# »  Solutions to puzzles in my National Review Online Diary

## May 2009

—————————

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.

—————————

[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

—————————

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:
— Ken marks the three dice like this —
A         1         7         10         12         13         14
B         2         3         4         15         16         17
C         5         6         8         9         11         18
— Dice chosen — — Result of 10,000 wars —
Bob Ken Ken wins Bob wins Tie
A         B         9468         353         179
C         362         9460         178
B         A         347         9454         199
C         9443         354         203
C         A         9434         376         190
B         341         9452         207

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.)
— Ken marks the three dice like this —
A         1         10         11         12         13         14
B         2         3         4         15         16         17
C         5         6         7         8         9         18
— Dice chosen — — Result of 10,000 wars —
Bob Ken Ken wins Bob wins Tie
A         B         9434         388         178
C         0         10000         0
B         A         371         9436         193
C         9411         383         206
C         A         10000         0         0
B         407         9390         203

[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.

—————————

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

— Ken marks the three dice like this —
A         1         6         8         12         13         17
B         2         4         9         11         15         16
C         3         5         7         10         14         18
— Dice chosen — — Result of 10,000 wars —
Bob Ken Ken wins Bob wins Tie
A         B         4596         4584         820
C         4638         4581         781
B         A         4600         4621         779
C         4573         4644         783
C         A         4559         4642         799
B         4519         4685         796

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

— Ken marks the three dice like this —
A         1         2         9         14         15         16
B         6         7         8         11         12         13
C         3         4         5         10         17         18
— Dice chosen — — Result of 10,000 wars —
Bob Ken Ken wins Bob wins Tie
A         B         419         9354         227
C         9473         334         193
B         A         9454         364         182
C         372         9441         187
C         A         344         9492         164
B         9431         373         196

[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!

—————————

— Ken marks the three dice like this —
A         1         2         9         10         17         18
B         5         6         7         8         15         16
C         3         4         11         12         13         14
— Dice chosen — — Result of 10,000 wars —
Bob Ken Ken wins Bob wins Tie
A         B         1115         8501         384
C         8450         1094         456
B         A         8488         1087         425
C         1123         8420         457
C         A         1170         8397         433
B         8485         1101         414

[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.

—————————

— Ken marks the three dice like this —
A         2         3         7         12         15         18
B         1         5         9         11         14         17
C         4         6         8         10         13         16
— Dice chosen — — Result of 10,000 wars —
Bob Ken Ken wins Bob wins Tie
A         B         2597         6748         655
C         6799         2502         699
B         A         6787         2542         671
C         2529         6766         705
C         A         2573         6795         632
B         6792         2526         682

[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.

—————————

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

— Ken marks the three dice like this —
A         1         2         11         12         15         16
B         3         4         7         8         17         18
C         5         6         9         10         13         14
— Dice chosen — — Result of 10,000 wars —
Bob Ken Ken wins Bob wins Tie
A         B         8484         1111         405
C         1186         8378         436
B         A         1104         8449         447
C         8439         1144         417
C         A         8489         1086         425
B         1140         8462         398

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

—————————

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:

— Ken marks the three dice like this —
A         1         6         7         12         13         18
B         2         5         8         11         14         17
C         3         4         9         10         15         16
— Dice chosen — — Result of 10,000 wars —
Bob Ken Ken wins Bob wins Tie
A         B         4581         4630         789
C         4639         4585         776
B         A         4535         4656         809
C         4607         4647         746
C         A         4649         4579         772
B         4616         4548         836

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

—————————

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

— Ken marks the three dice like this —
A         1         5         9         10         14         18
B         2         4         7         12         15         17
C         3         6         8         11         13         16
— Dice chosen — — Result of 10,000 wars —
Bob Ken Ken wins Bob wins Tie
A         B         4649         4571         780
C         4559         4646         795
B         A         4603         4596         801
C         4536         4618         846
C         A         4610         4611         779
B         4544         4658         798

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

—————————

•  Many thanks to all who contributed!