In cribbage, which four-card hand can gain the most points from the turned-up card?

competition/games/cribbage.improvement New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

In cribbage, which four-card hand can gain the most points from the
turned-up card?

In cribbage you are dealt 6 cards, you discard 2, and then the deck is
cut and a card is turned up which you will use in counting your hand.
The question is: what set of 4 cards can most benefit by the turned up
card?  Informally, which hand has the most "improvement potential"?

-- Heather Booth <booth@cs.utk.edu>

Show solution

I've found a 20-point improvement hand . . .

Start with 5 5 6 6 (suits irrelevant):  2 pair --> 4 points

Turn up a 4 and get 4 5 5 6 6 :  2 pair          -->  4 points
                                 4 3-card runs   --> 12 points
                                 4 sets of 15's  -->  8 points
                                     TOTAL           24 points
Can anyone beat that?

-- Brett Hughes <hughes@studsys.mscs.mu.edu>

This is my first guess, but I'd be very surprised if it isn't the best.

hand: two 4s and two 6s == 4 points

starter card: 5  increases the value to 24

Actually, any combination of two pair of 4, 5, 6, and the starter being
the one you don't have will give the same results.

The maximum cribbage hand is 29 (J5555 with nobs) and a 28 point hand
being the same without nobs.  Any 4-card subset of those hands is more
than 9 points, so this will not beat or match my answer.

I don't think 27-, 26-, and 25-point hands are possible, so to do
better, you'd have to find a 24-point hand that has a 4-card subset
with three or fewer points.  I can't think of any other combination
besides a quadruple run of 4, 5, 6 that produces 24 points.  Having the
starter be anything other than the singleton (for example, having 4566
and drawing a 4) will always give you 7 points in the hand.

Twenty-three can be gotten with JJ555 with nobs and 22 is the same hand
without nobs, but no 4-card subset of that have less than four points,
so that won't do it, either.

I don't think a 21-point hand is possible.  If I haven't forgotten a
21+ combination, the best one can do is match my answer by finding a
zero->twenty change.

[ Editor: 21 is possible: 4 5 6 6 6 scores three runs of 4-5-6 (9),
three pairs of sixes (6) and three fifteens 4+5+6 (6).  Its 4-card
subsets score at least 6 (4666 and 5666 score 6), so the conclusion is not affected. ]

-- Dan Tilque <dant@techbook.com>