»  Solutions to puzzles in my National Review Online Diary

  May 2009

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In my May diary I posed the following brain-teaser:

Ken and Bob find themselves in possession of three blank-sided dice. These are ordinary cubic dice, with six faces each.

Ken writes the numbers from 1 to 18 on the sides. No number is repeated. Each side of each of the three dice now shows a number from 1 to 18.

Bob then chooses one of the three dice. Ken chooses one of the other two. The third die is discarded.

The two men then play a game of dice war. The war consists of a hundred "rounds." In each round, first Ken rolls his die, then Bob rolls his. The man with the highest number showing on the topmost face of his die, wins the round.

Whichever man wins the larger number of rounds, wins the war.

Question:   If both men followed the strategy that gave them the best mathematical chance to win this war, what would the numbers on the dice look like?

This got a huge reader response. I am still getting emails (this written on June 27th).

About halfway through the month, with the emails in full flood, I posted the results below. Though I've had more results, they have all fallen into one of the two groups represented in that mid-month round-up:

• Readers arguing that no advantage is possible for either player, since one is choosing the numbers and the other the dice. The best either player can hope for is a 50-50 probability of winning the war.
• Readers arguing that by carefully numbering the dice, Ken can set the subsequent game up as a rock-paper-scissors sort of affair. Bob's choosing the first die will then be like a RPS player making his move. The second player, Ken, then has to riposte with whatever "kills" that first move —  you know:  Rock kills scissors, scissors kill paper, paper kills rock. So Ken just needs to number up the faces of the three dice so that they "kill" each other — statistically speaking, of course — in that same circular fashion: Die A "kills" die B, die B "kills" die C, die C "kills" die A.  There is a way for him to do this.

Indeed there is, as the solutions below demonstrate. However, shortly after I posted that interim discussion, a friend pointed out to me that this problem appeared in John Tierney's New York Times column on March 30. I had not known that. (Here is the downside of being a non-Times-reading conservative.) Tierney gave a full solution here.

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[Mid-month posting]:

Here's a selection from reader solutions. I ran each one through a Visual Basic simulation, fighting ten thousand "wars" (i.e. of a hundred rounds each) with each configuration of dice. Since a war can be tied, each player winning fifty rounds, it's not all win-loss. Ken's victories are in red.

 

Solutions

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Reader Michael T. used that straightforward enumerative approach:

Nice litle problem, first to ponder, then to simulate. Obviously, Ken's in complete control; Bob's only strategy is to minimize his disadvantage.

Given 36 possible results for each roll of a pair of dice, several dice arrangements result in a 21-to-15 advantage for whoever chooses the second die (Ken), regardless of which die is chosen first (Bob). One such example (of eight total, I believe) is as follows, with A having a 21-15 advantage over C, B having a 21-15 advantage over A, and C having a 21-15 advantage over B:

Whichever die Bob chooses, Ken can then make his choice in such a way as to give himself a massive — around 26 to 1 — statistical advantage. Michael T. further notes that:

Here's one dice arrangement that really surprised me. Without giving up his guaranteed advantage of 21-15, Ken can capitalize on Bob making the wrong choice with the following dice, giving himself a 25-11 advantage if Bob chooses C. (Although Bob, smart guy that he is, wouldn't choose quite so poorly.)

[Comment]:  Look at that! If Bob chooses C, Ken can totally shut him out by choosing A. (And vice versa.) Of course, Ken will lose a few rounds with that 1, and Bob will win a few with that 18, but absent totally sensational luck, Bob won't win any wars.

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Reader Guy came up with some good stuff on pure intuition. His first bite of the cherry:

[Comment]:  Hmm. But he bounced back with:

[Comment]:  Essentially the same stats Michael T. got. Then Guy went to the simulation, but started mumbling in UNIX and lost me. Doesn't reader-world know that I hate UNIX? For the record: I HATE UNIX!

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Reader Doug made a good fist of it:

[Comment]:  If Ken marked up the dice by Doug's method, he has a strong winning strategy whatever die Bob picks … just not as good as Michael's and Guy's.

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From reader Jonathon [sic]:

[Comment]:  Again, Ken has a good strategy whichever die Bob picks, though his strategy isn't as good as it would be if he followed Doug's mark-ups, let alone Michael's and Guy's.

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That same reader Jonathon [sic] came back later with another line of attack:

[Comment]:   … which puts him pretty much in Doug's territory.

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Those were the real winners on the simulation test. Many other readers got points for effort, but no cigar. Two different — and, so far as I can tell, unrelated — readers, Rich and Dave, both came up with this arrangement:

[Comment]:  Alas, didn't pass the simulation test. Neither did the next one.

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From reader's neighbor's son Matthew, 15 years old:

[Comment]:  Still a creditable try for a highschooler.

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•  Many thanks to all who contributed!