Sixes And Sevens I

Sixes And Sevens II

Good man! Take a drop of this aqua vitae and come join me at my table.

I have a fancy for a wager much enjoyed in the famed gaming houses of Atlantis, although I fear that, owing to some small misadventure, it shall be many years before I frequent those glorious halls once again.

I was attending the lavish ball that marks the end of the Atlantean summer festival; a month long celebration of their supremacy in all matters cultural and scientific during which the inhabitants attend parties night and day and nobody attends to their work, or at least nobody of any import.

I had retired to the roof of the senate house to observe the ceremonial lighting of the towering bonfire that had been constructed in its courtyard and, as the orchestra reached a crescendo and the assembled multitude began hurling their torches into its body, I perceived a distinct lurch of the structure and deduced that it would, at any moment, collapse on to the building.

Thinking only of the safety of the assembled nobility in the great hall beneath me I turned the ceremonial cannon on the monumental dyke that surrounds the island. After several vollies I succeeded in breaching it and the consequent deluge brought down the great mass of flaming timber, extinguishing it in an instant.

Unfortunately not another soul had recognised the impending peril and, given that the entire island was soon submerged by two feet of seawater, my actions were universally interpreted as malicious. Having through very great cunning escaped the island unmolested I felt that it would be unwise to return for the foreseeable future.

But I am keeping us from our sport!

Here, I have gathered together a single suit of cards, shuffled them some several times, and shall deal out six for you and seven for myself. For each of my cards I shall have from your purse two coins for each point of its face value less one coin, with aces counting for one, deuces for two and so on and so forth and picture cards counting for ten. Contrariwise, for each of your cards you shall have twice its value plus one coin from mine.

By way of an example, if I were to deal myself a hand of

leaving you a hand of

then I should have for my purse

\[
(2 \times 1 - 1) + (2 \times 6 - 1) + (2 \times Q(10) - 1) + (2 \times 2 - 1) + (2 \times 7 - 1) + (2 \times 4 - 1) + (2 \times 5 - 1)\\
= 1 + 11 + 19 + 3 + 13 + 7 + 9\\
= 63
\]

of your coins, and you shall have
\[
(2 \times 8 + 1) + (2 \times 9 + 1) + (2 \times J(10) + 1) + (2 \times K(10) + 1) + (2 \times 3 + 1) + (2 \times 10 + 1)\\
= 17 + 19 + 21 + 21 + 7 + 21\\
= 106
\]

of mine.If this seems too rich for your blood I have an alternative; if the sum of my hand exceeds that of yours I shall claim three of your coins, otherwise you may have seven of mine.

When I described these wagers to that boorish student of my acquaintance he ignored the subject entirely and started bemoaning that despite innumerable potential concubines his expectation was one of singlehood, as if I should be surprised that a milksop such as he should find no favour with the fairer sex!

But enough of him! Come take another drop and consider your chances!

October 13, 2013 10:52 AM

Hello,

Here's my analysis of the first game (as short as I can make it so as not to bore the Baron).

The expected number of points on card is E(c) = (1 + 2 + ... + 10 + 30)/13 = 85/13

The Baron can expect to receive 2*E(c) - 1 per coins per card

Sir R can can expect to receive 2*E(c) + 1 per coins per card

Thus the Baron can expect to receive B = 14*E(c) - 7 per game

and Sir R can can expect to receive R = 12*E(c) + 6 per game

B - R = 2*E(c) - 13 = 1/13, and so a win for the Baron in the long run.

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The second game's not so neat, unless I've missed a trick.

If sk is the sum of the points on k cards then s6 + s7 = s13 = 85.

So s6 is less than s7 iff s6 is less than s13/2 iff s6 s less than or equal to 42.

But that's as far as I'm going, as after playing about with it for a bit I cannot see a neat way to a solution other than to slog it out, and I'm far too lazy to do that! I look forward to the student's solution.

Regards, Louis.

October 15, 2013 7:22 PM

You have struck the nail squarely upon the head!

Your analysis of the first game is equivalent to my own, albeit from a slightly different perspective.

My instincts regarding the second are also the same as yours and my analysis is consequently focused upon finding a simple scheme for enumerating the hands.

If anyone discovers a more satisfying solution I should be extremely interested to see it!