PeterAlan

As I understand it: It seems that the second paragraph is the one relevant here, but from the first it appears that the Vice President becomes Acting President as soon as the written declaration is "transmitted", and there is no pre-emption mechanism, however short it may be before the President invokes the terms of the second. In this event, I don't suppose that the full 4 days + 48 hours + 21 days would be taken to resolve the matter. It would seem that the Vice President would remain Acting President in the event that a second declaration was transmitted; what's not clear to me is who is in office in the period between the President declaring himself able and the transmission of that second declaration. In the previous invocations of the 25th Amendment, when Reagan and Bush were in surgery, they resumed office almost immediately afterwards, but they had invoked section 3.

That one's easy: no-one can. I liked a comment someone added to a WaPo article: impeachment should be very quick; House in the morning, Senate in the afternoon - the Republicans have already made clear that they don't need any witnesses at an impeachment.

Quite apart from anything else, he's only referring to "an orderly transition on January 20th" (my emphasis), which hardly gets the job done properly (as we've already been seeing with, for example, Defense and Budget Management). What are the chances of Biden turning out to be President 47?

I remember reading this article at the time: it would appear that the work of Katalin Karikó and Drew Weissman was fundamental to all that followed.

My reply to Blackshoe crossed with your posting this. I wasn't aware of that detail of Culbertson's suggestion, but it's notable that it's only particular 6- and 7- card shapes that stand out: 7=3=2=1 (especially), 6=3=2=2, and to a lesser extent 6=4=2=1 and 6=3=3=1. Deals containing these shapes are noticeably commoner than those containing other 7- or 6-card shapes.

Well the effect asserted by Culbertson certainly isn't generally there. In the extreme cases of deals containing hand patterns 13=0=0=0 or 12=1=0=0 there are perforce the same suit patterns, and some residue of the same effect shows in the relatively high results for the very unusual 11=1=1=0 and 10=2=1=0 shapes. But in the realm of the common hand distributions the numbers largely reflect the preponderance of those shapes in the set of all deal distributions: for example, in the 4=4=3=2 case, 61.185% of all deals contain at least one suit distributed 4=4=3=2; when we're constraining the shapes of the deals in question by limiting them to those with at least one flat 4=4=3=2-shaped hand it's not too surprising that the proportion with a 4=4=3=2 suit rises from 61.185% to 76.127%, but that's as far as it gets in all but the most extreme cases. I don't suppose that Culbertson ever made any attempt at a proper analysis, either of this mathematical structure or of whatever anecdotal evidence he had of the (hand-shuffled) deals he played. It's natural to notice the cases where patterns occur, and to tune out the cases where they don't, which leads to distorted impressions. Beware of intuition: results are often counter-intuitive. For example, what do you suppose is the commonest result for the longest suit in a deal? In fact, it's 6 cards (43.3% of all deals), which beats 5 (40.23%), 7 (12.64%), 4 (2.93%) and 8 (1.83%).

In the unlikely event that anyone is interested, I extended my program to calculate the following statistics concerning the validity of the so-called "Law" of symmetry in the form that a particular hand shape in a deal is likely to lead to the same shape applying to one or more of the suits (Since there is a complete duality between the hand shape distributions of all possible deals and the corresponding suit distributions, any such "Law" would apply mutatis mutandis to say that a particular suit distribution would imply a corresponding hand distribution somewhere in the deal; did Culbertson make some remark to indicate that he recognised this?) The table gives, for each hand shape, the probability that a deal containing that hand shape will have one or more suits of the same shape: Proportion of deals with target hand shape that contain same suit shape: Target shape Proportion ============ ========== 4=3=3=3 59.372 % 4=4=3=2 76.127 % 4=4=4=1 30.810 % 5=3=3=2 65.237 % 5=4=2=2 56.652 % 5=4=3=1 63.130 % 5=4=4=0 29.634 % 5=5=2=1 31.655 % 5=5=3=0 23.434 % 6=3=2=2 50.414 % 6=3=3=1 35.340 % 6=4=2=1 42.194 % 6=4=3=0 32.441 % 6=5=1=1 12.984 % 6=5=2=0 17.255 % 6=6=1=0 3.236 % 7=2=2=2 18.922 % 7=3=2=1 56.865 % 7=3=3=0 12.686 % 7=4=1=1 15.143 % 7=4=2=0 16.513 % 7=5=1=0 5.459 % 7=6=0=0 0.618 % 8=2=2=1 42.049 % 8=3=1=1 26.175 % 8=3=2=0 24.625 % 8=4=1=0 10.496 % 8=5=0=0 0.910 % 9=2=1=1 48.209 % 9=2=2=0 22.315 % 9=3=1=0 27.239 % 9=4=0=0 2.676 % 10=1=1=1 24.048 % 10=2=1=0 66.576 % 10=3=0=0 9.397 % 11=1=1=0 68.421 % 11=2=0=0 31.579 % 12=1=0=0 100.000 % 13=0=0=0 100.000 % ============ ==========

The following table gives the answers for 5=4=3=1: Target shape: 5=4=3=1 # deals with 1, 2, 3, 4 or no instances of target: -- % of - --- % of --- # # deals all deals target deals -- -------------------------------------- --------- ------------ 1: 18,462,006,540,449,844,182,883,072,000 = 34.41532 % 80.66927 % 2: 3,997,264,604,908,323,574,425,600,000 = 7.45136 % 17.46595 % 3: 417,060,951,961,543,212,331,008,000 = 0.77745 % 1.82234 % 4: 9,713,344,598,876,987,228,160,000 = 0.01811 % 0.04244 % -- -------------------------------------- --------- --------- 22,886,045,441,918,587,956,867,840,000 = 42.66224 % 100.00000 % -- -------------------------------------- --------- --------- 0: 30,758,692,323,570,204,882,369,600,000 = 57.33776 % == ====================================== ========= 53,644,737,765,488,792,839,237,440,000 = 100.00000 % ====================================== ========= This is an extract from program output that gives these details for all 39 deal shapes, as well as the distributions of longest and shortest suits in the other 3 hands. If anyone is interested in the full output, I'm happy to send it out. The program calculates these quantities directly from the set of all the possible deal shapes, and does not involve simulation in any form. With that set as the staring point, it's a quick calculation: less than 1/4 second on my first-generation Surface Book.

I don't understand exactly what you're saying, but the numbers I gave above are from the complete 53,644,737,765,488,792,839,237,440,000 deal space and its detailed sub-structure, and not any sort of simulation. In that sense, it's exactly the sort of thing you're calling for.

Since I have the data, I spent a little while putting together a program that determines these basic statistics for each of the 39 generic hand shapes (4=3=3=3 to 13=0=0=0): (1) The numbers & proportions of deals that have 1, 2, 3, 4 and no hands of that shape; and, analysing the sets of deals that contain the shape in question, (2) The percentages (and numbers) of such deals that have 2 or more hands of that shape, and (3) The percentages (and numbers) of such deals categorised by the longest suit in one (or more) of the other hands. Results for the two shapes (7=2=2=2 & 4=3=3=3) you originally cited are: Total deals: 53,644,737,765,488,792,839,237,440,000 Target shape: 4=3=3=3 18,904,824,864,906,126,262,212,096,000 deals with 1, 2, 3 or 4 of target: 1: 15,538,600,726,161,191,018,436,096,000 = 82.19384 % of all such deals; 2: 3,078,920,993,459,221,886,976,000,000 = 16.28643 % 3: 237,337,380,888,197,990,400,000,000 = 1.25543 % 4: 49,965,764,397,515,366,400,000,000 = 0.26430 % Length of longest suit(s) in the other 3 hands: 4: 1,341,268,488,045,803,116,800,000,000 = 7.09485 % 5: 9,389,152,279,359,996,649,015,910,400 = 49.66538 % 6: 6,567,757,945,719,475,336,421,990,400 = 34.74117 % 7: 1,457,245,678,194,666,886,759,833,600 = 7.70833 % 8: 143,135,495,072,392,782,498,739,200 = 0.75714 % 9: 6,173,444,779,468,438,792,320,000 = 0.03266 % 10: 91,533,734,323,051,923,302,400 = 0.00048 % --------------------------------------------------------------------------------------------------------------------------------- Target shape: 7=2=2=2 1,091,600,331,330,190,676,219,596,800 deals with 1, 2, 3 or 4 of target: 1: 1,082,537,511,172,874,920,302,796,800 = 99.16977 % of all such deals; 2: 9,049,166,455,516,001,717,760,000 = 0.82898 % 3: 0 = 0.00000 % 4: 13,653,701,799,754,199,040,000 = 0.00125 % Length of longest suit(s) in the other 3 hands: 4: 34,329,156,878,471,495,040,000,000 = 3.14485 % 5: 465,402,880,409,496,995,338,813,440 = 42.63492 % 6: 444,973,635,287,073,217,557,872,640 = 40.76342 % 7: 126,372,546,662,990,136,932,966,400 = 11.57681 % 8: 19,117,164,834,880,139,265,638,400 = 1.75130 % 9: 1,360,547,526,432,774,670,848,000 = 0.12464 % 10: 43,938,720,545,275,887,851,520 = 0.00403 % 11: 461,010,300,641,525,606,400 = 0.00004 % For example: (1) of all the 1.9x10^28 deals with a 4=3=3=3 hand only 16.28643% have 2 such; for 7=2=2=2 the proportion is 0.82898%. (2) of all the deals with a 4=3=3=3 hand the longest suit in one of the other hands is 7 cards in 7.70833% of such deals; for 7=2=2=2 the proportion is marginally higher at 11.57681%. Whilst there are differences, if you plot the bar graph of the percentages for each length the patterns of the distribution for each shape are markedly similar. The data does not support any sort of "symmetry" law. Of course, this is what one would expect: one is concerned with the ways in which the 39 cards not in the "target" hand are distributed between the other 3 hands: in the 4=3=3=3 case those are 9 cards of one suit and 10 of each of the others; for 7=2=2=2 it's 6 of one and 11 of each of the others. Just as 5 cards in two hands tend to split 3-2, so n cards over 3 hands tend to distribute relatively evenly; it's the same effect. It's not surprising that finding 7+ in one of those hands is relatively uncommon. Edit: Tables put into 'code' format.

And presumably matchpoints since you have at least 9 top tricks (1+2+1+5) after the ♦ lead.

I agree with Cyberyeti.

I've been wondering for a while about posting this anecdote; it's fairly well-known, but may be new to some. The pre-eminent mathematical logician Kurt Gödel fled Austria just before the second world war, and settled at the IAS in Princeton. In 1947 he had applied for US citizenship, and was about to undertake the examination, with Albert Einstein and Oskar Morgenstern as his witnesses. Being the man he was, he took this very seriously, and gave the Constitution close study. Shortly before the examination, to Morgenstern's consternation, Gödel told him that he had found a logical flaw in the Constitution that could lead to the legal establishment of a dictatorship in the United States. Morgenstern was concerned that Gödel would pursue this at the examination, and enlisted Einstein's help in trying to dissuade him from it; they spent the journey to Trenton by telling one joke after another in an attempt to distract him. The examination was conducted by Judge Phillip Forman, who had done the same for Einstein a few years earlier and was friendly with him. Early on in the process, Gödel picked him up on the distinction between German and Austrian citizenship, and Forman made some remark about it being just the one dictatorship, adding that it "couldn't happen here." This, of course, triggered Gödel, but Forman was a sensible man and interrupted, hastily changing the subject, and the rest of the process passed off smoothly. The thing is, Morgenstern doesn't seem to have made a record of the point that Gödel had found, and I'm not aware of any suggestion of just what it was. Suddenly, though, it begins to seem more pertinent ...

You might think that this has been part of the Trumpian playbook since the Republican legislatures of Michigan, Wisconsin and Pennsylvania ensured that the counting of their mail-in ballots could not start before election day. I couldn't possibly comment.

Zel, I'm too tired to want to get into a lengthy debate with you, and disinclined to do so anyway, but just for the record I was in no way disputing the advantages of (good) proportional voting systems over FPTP. I'm doing no more than saying that in such unsatisfactory circumstances voting itself can be a critical activity. Goodnight.

I don't disagree with you about activism and other political activity being crucial elements, and ones to which one's individual contribution generally has proportionately more weight than voting itself, but voting remains the bedrock of political activity, and can be even more critical: for example, Florida 2000 or Georgia right now. Mine was an anecdote, not a detailed analysis of the comparative impacts of different aspects of political activity.

An anecdote from the UK. The 1970 General Election on 18 June was the first in which 18-year-olds had the vote. My 18th birthday was 3 days earlier, and I was determined to take part (as I have done in every election since). My grandmother lived with us, and I took her along to the polls. She was then nearly 90 (her birthday, in a happy chance for this thread, was 4th July), and one of the 'lucky' women to be first enfranchised in 1918, the year she was 38 - others had to wait another 10 years. I remember how fortunate I was to be exercising the right that so many had to struggle so hard to achieve. Decisions are made by those who show up and make a statement; it's trite but true to say that we can't all rely on someone else to do it for us, even in groups of millions.

Insofar as they had any agreement, it was to play a system in which "2D is our strongest bid". They incorrectly called their system "Reverse Benji"; since it is not, in fact, Reverse Benji there is no basis to conclude from these conflicting statements that the meaning of "2♣ is not "weak""; nor, indeed, anything else about it. You can't pick and choose, ignoring the one clear statement they actually made about their methods, especially as you are relying on the solecism of determining the meaning of a conventional bid by its name rather than by a statement of what it means.

It is difficult to see how this requirement of SB's can be satisfied. What is "actual" here? Reverse Benji as SB (and probably the rest of the bridge world) understands it? If so, and SB already understands it, then (1) why does SB need an explanation, and (2) why does SB not query RR's description of the 2♣ bid as "weak"? If, however, the "actual" methods are not Reverse Benji as SB (and probably the rest of the bridge world) understands it, then SB should accept RR's statement of what they are "actually" playing. In fact, as SB is surely aware, neither member of the partnership appears really to know what Reverse Benji is, since PP has said "2D is our strongest bid", which is the case in standard Benji but not the Reverse variety. Actually they appear to have no agreement, but RR is unaware of that; of the 3 players concerned, SB is likely to be the best-informed.

I don't see how RR "could have been aware at the time of his irregularity that it could well damage the non-offending side" when he appears not even to be aware of the methods that he's allegedly playing. SB surely knows by now that RR can't be supposed to be aware of anything specific at all. Moreover, if SB is supposedly aware of the whole conversation and knows that RR's description of his 2♣ opener as "weak" is not consistent with Reverse Benji as it is normally played, where the 2♣ opener, like Acol, is the game-force (unless followed by 2NT); he also knows that RR has admitted to not being sure that he knows what Reverse Benji is. SB's assertion that RR should have been aware that the 2♣ bid is game-forcing, but has nevertheless both psyched it and mis-described it as weak, is tantamount to accusing RR of cheating, and RR should instruct his lawyers accordingly. It appears from Mr Mollo's entertaining accounts that RR's pockets are quite deep, possibly significantly more so than SB's, and I look forward to hearing what m'learned friends make of it all.

What do you think I had wrong, Paul? I gave a solution for ♠s 5-1 and said that I was finding 6-0 problematic and why. I began to look at layouts like Nigel's, but at 4:30 wasn't going to spend more time on it before going to sleep. I'm very glad that he has come up with that construction, and congratulate him.

The trick target is 6, and a suit contract can't be the answer because of ruffing possibilities. I can't quickly see how to deal with 6-0 ♠s - 3 discards is the problem - and need my sleep, but no worse than 5-1 can be dealt with. Say ♠s 5-1. The lost tempo means that it does the defence no good to switch, so assume the hand with 5 ♠s has the other Aces too and persists with ♠s whenever in. When declarer first wins a ♠, play a ♣: If this is ducked, switch to a red suit; defence must win (or declarer switches back to ♣s, making 2 ♠s, 1 red card and 3 ♣s). On winning the next ♠, play the other red suit, which the defence must also win. Discard one red card and one club on the defence's 2 long ♠s and make 2 ♠s, 3 red cards and 1 ♣ If the ♣ is not ducked, win the next ♠ and play a red suit, which the defence must win (or declarer cashes 3 ♣s for 6 tricks). Discard one card from each red suit on the defence's 2 long ♠s, and make a minimum of 2 ♠s, 1 card in the red suit played and 3 ♣s NB: With ♠s 6-0 the defence still need one trick in the other suits in addition to their Aces to get to 8.

I'm neither American nor a lawyer, but I found this New York Times op-ed interesting:

If you're thinking of PoorBridge it's still there.

Beware: Wikipedia is not reliable. In particular, the statistics on the numbers of hands with X losers were hopelessly wrong when I last looked a while ago.