Crooked Swiss

Once a month I run a Swiss team game in our ward, which is quite a challenge. With all the subterfuge and unethical behavior, I get many director calls, for which I seldom need a rule book but only a formidable presence to inhibit fist fights. On the first round of our February game, a bellowing “DI—REC—TOR” call came from Herman’s table.

Board 25	4		West	North	East	South
E-W vul	A K Q 7		Clyde	Sharko	Rocco	Herman
	A 10 8 6 3			1	1 !	1 NT
	A J 9		2	3	Dbl!	3 NT
A 10 8 7 6 5		Q 2	Pass	Pass	Pass
9 8 2		6 5 3
5 4 2		Q J 9
4		K Q 10 8 3
	K J 9 3
	J 10 4
3 NT South	K 7
West leads	7 6 5 2

I rushed to the table to find Herman was livid.

“You won’t believe these animals!” exclaimed Herman. “Rocco overcalls a spade on a doubleton vulnerable, and then doubles my partner’s cue-bid for the lead. Not only does Clyde make a wimpy raise to just two spades, but against three notrump he leads his singleton club, which was the only lead to set me! They have to be cheating, and I demand a score adjustment.”

“I forgot the contract,” Clyde explained, “and was trying for a ruff.”

“Of course!” confirmed Rocco. “My partner is very forgetful at times.”

“Yeah, sure,” countered Sharko, “and I say Clyde is a lying piece of…”

“Whoa!” I broke in quickly. “Let’s all calm down. Everyone knows that Rocco is a chronic psycher, so his bids and doubles don’t mean much, and Clyde may be telling the truth. Therefore, I must let the result stand.”

“Thank you,” acknowledged Rocco, “and while you’re here, the real cheater at this table is Sharko! He shuffled and dealt two boards, 22 and 25, and ends up with a very strong hand on both. Coincidence? I don’t think so. He is stacking the deck!”

Deja vu, I thought to myself, as I’ve heard this complaint before. “There is no proof of your allegation against Sharko, but you may be on to something. I’ll think about it.” When the round finished, I made an announcement over the loudspeaker, “Find your places please for Round 2! And if you play against Sharko, don’t let him make any boards!”

After publishing this story, I received a stern warning from the FCC (Federal Communications Commission) about fair journalism. Because my players are borderline crazy, I must offer equal time to someone with ‘borderline’ removed. So be it:

Tina Denlee: When I sat South, the bidding went 2 (strong) Dbl (clubs) 2 (relay) 2 ! … Then on a takeout double, I just could not sit and bid the ill-fated 3NT… But wait! West leads East’s suit; the 10 wins then K ducked (count rectification) and 3 won in dummy, West pitching spades. Next the 4 queen-king-ace, and West returns a diamond won in dummy (key play). Then four rounds of hearts (pitching the club) pseudo-squeeze West, who erroneously discards one of his ‘idle’ diamonds. Now  K, 3! West is endplayed and concedes the contract. What a save!

Puzzle time! Suppose Sharko stacks the deck to deal himself three aces and a king, and the remaining 48 cards are distributed at random. For the stacked deals, test your analytic ability by answering two questions:

1. What would be Sharko’s average (mean) number of HCP?
  18.67 18.75 19.12 19.33 19.69 20.25

2. What would be Sharko’s most common (mode) number of HCP?
  16 17 18 19 20 21

Now forgot about that and consider this: In a seven-board Swiss match, Sharko had all plus scores, and so did his teammates except for one board, yet they lost the match. There were no doubled or redoubled contracts.

3. What was the board number of their minus score?
  4 7 10 11 13 20 23 26

Quit Top Crooked Swiss

Cyrus Hettle Three-peats

*Solvers had to calculate their answers. Multiple choice was only added for this writeup.

This puzzle was more mathematical than bridge related, but anything providing a mental challenge is fair game around here — notwithstanding the old story that a bridge player makes his contract on a finesse, while the mathematician goes down on 50.89-percent squeeze.

The perfect scorers are ranked below by date and time of entry.

Winner List

Rank	Name	Location	Q1	Q2	Q3
1	Cyrus Hettle	Kentucky	19.69	18	20
2	Jim Munday	New Mexico	19.69	18	20
3	Tina Denlee	Quebec	19.69	18	20
4	Andrew Spooner	Australia	19.69	18	20
5	Charles Blair	Illinois	19.69	18	20
6	Dan Baker	Texas	19.69	18	29
7	Richard Stein	Washington	19.69	18	20

Puzzle 8Q59 Main

Top Crooked Swiss

Solutions

1. Average (mean) HCP

Sharko stacks the deck to deal himself three aces and a king (15 HCP) while the remaining 48 cards are distributed at random. What would be Sharko’s average (mean) number of HCP?

An intuitive way to solve this is to consider that 25 HCP must be distributed among 48 slots, 13 of which belong to each other player, and 9 belong to Sharko. Hence, Sharko should average 9/48 of the 25 HCP, or 4.6875 HCP. Adding this to the 15 assured HCP makes 19.6875, or 19.69 rounded. But of course! Nineteen sixty-nine was the year of the moon landing, so it’s all beginning to make sense.

I was skeptical that 19.6875 is the exact answer because of granularity issues (fitting specific high cards into 9 vacant slots versus 13) so I did a complete analysis of every possible high-card arrangement. Stand back!

HCP	Extra HCP	Ways	Comb	Hands	Hands × HCP	Hands w/HCP
15	None	1	36c9	94,143,280	1,412,149,200	94,143,280
16	1J	4	36c8	121,041,360	1,936,661,760	121,041,360
17	1Q	4	36c8	121,041,360	2,057,703,120	171,127,440
	2J	6	36c7	50,086,080	851,463,360
18	1K	3	36c8	90,781,020	1,634,058,360	232,135,068
	1Q 1J	16	36c7	133,562,880	2,404,131,840
	3J	4	36c6	7,791,168	140,241,024
19	1A	1	36c8	30,260,340	574,946,460	227,642,580
	2Q	6	36c7	50,086,080	951,635,520
	1K 1J	12	36c7	100,172,160	1,903,271,040
	1Q 2J	24	36c6	46,747,008	888,193,152
	4J	1	36c5	376,992	7,162,848
20	1K 1Q	12	36c7	100,172,160	2,003,443,200	221,402,016
	1A 1J	4	36c7	33,390,720	667,814,400
	2Q 1J	24	36c6	46,747,008	934,940,160
	1K 2J	18	36c6	35,060,256	701,205,120
	1Q 3J	16	36c5	6,031,872	120,637,440
21	2K	3	36c7	25,043,040	525,903,840	189,736,932
	1A 1Q	4	36c7	33,390,720	701,205,120
	3Q	4	36c6	7,791,168	163,614,528
	1K 1Q 1J	48	36c6	93,494,016	1,963,374,336
	1A 2J	6	36c6	11,686,752	245,421,792
	2Q 2J	36	36c5	13,571,712	285,005,952
	1K 3J	12	36c5	4,523,904	95,001,984
	1Q 4J	4	36c4	235,620	4,948,020
22	1A 1K	3	36c7	25,043,040	550,946,880	150,915,171
	1K 2Q	18	36c6	35,060,256	771,325,632
	2K 1J	12	36c6	23,373,504	514,217,088
	1A 1Q 1J	16	36c6	31,164,672	685,622,784
	3Q 1J	16	36c5	6,031,872	132,701,184
	1K 1Q 2J	72	36c5	27,143,424	597,155,328
	1A 3J	4	36c5	1,507,968	33,175,296
	2Q 3J	24	36c4	1,413,720	31,101,840
	1K 4J	3	36c4	176,715	3,887,730
23	2K 1Q	12	36c6	23,373,504	537,590,592	106,130,745
	1A 2Q	6	36c6	11,686,752	268,795,296
	1A 1K 1J	12	36c6	23,373,504	537,590,592
	4Q	1	36c5	376,992	8,670,816
	1K 2Q 1J	72	36c5	27,143,424	624,298,752
	2K 2J	18	36c5	6,785,856	156,074,688
	1A 1Q 2J	24	36c5	9,047,808	208,099,584
	3Q 2J	24	36c4	1,413,720	32,515,560
	1K 1Q 3J	48	36c4	2,827,440	65,031,120
	1A 4J	1	36c4	58,905	1,354,815
	2Q 4J	6	36c3	42,840	985,320
24	3K	1	36c6	1,947,792	46,747,008	72,221,100
	1A 1K 1Q	12	36c6	23,373,504	560,964,096
	1K 3Q	12	36c5	4,523,904	108,573,696
	2K 1Q 1J	48	36c5	18,095,616	434,294,784
	1A 2Q 1J	24	36c5	9,047,808	217,147,392
	1A 1K 2J	18	36c5	6,785,856	162,860,544
	4Q 1J	4	36c4	235,620	5,654,880
	1K 2Q 2J	108	36c4	6,361,740	152,681,760
	2K 3J	12	36c4	706,860	16,964,640
	1A 1Q 3J	16	36c4	942,480	22,619,520
	3Q 3J	16	36c3	114,240	2,741,760
	1K 1Q 4J	12	36c3	85,680	2,056,320
25	1A 2K	3	36c6	5,843,376	146,084,400	44,246,244
	2K 2Q	18	36c5	6,785,856	169,646,400
	3K 1J	4	36c5	1,507,968	37,699,200
	1A 3Q	4	36c5	1,507,968	37,699,200
	1A 1K 1Q 1J	48	36c5	18,095,616	452,390,400
	1K 3Q 1J	48	36c4	2,827,440	70,686,000
	2K 1Q 2J	72	36c4	4,241,160	106,029,000
	1A 2Q 2J	36	36c4	2,120,580	53,014,500
	1A 1K 3J	12	36c4	706,860	17,671,500
	4Q 2J	6	36c3	42,840	1,071,000
	1K 2Q 3J	72	36c3	514,080	12,852,000
	2K 4J	3	36c3	21,420	535,500
	1A 1Q 4J	4	36c3	28,560	714,000
	3Q 4J	4	36c2	2520	63,000
26	3K 1Q	4	36c5	1,507,968	39,207,168	23,836,113
	1A 1K 2Q	18	36c5	6,785,856	176,432,256
	1A 2K 1J	12	36c5	4,523,904	117,621,504
	1K 4Q	3	36c4	176,715	4,594,590
	2K 2Q 1J	72	36c4	4,241,160	110,270,160
	3K 2J	6	36c4	353,430	9,189,180
	1A 3Q 1J	16	36c4	942,480	24,504,480
	1A 1K 1Q 2J	72	36c4	4,241,160	110,270,160
	1K 3Q 2J	72	36c3	514,080	13,366,080
	2K 1Q 3J	48	36c3	342,720	8,910,720
	1A 2Q 3J	24	36c3	171,360	4,455,360
	1A 1K 4J	3	36c3	21,420	556,920
	4Q 3J	4	36c2	2520	65,520
	1K 2Q 4J	18	36c2	11,340	294,840
27	1A 2K 1Q	12	36c5	4,523,904	122,145,408	12,974,655
	2K 3Q	12	36c4	706,860	19,085,220
	3K 1Q 1J	16	36c4	942,480	25,446,960
	1A 4Q	1	36c4	58,905	1,590,435
	1A 1K 2Q 1J	72	36c4	4,241,160	114,511,320
	1A 2K 2J	18	36c4	1,060,290	28,627,830
	1K 4Q 1J	12	36c3	85,680	2,313,360
	2K 2Q 2J	108	36c3	771,120	20,820,240
	3K 3J	4	36c3	28,560	771,120
	1A 3Q 2J	24	36c3	171,360	4,626,720
	1A 1K 1Q 3J	48	36c3	342,720	9,253,440
	1K 3Q 3J	48	36c2	30,240	816,480
	2K 1Q 4J	12	36c2	7560	204,120
	1A 2Q 4J	6	36c2	3780	102,060
	4Q 4J	1	36c1	36	972
28	1A 3K	1	36c5	376,992	10,555,776	5,739,564
	3K 2Q	6	36c4	353,430	9,896,040
	1A 1K 3Q	12	36c4	706,860	19,792,080
	1A 2K 1Q 1J	48	36c4	2,827,440	79,168,320
	2K 3Q 1J	48	36c3	342,720	9,596,160
	3K 1Q 2J	24	36c3	171,360	4,798,080
	1A 4Q 1J	4	36c3	28,560	799,680
	1A 1K 2Q 2J	108	36c3	771,120	21,591,360
	1A 2K 3J	12	36c3	85,680	2,399,040
	1K 4Q 2J	18	36c2	11,340	317,520
	2K 2Q 3J	72	36c2	45,360	1,270,080
	3K 4J	1	36c2	630	17,640
	1A 3Q 3J	16	36c2	10,080	282,240
	1A 1K 1Q 4J	12	36c2	7560	211,680
	1K 3Q 4J	12	36c1	432	12,096
29	1A 2K 2Q	18	36c4	1,060,290	30,748,410	2,453,184
	1A 3K 1J	4	36c4	235,620	6,832,980
	2K 4Q	3	36c3	21,420	621,180
	3K 2Q 1J	24	36c3	171,360	4,969,440
	1A 1K 3Q 1J	48	36c3	342,720	9,938,880
	1A 2K 1Q 2J	72	36c3	514,080	14,908,320
	2K 3Q 2J	72	36c2	45,360	1,315,440
	3K 1Q 3J	16	36c2	10,080	292,320
	1A 4Q 2J	6	36c2	3780	109,620
	1A 1K 2Q 3J	72	36c2	45,360	1,315,440
	1A 2K 4J	3	36c2	1890	54,810
	1K 4Q 3J	12	36c1	432	12,528
	2K 2Q 4J	18	36c1	648	18,792
	1A 3Q 4J	4	36c1	144	4176
30	1A 3K 1Q	4	36c4	235,620	7,068,600	951,027
	3K 3Q	4	36c3	28,560	856,800
	1A 1K 4Q	3	36c3	21,420	642,600
	1A 2K 2Q 1J	72	36c3	514,080	15,422,400
	1A 3K 2J	6	36c3	42,840	1,285,200
	2K 4Q 1J	12	36c2	7560	226,800
	3K 2Q 2J	36	36c2	22,680	680,400
	1A 1K 3Q 2J	72	36c2	45,360	1,360,800
	1A 2K 1Q 3J	48	36c2	30,240	907,200
	2K 3Q 3J	48	36c1	1728	51,840
	3K 1Q 4J	4	36c1	144	4320
	1A 4Q 3J	4	36c1	144	4320
	1A 1K 2Q 4J	18	36c1	648	19,440
	1K 4Q 4J	3	36c0	3	90
31	1A 2K 3Q	12	36c3	85,680	2,656,080	291,805
	1A 3K 1Q 1J	16	36c3	114,240	3,541,440
	3K 3Q 1J	16	36c2	10,080	312,480
	1A 1K 4Q 1J	12	36c2	7560	234,360
	1A 2K 2Q 2J	108	36c2	68,040	2,109,240
	1A 3K 3J	4	36c2	2520	78,120
	2K 4Q 2J	18	36c1	648	20,088
	3K 2Q 3J	24	36c1	864	26,784
	1A 1K 3Q 3J	48	36c1	1728	53,568
	1A 2K 1Q 4J	12	36c1	432	13,392
	2K 3Q 4J	12	36c0	12	372
	1A 4Q 4J	1	36c0	1	31
32	1A 3K 2Q	6	36c3	42,840	1,370,880	93,000
	3K 4Q	1	36c2	630	20,160
	1A 2K 3Q 1J	48	36c2	30,240	967,680
	1A 3K 1Q 2J	24	36c2	15,120	483,840
	3K 3Q 2J	24	36c1	864	27,648
	1A 1K 4Q 2J	18	36c1	648	20,736
	1A 2K 2Q 3J	72	36c1	2592	82,944
	1A 3K 4J	1	36c1	36	1152
	2K 4Q 3J	12	36c0	12	384
	3K 2Q 4J	6	36c0	6	192
	1A 1K 3Q 4J	12	36c0	12	384
33	1A 2K 4Q	3	36c2	1890	62,370	20,368
	1A 3K 2Q 1J	24	36c2	15,120	498,960
	3K 4Q 1J	4	36c1	144	4752
	1A 2K 3Q 2J	72	36c1	2592	85,536
	1A 3K 1Q 3J	16	36c1	576	19,008
	3K 3Q 3J	16	36c0	16	528
	1A 1K 4Q 3J	12	36c0	12	396
	1A 2K 2Q 4J	18	36c0	18	594
34	1A 3K 3Q	4	36c2	2520	85,680	4306
	1A 2K 4Q 1J	12	36c1	432	14,688
	1A 3K 2Q 2J	36	36c1	1296	44,064
	3K 4Q 2J	6	36c0	6	204
	1A 2K 3Q 3J	48	36c0	48	1632
	1A 3K 1Q 4J	4	36c0	4	136
35	1A 3K 3Q 1J	16	36c1	576	20,160	618
	1A 2K 4Q 2J	18	36c0	18	630
	1A 3K 2Q 3J	24	36c0	24	840
36	1A 3K 4Q	1	36c1	36	1296	60
	1A 3K 3Q 2J	24	36c0	24	864
37	1A 3K 4Q 1J	4	36c0	4	148	4
Totals				1,677,106,640	33,018,036,975	1,677,106,640
Average				Total HCP / Total Hands = 19.6875

Note that the total number of hands (1,677,106,640) is equal to 48c9 as expected.

Lo and behold, the intuitive answer is indeed exact! A real mathematician evidently knew this:

Dan Baker: Each missing card has a 9/48 chance of being in Sharko’s hand.
By “linearity of expected value,” average HCP = 15 + 25 × 9/48 = 19.6875.

The rounded average could also be found empirically:

Andrew Spooner: I wrote a simulation, but I am not sure whether 19.68 or 19.69 is correct.
It seemed very close, so my tiebreaker was the simple 15 + 25 × 9/48 = 19.6875, or 19.69 rounded.

Richard Stein: I wrote an R simulation for 1 million hands and ran it 10 times…

2. Most Common (mode) HCP

Sharko stacks the deck to deal himself three aces and a king (15 HCP) while the remaining 48 cards are distributed at random. What would be Sharko’s most common (mode) number of HCP?

This was the incongruity that prompted my puzzle. With an average of 19.69 HCP, one would surmise that the most common occurrence would be either 19 or 20 HCP. But no! Surprisingly the mode is 18. As shown by the green box in the table, there are 232,135,068 hands with 18 HCP, while 19 HCP produces only 227,642,580; and 20 HCP even fewer with 221,402,016.

Obviously, the mode of 18 is heavily influenced by the inclusion of one queen and one jack, the most common high-card arrangement, occurring in 133,562,880 hands.

Dan Baker: Most common is 18 HCP with 232,135,068 hands, ahead of 19 with 227,642,580 and 20 with 221,402,016.

Puzzle 8Q59 Main

Top Crooked Swiss

3. Board Number of Minus Score

In a seven-board Swiss match, Sharko had all plus scores, and so did his teammates except for one board, yet they lost the match. There were no doubled or redoubled contracts. What was the board number of their minus score?

These solvers explain it as well as I could:

Andrew Spooner: Under the given conditions, the maximum that can be lost on a single board is 19 IMPs (+70, -2220) and the minimum that can be gained is 3 IMPs (+70, +50) except with both sides vulnerable, in which case it is 4 IMPs (+70, +70). Therefore, the match requires a board set in which both vul occurs only once, which is board 20 of set 15-21.

Jim Munday: The 7-board match must be lost 18-19. The most we can lose on a board with one pair plus is 19 IMPs (-2210 with an unsatisfactory +100). To win only 3 IMPs with two plus scores needs at least one side to be nonvul (+70 with +50 setting the nonvul opponents). Board set 15-21 has only one board with both vul (20). I believe this would also apply to board 29.

Indeed it would, though board set 29-35 is rarely used, and my puzzle answer was restricted to 1-28.

Dan Baker: The biggest possible loss is 19 IMPs (small plus vs -2220). With both sides plus, the minimum gain is 3 IMPs with either side nonvul (+50/+50 or +50/+70) else 4 IMPs with both vul (+70/+70). We need only one board with both vul: 20 (in 15-21) and 29 (in 29-35) both work.

Charles Blair: I assume the match used boards 15-21. Otherwise there would be more than one board with both sides vulnerable.

Richard Stein: They played boards 15-21 and lost the crusher on the only board with both vul.

Guilty as Charged

Regarding the complaint against Sharko, I submitted 24 boards he is known to have shuffled and dealt to the IBA (Institute for Bridge Arbitration). Using a detection tool called EDGAR (Extrema Detection Gauge for Asylum Records) they calculated his chance of being honest as 0.0037 percent, which is convincing enough for our purposes. Sharko will not be making any more boards! Nonetheless, his presence in the ward is most welcome for upholding its principles: Everyone’s a crook, so who really cares!

These solvers have been barred from the ward for too many licit activities:

Charles Blair: No doubles and redoubles? I suspect Quaalude abuse in the ward!

Jim Munday: The only thing worse than my statistical acumen is my leads (and bidding?) but I gave it a shot.

And a perfect shot it was! Now if he could only apply this to his leads. (I can vouch that his bidding is incurable.)

Puzzle 8Q59 Main

Top Crooked Swiss

© 2026 Richard Pavlicek