# Problem of the Month (March 2006)

This month we consider some problems involving playing cards.

Problem #1: In Cribbage, a collection of cards scores as follows:

2 points for each set of cards that totals 15 (face cards count 10, aces count 1)
2 points for each pair (this means 3-of-a-kind is worth 6 points and 4-of-a-kind is worth 12 points)
n points for each maximal straight containing n≥3 cards
n points for each maximal flush containing n≥4 cards
1 point for the jack of trump

Usually, Cribbage is played with 5 cards. It is well known that the highest scoring 5 card cribbage hand is J5555, worth 29 points. What is the highest scoring n card Cribbage hand, for other values of n?

Problem #2: In a Poker variant called Guts, each player is dealt cards, and then each player declares simultaneously whether he is in or out. Of all the players who are in, the best poker hand wins the pot and all the others match the pot.

If there are p players, and n cards dealt to each player, what is the worst hand that a player should stay in with?

Problem #3: My favorite magic trick involving cards is the following. Alice gives Bob any 5 cards without Chuck seeing any of them. Bob reveals 4 of those cards face up, one at a time. Chuck, who is working with Bob, names the fifth card without seeing it.

How do Bob and Chuck do this trick? Here's a more general question: If Alice gives Bob n cards, and Bob can always play k of them so that Chuck can predict the other cards, what is the maximum number of cards in the deck? If the cards can be labeled in any way, what is the "easiest" strategy for achieving this trick?

Problem #4: Pinochle is played with a 48 card Pinochle deck: the aces, kings, queens, jacks, tens and nines from a double deck. A hand scores as follows:

These points are multiplied by 10 if you have both sets:
15 points for a run (A K Q J T of trump)
10 points for aces around (ace of each suit)
8 points for kings around
6 points for queens around
4 points for jacks around
4 points for pinochle (Q of spades and J of diamonds)

These points are multiplied by 2 if you have both sets:
2 points for non-trump KQ
4 points for KQ of trump (not already part of a run)
1 point for 9 of trump

Unlike Cribbage, a card in Pinochle can only be used once within any given category. Usually, Pinochle is played with 12 cards. I think the maximal score with 12 cards is 190, for a double run in spades and two jacks of diamonds. What is the highest scoring n card Pinochle hand, for various values of n?

Problem #1:

Gavin Theobald found the highest scoring hands for n≤20, including one improvement.

Jeremy Galvagni found the highest scoring hands for n≤6.

Joe DeVincentis found the highest scoring hands for n≤7, but proved his assertions for n≤4.

Philippe Fondanaiche found the highest scoring hands for n=6 and n=7.

Highest Known Scoring
Cribbage Hand with n Cards
CardsPointsHand
11J
23JJ   or   5J
38555
4205555
5295555J
646445566
7724455566
811233334455
9188AA2233444
10332AA22334455
11576AAA22233344   or   AAA22233444   or   AAA22333444   or   AA222333444 (GT)
121,024AAA222333444
131,836(A2)33344
143,206(A23)44
155,086(A23)444
167,440(A234)
179,601(A234)5
1812,980(A234)55
1916,555(A234)555
2020,328(A2345)
.........
52872,449,969(A23456789TJQK)

Problem #2:

Joe DeVincentis sent a well thought out analysis. You should stay in when your expected winning is larger staying in than going out. Ignoring that the same cards can not be dealt twice, he found out that you should stay in with probability .5 with 2 players, .293 with 3 players, and .207 with 4 players.

Worst Approximate Hand to Stay
in Guts With p Players and n Cards
n \ p234
18JQ
2J7K5 (JD)KQ (JD)
3K63A85 (TG)AQT (JD)
4A875
5AKQJ7
666A74

Trevor Green did not ignore that cards can not be dealt twice, and found that the critical hands changed slightly. His complete analysis for n=5 and p=2 convinced me that this was a VERY HARD problem.

Worst Exact Hand to Stay
in Guts With p Players and n Cards
n \ p234
18JQ
2J8 (TG)K4 (TG)A2 (TG)
3K42 (TG)A82 (TG)
4A532 (TG)
5AKQ98* (TG)
* with AKQ98, call if 4-suited with A-9, A-8 or K-8 matching,
and call 4/5 of the time if 4-suited with K-9 or Q-8 matching.

Then in 2013, Trevor Green gave an even more detailed analysis of the 2-player game. His final results were:

Worst Exact Hand to Stay
in Guts With 2 Players and n Cards
nhand
1T (TG)
2K2 (TG)
3A32 (TG)
4AKJ4 (TG)
577982 or 77A32 (TG)

Problem #3:

Joe DeVincentis solved the problem with n=5 and k=4. He also found the upper bounds below.

The method for 124 cards for n=5 and k=4 is basically this: The cards are numbered 0 to 123, given 5 cards c0 < c1 < c2 < c3 < c4, the assistant hides ci where i = c0 + c1 + c2 + c3 + c4 (mod 5). Let the sum of the displayed cards be s (mod 5). If you renumber the 120 non-displayed cards consecutively 0 to 119, then the hidden card must be -s (mod 5), limiting it to precisely 24 of the non-displayed cards. The order of the four cards then determines which of the 24 it is. Philippe Fondanaiche also mentioned that it is possible to predict the last card when the deck has size d = n! + n – 1 cards.

Trevor Green found explicit strategies for k=2 and 3≤n≤5, and the lower bound d≥n+k.

Here are (rotationally symmetric) strategies for k=2.

n=3: 123: play 32 / 124: play 42 / 125: play 52 / 126: play 62 / 134: play 34 / 135: play 35 / 136: play 36

n=4: 1234: play 12 / 1235: play 53 / 1236: play 63 / 1245: play 54 / 1246: play 46

n=5: 45678: play 54 / 35678: play 53 / 34678: play 63 / 34578: play 73 /
25678: play 56 / 24678: play 46 / 24578: play 47

Bounds on Largest Deck Size
n \ k123456
23
348
45727
5688-14124
6789-1310-31725
78910-1311-2212-765046

Problem #4:

Highest Known Scoring
Pinochle Hand with n Cards
CardsPointsHand
119
24Q J   or   KQ
38KQ J (JD)
440QQ JJ
544KQQ JJ
648KKQQ JJ
749KKQQ9 JJ
8100AA AA AA AA
9101AA9 AA AA AA
10150AAKKQQJJTT
11154AAKKQQJJTT J   or   AAKKQQJJTT Q
12190AAKKQQJJTT JJ   or   AAKKQQJJTT QQ
13192AAKKQQJJTT KQQ
14196AAKKQQJJTT QQ Q Q (JD)
15200AAKKQQJJTT AJJ A A   or   AAKKQQJJTT AQQ A A
16250AAKKQQJJTT AA AA AA   or   AAKKQQJJTT QQ QQ QQ
17254AAKKQQJJTT AAJ AA AA   or   AAKKQQJJTT AAQ AA AA
18290AAKKQQJJTT AAJJ AA AA   or   AAKKQQJJTT AAQQ AA AA
19292AAKKQQJJTT AAKQQ AA AA
20296AAKKQQJJTT AAQQ AAQ AAQ
21298AAKKQQJJTT AAQQ AAKQ AAQ (JD)
22350AAKKQQJJTT AAQQ AAQQ AAQQ (JD)
23352AAKKQQJJTT AAKQQ AAQQ AAQQ (GT)
24374AAKKQQJJTT AAKKQQ AAKK AAKK (GT)
25376AAKKQQJJTT AAKKQQ AAKKQ AAKK (GT)
26384AAKKQQJJTT AAKKQQ AAKKQ AAKKQ (GT)
27386AAKKQQJJTT AAKKQQ AAKKQQ AAKKQ (GT)
28442AAKKQQJJTT AAKKQQ AAKKQQ AAKKQQ (GT)
29443AAKKQQJJTT9 AAKKQQ AAKKQQ AAKKQQ (GT)
30444AAKKQQJJTT99 AAKKQQ AAKKQQ AAKKQQ (GT)
31446AAKKQQJJTT AAKKQQJ AAKKQQJ AAKKQQJ (GT)
32447AAKKQQJJTT9 AAKKQQJ AAKKQQJ AAKKQQJ (GT)
33448AAKKQQJJTT99 AAKKQQJ AAKKQQJ AAKKQQJ (GT)
34482AAKKQQJJTT AAKKQQJJ AAKKQQJJ AAKKQQJJ (GT)
35483AAKKQQJJTT9 AAKKQQJJ AAKKQQJJ AAKKQQJJ (GT)
36-48484AAKKQQJJTT99 AAKKQQJJ AAKKQQJJ AAKKQQJJ (GT)
(bold are spades, italics are diamonds)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 7/27/13.