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

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Modern Puzzles by Henry Ernest Dudeney: $204$

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.


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Sources

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