April 1, 2017: PavCo Enterprises files a class-action suit against Walt Disney Studios for its unlawful appropriation of a bridge term.
Financial recovery is expected to exceed $500 million. Claimants must register before midnight GMT, April 30. To be eligible you must be a member in good standing of a national bridge organization, or a homeless derelict; we don’t really care. The important thing is to be alive, which puts us one up on Mr. Disney. Register now and we’ll turn this ice-capade into our own Scrooge McDuck.
Card players have been aware of frozen suits before Mickey Mouse even hit the drawing board.
A suit is considered frozen if leading the suit by any player allows the opponents to win a trick they could not win on their own. It’s like the zugzwang in chess; whoever moves first loses. Consider the major suits on the following deal:
The spade suit is frozen. Declarer gains a trick if either defender leads a spade, and the defense gains a trick if declarer leads a spade from either hand. More specifically, the side that first leads a spade wins only one trick, and the other side wins two. (This ignores North’s fourth spade, which will always win a trick if established in time, regardless of what happened previously.)
The heart suit might also look frozen, but not quite. Call it semi-frozen, because it is frozen only to one side, in this case the defenders, who cannot lead a heart without losing a trick. Note that declarer is able to lead hearts once (low to the jack) without disturbing its frozen state to the defense, after which the suit becomes truly frozen.
The difference between frozen and semi-frozen is often slight. For instance, swapping the 7 and 9 makes the spade suit semi-frozen, as declarer could then run the jack (losing to the king) after which the suit remains frozen to the defense and is truly frozen.
The play in 3 NT is irrelevant to this puzzle, but even after the best lead of a diamond, declarer can succeed by availing his frozen assets.
Now it’s your turn! Using different high-card arrangements than the above deal:
Construct (1) a frozen spade suit, and (2) a semi-frozen heart suit.
The semi-frozen hearts must be frozen to East-West as in my example. A further goal (contest tiebreaker) is to keep the rank sum of each holding as close to average (26) as possible, and the North-South HCP as low as possible.
Before reading further, test yourself or make your best guesses:
1. Which spade suit (WNES order) was the winner?
2. Which heart suit (WNES order) was the winner?
Quit
Congratulations to Duncan Bell, England, who is on the hottest RPbridge streak of all-time, winning three months in a row, and four of the last five. Being fast and clever, the only way to beat this guy might be to avail our frozen assets and freeze his hard drive. Also achieving a perfect score were Tina Denlee, Foster Tom and Lief-Erik Stabell.
Countrywise, the win goes to Canada with three placings, versus only two each for the U.K. and U.S. Considering the puzzle theme, I guess this was predictable, as I envision their entries being delivered by dogsled along the Iditerod Trail.
Before showing any solutions, let’s theorize a bit. There are 67,108,864 (413) possible distributions of a single suit among four hands, all of which have a rank sum of 104 (counting cards at face value, or jack = 11, queen = 12, king = 13, ace = 14). Therefore, the average rank sum of the suit holding for each hand is 104/4, or 26. Winning this contest required each hand to be exactly that.
My count shows 1728 of the 67+ million layouts to be equivalued, but only one doubleton (A-Q) qualifies, which cannot exist in a frozen or semi-frozen state, so the distribution must be 4-3-3-3. This reduces the count to 840, which includes 24 permutations of each combination, so there are only 35 distinct layouts where each hand adds to 26. (I would list them here but am afraid I might be committed.)
An interpretation problem arose when the same layout was submitted by two respondents, one as frozen spades, the other as semi-frozen hearts. To be impartial, I’ll show it below as clubs:
Tom Slater: [spades] This suit is doubly frozen. Declarer has nothing by force, and the defense has just the ace.
Charles Blair: [hearts] The suit will be frozen after South leads and plays any card from dummy.
PavCo attorneys assured me that whichever way I ruled, they would manufacture the legalese to prevail in court. After all, they put Disney on ice*, so what chance could Slater and Blair have? Nonetheless, since my leaderboard was desolate, and this was not the optimal solution for either case, I decided to accept both.
*Bringing to mind the legend that Walt Disney was cryonically preserved after death. While untrue, I keep my house temperature very low to get a head start just in case.
Technically the suit is frozen, as any lead by any player allows the other side to win a trick they could not win on their own. Once this happens, the suit is not released from the freezer as usual but remains frozen, hence the description doubly frozen is spot on.
About a dozen respondents lost out because they overlooked a finessing option by declarer, typically a ‘backward’ finesse. Most of these layouts were perfectly balanced (zero variance) — but not frozen. For example:
North-South can establish a trick by force if North leads the nine, which East covers: 10, queen, ace. On the second round South leads and finesses against the eight, so E-W can win only their two tops. Even if you swap the six and four to foil the backward finesse, the suit wouldn’t be frozen but only semi-frozen, since low to the jack (or simply leading the jack) keeps the suit frozen to the defense.
All it takes is a slight change in the previous layout to effect a perfect solution. The following spade suit was submitted by Tina Denlee and Foster Tom, both of whom live in the “great white north” (albeit 5000 kilometers apart) so they should know all about freezing.
Each hand totals 26, and the suit is completely frozen. If North or South leads any card from either hand, they lose three fast tricks; but if East or West starts the suit, N-S must score the queen or jack.
Foster Tom: East-West have two tricks and N-S only have their 13th card, unless someone breaks the ice. In the play, one need only remember “second hand low” and “cover an honor.” It is barely possible to give N-S only 2 HCP [with 26 sums all around], but I could only semi-freeze that.
The above layout contains the fewest N-S HCP for a frozen suit with four 26-sum holdings, but an equally good alternative is J-8-7 opposite Q-6-5-3, as discovered by Duncan Bell, Lief-Erik Stabell and Tim Broeken. In either case, the N-S and/or E-W hands can be swapped.
Duncan Bell: If N-S lead, E-W cover honors to set up a finessing position at trick 2. If E-W lead, N-S play low as second hand to guarantee a trick.
For Part 2, the object was to create a layout frozen only to East-West, i.e., semi-frozen such that North or South can lead the suit once without disturbing its frozen state to the defense. Again the goal was to equalize the rank totals of each hand at 26, which Dan Gheorghiu, British Columbia, achieved with this curious layout:
North-South are entitled to one trick (ignoring South’s long card) if only they lead; but if East or West starts the suit with any card, N-S can develop two tricks with direct play, e.g., if East leads the five, South covers with the six. What deems the suit semi-frozen is that South can lead it once without loss, specifically the nine (or eight) covered by West and North, then the suit remains frozen whether East takes the ace or not.
Dan Gheorghiu: If either defender leads, the Q-10 can be double-finessed for two tricks. If South leads (i.e., nine, 10, jack) the finesse assures only one trick, but the remaining two tricks are frozen.
Only one semi-frozen layout fits the optimal conditions (balanced pips, fewest N-S HCP) except for the obvious 180-degree rotation that applies to any layout in this puzzle. This was discovered by the top four solvers, and a few others who missed out on Part 1. Our winner, Duncan Bell, England, gave the 2 HCP to South:
North-South are entitled to zero tricks (ignoring South’s long card) if only they lead. If East or West leads first (any card), N-S can establish a trick by force, so the suit is clearly frozen to E-W. It is not frozen to N-S, however, because North can lead the 10 or nine (East covers with the king) after which the suit remains frozen to E-W. Hence, semi-frozen.
Duncan Bell: North can lead the 10 or nine, which must be taken with the king to set up a frozen position. (If East plays the J or 2, N-S can guarantee one of the first three tricks.)
Leif-Erik Stabell: North can lead the 10 to the king, then the suit is frozen.
Foster Tom: North can lead the ten or nine, which West must carefully cover with the king to keep the position intact. Giving N-S only 1 HCP would leave the A-K-Q for E-W, forcing A-Q doubleton for a 26 rank sum, which I’m sure admits no frozen position.
Tina Denlee: I imagine a deal where N-S cash seven minor winners while E-W follow suit, then exit in a major to force E-W to break the ice, but E-W exit in the same major to zugzwang declarer in the other. Down one!
And the rest of us imagine Tina in a cage — except for Trump who would just build a wall around Canada.
Dan Gheorghiu: Since I can still fog a mirror, I signed up for the PavCo class-action suit, and I eagerly await my cut from the Donald Duck family.
That about does it… the duck stops here.
Apologies to Donald and nephews Huey, Dewey and Louie© 2017 Richard Pavlicek
by Richard Pavlicek by Richard Pavlicek