Can a game of cribbage be won 121-0?
competition/games/cribbage.puzzles New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
Can a game of cribbage be won 121-0? And other cribbage puzzles.
I only recently learned how to play Cribbage, and have thought of a few
neat puzzles. These all refer to standard two-player, six-card
cribbage (six cards dealt to each person; two thrown into crib; leaves
four in crib, four in each hand, to which one card is added (the
'turn-up' or 'cut' card)). Remember that fifteens and pairs count two,
a flush counts as the number of cards (at least four in hand, or all
five in the crib hand), and a run counts as number of cards (at least
three). For further info, consult a Hoyle's Book of Card Games. {And
yes, the highest five-card hand is 28: four fives and a ten-card
(neglecting the ten-card being a jack of the same suit as the turn-up
card).}
1. What four-card hand will not have its point value increased by the
cut card?
2. How many four-card hands are there that *will* be helped by any cut?
(You can ignore permutations contributed by suit differences.)
3. What is the highest valued four-card hand, and what is the point
total?
4. Is it possible for a person to win a game 121-0? If so, give an
example of a scenario.
5. What is the highest possible score for a person to earn on their
turn, not including pegging? (I don't as yet have an answer to this
one.)
6. If the entire deck of 52 cards were to be counted as one cribbage
hand, would the total points be around 1000, 100000, or more than a
million? (Hint: Don't bother with totals to fifteen.)
[ Editor: posted by Morris Maduro <Morris_maduro@biology.ualberta.ca>
with the answers, August 1995; the author of the puzzles is not named.
The title sentence is the editor's. ]
Show solution
> {And yes, the highest five-card hand is 28: four fives and a ten-card
> (neglecting the ten-card being a jack of the same suit as the turn-up
> card).}
You can get 29; J555, and the turn-up card is the 5 of the same suit as
the jack. The one point for his nobs counts as part of the hand, while
2 for his heels doesn't.
> 1. What four-card hand will not have its point value increased by the
> cut card?
Four aces; any other four-of-a-kind can make 15 with the starter.
> 2. How many four-card hands are there that *will* be helped by any
> cut?
I can't guarantee that this list is complete, but:
AA34, AA35, AA36, AA46, A226, A234, A235, A236, A237, A238, A244,
A245, A246, A247, A248, A256, A258, A259, A344, A345, A346, A347,
A356, A357, A358, A457, A458, 2236, 2237, 2337, 2344, 2345, 2346,
2347, 2348, 2349, 234T, 234J, 234Q, 234K, 2356, 2359, 2457, 2458,
2567, 3456, 3459, 4567, 4568.
> 3. What is the highest valued four-card hand, and what is the point
> total?
5555 scores 20.
> 4. Is it possible for a person to win a game 121-0? If so, give an
> example of a scenario.
No. Dealer must always score at least one GO or 31. The dealer of the
first hand can score at most two hands and some pegging before this
happens.
61-0 is possible. Non-dealer has KJT7. Dealer has 6655 in hand, 6654
in crib, and turns up a 4 as starter. K-5(peg 2)-7-6(peg 3+1). J-5(peg
2)-T-6(peg 2). Dealer now has 10, and 24 each for hand and crib makes
58.
Next round, as non-dealer, he begins by leading a 4, dealer plays 6, and
he continues with 5 to peg 5 for 63. There's some slack.
> 5. What is the highest possible score for a person to earn on their
> turn, not including pegging? (I don't as yet have an answer to this
> one.)
Do you mean "including pegging"? If so, I think the best is 6644 in
hand, J555 in crib, 7664 held by opponent, starter the 5 of the same
suit as the jack.
6-6(peg 2)-6(peg 6)-6(peg 12)-7(peg 2); 4-4(peg 2)-4(peg6+1).
Dealer has pegged 21, and scores 24 for his hand, 29 for his crib,
totaling 74.
> 6. If the entire deck of 52 cards were to be counted as one cribbage
> hand, would the total points be around 1000, 100000, or more than a
> million? (Hint: Don't bother with totals to fifteen.)
There are 4^13 possible 13-card runs, which are worth almost one
billion points.
-- David Grabiner <grabiner@math.harvard.edu>
[ Editor: from a January 1997 thread "cribbage puzzle". ]
Matthew Daly wrote:
> I know I should have saved it, but there were a pair of Panthera
> puzzle contests that illustrated that it is possible for one player
> to win the game in two deals. When the player was dealing, I think
> that it turned out to be 82-2 after the play, hand, and crib were
> all counted. I don't remember the details at all, except that it
> wasn't especially intuitive.
Cribbage folklore seems to suggest that the highest point score
possible for dealer is 75 with pone scoring 32. The highest spread of
points is 69 where dealer scores 74 and pone scores 5. See "How to win
at Cribbage" by Joseph Petrus Wergin (Winchester Press 1980). As you
can see the book is quite old but nevertheless I would be suspicious of
the 82-2 claim without seeing the details.
-- Martin Chlond <martin@chlond.demon.co.uk>