Henry Ernest Dudeney/Modern Puzzles/204 - Turning the Die/Solution
Jump to navigation
Jump to search
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.
This theorem requires a proof. In particular: Analysis of the above to be done. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $204$. -- Turning the Die
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $473$. Turning the Die