make in order to have the best chance at winning?
 * Remember that the numbers on opposite sides of a correct die always sum to $7$, that is, $1 - 6$, $2 - 5$, $3 - 4$.

Solution
The best call is either $2$ or $3$.

In either case, only one specific throw will defeat him.

If he calls $1$ then either $3$ or $6$ defeats him.

If he calls $2$, then only $5$ defeats him.

If he calls $3$, then only $4$ defeats him.

If he calls $4$, then $3$ or $4$ defeats him.

If he calls $5$, then $2$ or $3$ defeats him.

If he calls $6$, then $1$ or $5$ defeats him.

Proof
The basic idea is that if at any time you score $5$, $6$, $9$, $10$, $14$, $15$, $18$, $19$ or $23$ with the die any side up, you should lose.

If you score $7$ or $16$ with any side up, you win.

The chances of winning with any other score depends on how the die lies.