Natty Bumppo’s euchre columns from the publisher ofThe Columbus Book of Euchre How would you play this hand?
 Presented here are archives of euchre columns by Natty Bumppo, author of The Columbus Book of Euchre, published on line.

 “Double suited,” part 2 – July 5, 2002 The dealer picks up a club and holds A-K-10 of clubs and K-9 of diamonds. Itappears that no one else had reason to or-der the club (e.g., none had three clubsheaded by both bowers) or that if he did(according to a common precept to orderwith two dry bowers), he blew it.
 “I would (by instinct) lead the 9 of diamonds. I also believe that it is the correct call mathematically because there seem to be more possibilities to succeed by getting the off suit out there. The perfect scenario would be your partner’s having an ace of diamonds that walks and a suit lead back that you can ruff. Making the defenders spend a bower stopping the 9 of diamonds would be probably second best. The king of diamonds is a nice card for an end play. “My friend Henry and I tried a few duplicate style samples of the hand. As with any common hand in euchre, we found that almost anything is possible. Sometimes the maker won and sometimes he got euchred. We tried leading trump and throwing off, but there seems to be no definite answer. I would simply guess that you have a 55 to 60 per cent chance of making a point with the hand, with the play siding in favor of leading the 9 of diamonds.”
 Our mathematician – my little brother the Ph.D. at Motorola, who tells engineers where and when (and how high) to jump (we call him “Dr. Math”) – mused as follows before digging in:
 “This is reminiscent of what happens to me when I contract to model some gizmo at work. I always think the model will be relatively simple. Then, as the people I contract with start filling me in on gizmo details, I realize how complicated the model must be. The result for me is always overly optimistic deadlines that I can’t meet. “I can see the potential for this problem to grow in complexity very rapidly. And I don’t think we can get a definitive answer soon.”
 The opponents’ holding that would most favor leading trump would be an unguarded bower in each opponent’s hand. And the holding I most easily envision as almost requiring a trump lead is an unguarded bower in each opponent’s hand and the ace of diamonds also in one of the two opponents’ hands. But a 9 of diamonds lead will work even against that if the dealer’s partner can trump the diamond and neither of the opponents can or does, or if it is the right hand opponent who has a lone ace of diamonds and the left hand opponent lays down a bower. But what is the probability each opponent will hold an unguarded bower? If the probability is infinitesimal (definition: Significantly less than 5 per cent), then my conclusion is either (a) lead the 9 of diamonds, or (b) it doesn’t make any difference. I would have liked to have withheld calculation of the probabilities of the dealer’s partner’s holding as long as possible, but ultimately we could not. E.g., the probability that his partner has a bower is almost 4/15, but a little less, since he didn’t order (I do not factor in the knowledge that the lead opponent holds or held the ace of hearts because only an idiot opponent will lead a bower in this situation). The probability the dealer’s partner has the ace of diamonds is about the same – less than 4/15 because he didn’t order, but a little closer to 4/15 because it was not led. I e-mailed these thoughts to Dr. Math just to get his juices flowing, and he got definitive. He replied:
 “The probability that neither opponent has a guarded bower is 57 per cent – which means that 57 per cent of the time, a trump lead will leave the dealer with the highest trump. “With that exercise complete, the probability that both opponents hold unguarded bowers (“simultaneous unguarded bowers”) is fairly easy: There are two classes of pairs of opponents’ hands – right bower or left bower in West’s hand. Assume West has the right. The rest of West’s hand is formed by drawing three cards from eight (since none can be a trump). This makes C(8,3) such hands. Then East’s hand is formed by drawing three cards from a deck lacking both trumps and West’s other three cards. This makes C(5,3) hands. So we have C(8,3)C(5,3) pairs of hands with the unguarded right in West’s hand and unguarded left in East’s. Since we have divided the totality of pairs of hands with unguarded bowers into two equally sized classes, the number of such pairs is 2C(8,3)C(5,3). The total number of pairs of any kind is C(15,4)C(11,4), so the probability of simultaneous unguarded bowers is 2C(8,3)C(5,3)/[C(15,4)C(11,4)] = 0.0025. “Or one-fourth of 1 per cent. Definitely infinitesimal.”
 Yeah, that’s infinitesimal. And now it was all making sense. And we were back on the precipices of intuition and subliminal perception, as in, “Did the guy on my right raise his left eyebrow, or did I just imagine that?” Because even though a trump lead would establish high trump in the dealer’s hand 57 per cent of the time, it would do nothing to establish his king of diamonds; and it would guarantee him only two tricks – the one he already had taken with the 10 of clubs on the first lead, and the one he eventually would take with the king of clubs. And it could jerk an unguarded left bower or other lone trump from his partner that his partner could have trumped the 9 of diamonds with. The quest went on. And it brought the kicker from Dr. Math:
 “I have found that there are few mathematical solutions, but often mathematical guidelines. We could do better by doing some behavior modeling, but our answers would be only as good as our assessment of the behavior. In the engineering world, where I have served for almost 30 years, I have found that the best I can do is put the engineers in the ball park. They do the rest by trial and error. More than two decimal points of accuracy is wasted compute time.”

 “Double suited,” part 1 – June 28, 2002 John Ellis, author of Euchre: TheGrandpa Lou Way, thinks that be-ing “double suited” means having twocards in one suit.
 We know that it means holding a hand containing only two suits, and it can be pretty valuable. My friend Ron told me that if, as dealer, he holds the ace, king and 10 of diamonds, the queen of clubs and the 10 of hearts, and he turns up the 10 of clubs, he will pick up the 10 of clubs and discard the 10 of hearts. Well, I don’t think so – unless your opponents have six or seven points and could go out on a loner – or you see your left-hand opponent, who never has managed to hold a poker face, drooling and panting, waiting for you to turn down the heart. Or maybe you are getting your butt kicked 9 to 2, and you have to try something outrageous just to shift the momentum (but if that’s the case, why not go alone with that Q-10 of clubs?!). Ron made a pretty good pitch, and he’s a good player (58 per cent winning percentage on Yahoo!, with the big boys, last time I looked); but I took his proposition to Gerry Blue’s Euchre Laboratory to test it. I played Ron’s scenario 100 times – playing the opponents’ cards optimally but without taking advantage of the X-ray vision the Euchre Lab gives you (and I threw out the hands on which an opponent obviously would have ordered up the club). These were the results:
 On 36 of the 100 hands, Ron made a point picking up the little club; and on 13 of the hands he made two points – i.e., he scored on 49 hands, or almost half of them – and he scored big more than an eighth of the time. It means also, however, that he got euchred 51 per cent of the time. But also he stopped nine loners held by the opponents, including two hands on which he scored two points – those were 6-point turnarounds! Ron was euchred on the other seven stoppers, saving his team 2 points on each hand. Also he stopped one lone hand his partner would have made: That one cost his team 3 points. In sum, Ron scored a net of -23 points in the 100 hands (that’s minus 23, or an average of nearly a fourth of a point lost every time he picked up that 10 of clubs).

 Not the perfect hand – June 14, 2002 The logo on the Yahoo! group EuchreScience is not the standard “perfect”hand you see on so many euchre websites, but consists of ace and jack ofdiamonds, 10 of clubs, and queen and10 of hearts.

 B Woods’ loner – June 7, 2002 In a question posed in a poll on Yahoo!’sEuchreScience group, B Woods and hispartner, the dealer, trailed 8 to 6. The dealerturned down a small club, and B called heartstrump alone.

 The “perfect” hand – May 24, 2002 [Note: This column was revised July 24, 2003, for corrections that make the “pefect hand” a- bout 22 per cent rarer than initially reported.]
 Harvey Lapp, Euchre Central webmaster and author of the “Dear Abby”ish euchre advice column “Ask Harv,” forwarded the following query to me: “Here is a question I received that I also would be interested in finding out the answer to. Perhaps Dr. Math knows?” [Dr. Math is my brother the mathematics Ph.D. at Motorola who tells the engineers to jump, and how high.]
 I'm doing a speech on euchre and I need some statistics. I was searching all over the internet, but still couldn't find what the probability of getting a "perfect" hand is. By "perfect hand" I mean jack, jack, queen, king, ace. I know it's very, very rare but I was hoping you would know the answer. A . . . A . . .Baltimore, Md.
 Well, I said, shucks, Harv, I think I can do this without Dr. Math (oops! Thought I could! But I ran it by Dr. Math first, and he caught a mistake I made. Another mistake was caught July 21, 2003, by a reader, Eric Reid; and the original of this column has been revised to take his correction into account. Here is the corrected poop). Start with any jack: The probability of getting a jackless hand is 20/24 times 19/23 times 18/22 times 17/21 times 16/20 equals 36.48 per cent; so the probability of getting a jack is the complement of that, 63.52 per cent (another way to look at this is that if you hold a hand without a jack, that’s fairly unlucky). Now we need the other jack, same color. Four chances out of 23, 4/23. So, the probability of holding both bowers – if their color becomes trump – is 4/23 of 63½ per cent equals 11.05 per cent. Not so rare – a little better than one in ten (but keep in mind, for more than three out of four of the scenarios depicted below, it depends on everyone else’ keeping his mouth shut – i.e., not ordering or calling trump before you get the opportunity. Not presently practically calculable, even in the 21st century). And on down:
 Ace of suit, three out of 22; all three, 3/22 of 11.05 per cent equals 1.5 per cent (thus you have about one chance in sixty-seven of getting the top three cards of a suit). King of suit, 2/21; all four, 2/21 of 1.5 per cent equals 0.14 per cent – (one chance in 714 of getting the top four). And queen, 1/20; all five, 1/20 of 0.14 per cent equals 0.007 per cent – i.e., less than a hundredth of 1 per cent, or a little less than one chance out of 14,000 hands of getting a “perfect” hand (without consideration of the suit turned or another player’s making trump. N.B.: The original calculation, before Mr. Reid’s correction, was 0.009 per cent, or one chance in a little over 11,000 hands).
 But let’s back up a little. You don’t need the king or queen for a “perfect” hand. Both bowers and ace and any two other trump constitute a perfect hand: Played correctly (i.e., ruffing with a bower or ace if it is necessary to ruff), such a hand cannot fail to take all five tricks, from any position. So:
 Fourth trump, 4, not 2, out of 21: 4/21 of 1.5 per cent equals 0.29 per cent – i.e., there is about one chance in 266 hands of getting the top three cards plus one in a suit. Fifth trump: 3/20 of 0.29 per cent equals 0.043 per cent – i.e., the probability that you will get a “perfect” hand (but not necessarily a royal flush) is once in every 2,323 hands (again, without consideration of the suit turned or another player’s making trump. N.B.: The original calculation, before Mr. Reid’s correction, was 0.06 per cent, or once in every 1,771 hands).
 Now, that does not seem amazingly rare to me, and especially considering this: A typical game consists of ten hands or more (it would take Dr. Math to give us a mean of hands per game, but consider: Three hands is the minimum for a game, and that is quite rare; nineteen is the maximum, and that is probably even rarer. And it all depends on the aggressiveness of the players, which is not calculable except statistically: What does it take to get them to go alone? How willing are they to risk being euchred?). Let’s take 10 for a conservative average of hands per game. If you play ten games of euchre a day (not an awful lot for a euchreholic, particularly for an internet euchreholic), that’s a hundred hands a day. At that rate you will get a “perfect hand” every 23.23 days – i.e., roughly once every three weeks. If your correspondent still insists on euchre’s version of a “royal flush” (i.e., two jacks, no ten and no nine), he’ll still get his “perfect hand"“ once every 140 days, or nearly three a year. Now, just for fun and nausea, let’s revisit that “perfect hand” one more time: Actually, only the player to the left of the dealer can get a “perfect” hand, because of the possibility of sabotage – intentional or accidental – in the making of trump (i.e., someone else’ ordering or calling before you get the chance). And because the player to the left of the dealer has the lead, all he needs is either (1) the two bowers and ace and any other trump plus a suit ace, or (2) the two bowers and any three other trump. Those – held by the player left of the dealer, and only by that player – are the real “perfect hands.” But he needs also the right suit turned. The combined probability of J-J-A-x-x (all trump) and J-J-A-x + A (four trump plus outside ace) – with the probability that the dealer will turn a card of the desired suit factored in – is 0.00707 per cent, or once in every 14,144 hands – once every 141 days at ten games a day; you’ll get about two-and-a-half such hands a year. But then you have to divide by four because you sit to the left of the dealer only a fourth of the time. That makes the probability 0.00176 per cent, or once in every 56,818 hands. By this definition you will get a “real” perfect hand about every 568 days – i.e., about every year-and-a-half (if you play ten games of euchre every day for a year-and-a-half!). [Caveat: Some of the decimal numbers above are rounded slightly; so you won’t get the exact results I got if you crunch them.] Natty Bumppo, author,The Columbus Book of Euchre P.S. Where does one give “speeches” on euchre???!!! Borf Books http://www.borfents.com Box 413 Brownsville KY 42210 270-597-2187 [copyright 2002] [next]

The A-K-10-9 loner revisited – April 19, 2002
 A critic wrote in response to lastweek’s column, “The odds that thebowers both are not in your oppo-nents’ hand are 8/18 times 8/18 =.198 or about 20 per cent.”

Odds & probabilities, part 2 – April 12, 2002
 Dwend_98 wrote: Just an opinion on your recent es-say “Dumb and dumber”: Ron shouldhave assumed that the dealer had theright bower when the dealer called a- lone and led ace of trump on the sec-ond trick. Otherwise the best thedealer could have had is ace, king,10 and 9 of trump and an off ace.Certainly not enough to go alone onwith the dealer’s partner’s havingpassed. Dwend Dealer’s hand Ron’s hand

 Odds & probabilities, part 1 – April 5, 2002 Carl Payne (culinarytracker@hotmail.com) wrote: I am always impressed by NattyBumppo’s euchre statistics. Manytimes I have noticed posts that includevery interesting stats, such as the recent13.07189 per cent probablity of youropponent’s having a guarded left. Iwould love to see a large list of theseif you have one made. Carl Gerry’s loner

 “Dumb and Dumber” – March 29, 2002 As dealer I hold jack-nine of diamonds,pick up the ace of diamonds, and go alone.After discard I hold the right-ace-nine oftrump and the king and ten of clubs.
 Ron, on my right, holds left-queen oftrump (i.e. the jack of hearts and queenof diamonds), the queen of clubs, andthe king and ten of hearts..

 Euchre on line – March 22, 2002 Gerry Blue, EuchreScience webmaster, wrote: I was introduced to euchre while going to school at Michigan State and have missed it terribly since moving back to California. There’s not a lot of euchre players out here. I’ve tried to teach my wife, but wives seem to be unteachable. When the internet began to blossom and I discovered euchre on Excite (now Pogo), I was hooked. But I’d prefer to sit at a real table with real people, and steal the deal, and renege, and kick my partner when he trumps my ace, and . . . and . . . and . . . . But I’m stuck with watered down gaming on line for the time being. There’s a group of transplanted Midwesterners that get together in San Francisco occasionally, but I just can’t seem to get fired up about driving an hour-and-a-half through Sonoma, Marin, and the City to play a few games. Gerry Blue, inventor, the Euchre Laboratory Natty Bumppo, author of The Columbus Book of Euchre, replies: And . . . and . . . and . . . flailing elbows! Skewing the markers! Spilling beer on claimed loners! Euchre on line is fine, but – see the comments on my Euchre links page. I feel your frustration in looking for a “table” game in California (if we were talking about hearts and my name were Bill Clinton, I would “feel your pain”). It’s not easy to find a game of euchre here in the South, either, where the hoi polloi think Rook is a form of bridge (the sophsticates play spades). Perhaps you should move to L.A. A friend of mine described it as “not really a city – it’s a collection of Midwestern towns.” And as for wives, maybe you just need a new wife! My lovely second wife did not know how to play euchre before we married but was a good bridge player. She learned euchre so well that she remains, even these many years after my lovely second divorce, the best euchre player I have ever known. Natty Bumppo, author,The Columbus Book of Euchre Borf Books http://www.borfents.com Box 413 Brownsville KY 42210   270-597-2187 [copyright 2002] [next]