In our modern day, a certain Jack Barker has written a book about "The Conjoined Triangles of Success" which supposedly will guide you to running a startup company that thrives. Engineering and Manufacturing meet Sales and Growth along the hypotenuse of Compromise.

We can't really speak to this methodology, but there is another Barker and another triangle that predates the modern interpretation by over 100 years. It's a study by the famed checkerist C. F. Barker, and his position, shown below, is often known as Barker's Triangle.

We've got to say this problem is one of the most instructive we've ever come across. Why it's called a 'triangle' will become clear when you find the solution. So, give it a good effort and then triangulate your mouse on Read More to see how it's done.

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The holidays were over and now North Dakota was in the depths of winter. With even daytime temperatures often well below zero and plenty of snowy days, residents of Bismarck tended to spend as little time out of doors as possible.

Sal had arrived early at the Beacon Cafe for the regular Saturday afternoon session of the Coffee and Cake Checker Club, and he was warming himself with a steaming hot cup of Deana's coffee. Deana, the proprietor, made good coffee and even better desserts, surely the best for miles around.

The boys--- that's what Sal called the other members of the club, even though they were all over fifty--- were still stinging a little from the beating they took when Sal challenged them, just before the holidays, with his friend Brian's 6x6. Sal chuckled to himself. They were going to get another surprise today.

The boys started to file in out of the cold, all of them wearing winter parkas, wool caps, heavy gloves, and fur-lined boots. One by one they joined Sal at the big booth in the back. Wayne, Louie, Larry, and Delmer were there, and to everyone's surprise, so was Kevin, who only showed up a couple of times a year.

"Hey Spooler, seeing you're here for a change, you buying today?" Wayne asked. Kevin went by the nickname Spooler for reasons not really known to anyone.

"Buying what?" Kevin asked innocently.

"Caramel rolls," Deana called over from behind her counter. "Just baked a fresh tray of them. Great with coffee, especially on a day like this." Deana was, among many other things, a great promoter of her wares.

Everyone looked at Sal. "Okay, what've you got?" Larry asked. "Make it a good one so Spooler can buy for everyone."

"Just so happens," Sal said, "there's one from Ed that he calls 'Code Breaker.'" Ed, from Pennsylvania, was Sal's other checker penpal.

"Is it as hard as that one from Brian last month?" Delmer asked.

"See for yourself," Sal said, as he set up the following position on his favorite checkerboard.

"Oh, that doesn't look too ..." Spooler began, but then he stopped. "Uh ... wait a minute ..."

Everyone laughed. But only for a moment, for they were all soon busy examining the checkerboard.

"Fifteen minutes I'll give you," Sal said, but the concentration was so deep his words went unheard.

"Time's up," Sal said a quarter of an hour later.

"I've got it," Larry said. "Let me show you. It's just like ..."

"NO!" Wayne, Delmer, and Louie shouted all at once. "We want Spooler to pay today," Delmer pointed out.

"Aw, c'mon guys, I don't know how to do it," Spooler said. "You just want me to pay because I don't come every week.

Heads nodded in unison. "Not even every month," Wayne pointed out.

"But, okay," Spooler said. "I'll buy ... if Larry really has it right."

---

This really fine problem by master problemist Ed Atkinson is challenging but solvable, and the promise of one of Deana's caramel rolls would be too much for anyone to resist. Can you solve it? We urge you to give it a good trial. Now, we don't know if anything like The Beacon Cafe is in your area, but coffee and cake are certainly in order if you make a genuine effort. When you're ready, click on Read More for the solution and the rest of the story.

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The New Year 2020 is here and we've already been subject to all too many jokes about 2020 vision and so on. So let's skip all that and, while wishing all our readers the best of the New Year, we'll jump right into a nice little speed problem sent by regular contributors Lloyd and Josh Gordon. We won't use our Javascript clock this month. The problem isn't at all difficult but we thought we'd give you a chance to "ease" into the New Year.

Black is a piece up but White is about to even the count. How can Black win?

Even if you're still feeling the results of your New Year's Eve party, you can solve this one. Find the solution and then click on Read More to check your work.

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Priscilla Snelson, Chief Operating Officer for the international conglomerate Rust Belt Holdings in Detroit, was putting on an exclusive New Year's Eve party at her upscale condo in the swanky suburb where she resided.

The party was what you would expect from someone of her business standing. Only top executives and wealthy friends were invited. The food was catered from a company run by a three-star Michelin chef. The music came from Priscilla's ten thousand dollar custom audio system.

Of course, Marvin J. Mavin, Priscilla's long-time boyfriend--- although she referred to him as her "beau"--- was there. Not exactly by invitation; high end parties weren't his thing. More by fiat. Priscilla had told him that he was to be there or else, and that was the end of the discussion.

Everyone was circulating around the huge living room and kitchen, drinks in hand, making small chitchat.

Marvin was bored out of his mind. As Captain of the Detroit Doublejumpers, one of the top teams in the National Checker League, his interests definitely didn't run to the price of scrap tin in Kurdistan or the molybdenum futures market.

No, Marvin liked a good beer or two. Or more. And this party was definitely one of those "more" occasions.

So, at some point into the evening, Marvin was on perhaps his fifth beer--- he had lost track of the exact count--- and he was starting to feel pretty loose and relaxed.

At that moment Priscilla, who until now had been far too busy with her guests to pay much attention to Marvin, came over to where he was standing, a young girl in tow.

"Marv," she said, "I'd like you to meet ..." Priscilla stopped in mid-sentence when she saw that Marvin was in a somewhat zozzled state.

"Oh, dear," she said, "maybe another time."

"Oh, no, no, Auntie Prissy!" the little girl piped up. "I want to meet Captain Marvin! He's my hero! Some day I'm going to grow up to be a checker star just like him!"

"Another time, dear," Priscilla said, but Marvin interjected, "Yeah, Prissy! She wants to meet her hero! Ain'tcha gonna let her?"

Priscilla scowled, but she knew she was trapped. "Very well, then," she said, "Marvin, this is Harriet Liang. She plays for her grade school checker team over in Dearborn, and she's very good."

Marvin leaned down to shake hands. "Nice to meet ya," he said, his voice a bit slurred.

"Auntie Prissy, why is Captain Marvin talking so funny?" Harriet asked.

"Oh, it's just because ... well, never mind that," Priscilla said. "Now, why don't you show Uncle Marvin your problem."

Harriet beamed, "Oh wow, yes!" She pulled a sheet of paper from the pocket of her dress. "Here, Captain Marvin. It's a checker problem, and my Uncle Brian said to try it out on you!"

Marvin looked at the diagram and then looked puzzled. "Hey, what's a little kid like you doing at an adult party ..."

"Her Uncle Brian, from St. Louis, is working on a major deal with us, and he asked me if I would introduce his niece to you."

"I see," Marvin said. "The old you scratch my back I scratch yours routine, hey?" He reached for Priscilla's arm but she stepped back away.

"Okay, okay." Marvin turned to Harriet. "Well, let's see now, this should be easy, you just ... hmm, no, I guess you don't. Suppose you ... nope, doesn't work either."

"Captain Marvin, I solved in four minutes and my coach told me I was very smart!" Harriet said.

Marvin now regretted having those last couple of beers. If only he could think straight! Then Priscilla, as if she had read his mind, said, "I hope you learn something from this. Come on, Harriet, let's get you a soda and a snack. I think Captain Marvin is going to need a while."

---

It's pretty clear that drinking too much didn't lead to a good outcome; after all, it never does. Can you keep a clear head and solve the checker problem that Marvin, in his "zozzled" state, couldn't handle? Little Harriet won praise for her rapid solution. How well can you do?

Try it out and then click on Read More to see the solution and the conclusion of our story.

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The holiday season is in full swing and we extend greetings and best wishes to all of our readers. No matter which holiday you celebrate, may you find happiness and blessing.

We thought we'd present a little longer study this week. Perhaps you'll have some extra spare time over the holidays (or perhaps not!), but in any case it's a good one. The problem situation is based on a game contested in Markham, Ontario, on April 15, 1889. The players were checkerists Fleming and Wright, and the game and variations appeared in a relatively short-lived periodical called The Chicago Evening Lamp.

1.	11-15	23-19
2.	8-11	22-17
3.	9-14	25-22
4.	11-16	24-20
5.	16x23	27x11
6.	7x16	20x11
7.	3-7	28-24
8.	7x16	24-20
9.	16-19	29-25
10.	4-8	22-18
11.	14x23	17-14
12.	10x17	21x14

Perfectly played thus far, and we've reached the end of the KingsRow opening book for the game as actually played.

13.	8-11	31-27
14.	19-24	27x18
15.	12-16	32-28

This seemingly natural move actually loses. 25-22 would have been correct to draw.

16.	24-27	28-24
17.	27-31	25-22
18.	31-27	24-19
19.	16x23	26x19

Here the Lamp called the game a draw and showed a drawing line. But it's actually a Black win.

Can you do better than the Lamp did, over 130 years ago? Shed some light on this interesting position, and then click on Read More to illuminate the solution and notes.

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Four-In-A-Row, sometimes called "Connect Four" is a strategy board game that is actually quite good. Shown above is a large outdoor game set that you can buy for your backyard for only ... um ... $300.00 plus shipping.

At The Checker Maven, though, our game is checkers, and nothing will cost you $300.00 (or anything at all, as our content is completely free). However, our theme for this column is indeed "Four In A Row."

In today's Checker School entry, we're trying something a little different. When you click on the link below, you'll be presented with not one, not two, nor even three, but four elementary checker problems. Your job? Solve them as quickly as possible, and our Javascript clock will "help" you keep track of your time. Can you solve four in a row?

A top player won't require more than a minute or two, which includes time to read the problem terms and look at the diagrams. Those of us with lesser skills will require more time, of course, and perhaps quite a bit more.

But no matter; do the best you can. Are you ready?

Click to display problems

When you're done, click the "back" button on your browser and then Read More to check your solutions, all in a row.

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With this column, the Checker Maven completes 15 years of uninterrupted on-time publication with no missed Saturdays and no missed deadlines. Each weekend we've put something to do with checkers before our readers, who continue to number in the thousands. From what we can tell, you've by and large been pleased with our efforts.

It all started from a discussion with Brian Hinkle, and things went on from there.

Originally we were going to publish for 10 years. We extended that to 15. But we won't stop here. Although we can't give a timeline--- health and age have crept up on us and your editor has serious eyesight issues--- we'll go on as long as physically possible. We know one day we'll have to quit, but we hope that it won't be very soon.

One of the unique things we've done is to tell checker stories, and it's likely we've written more checker fiction than everyone else in the history of the game put together. So for our 15th anniversary, we have a story and a problem. The problem is by Brian (one of his best ever), and the story is set at The Beacon Cafe. It all somehow seems fitting.

---

It was the first Saturday in December, a sunny, crisp and cold day. Sal was all smiles as he walked from his home on 7th Street over to the Beacon Cafe.

Certainly, he was bundled up against the cold. His wife Sylvia wouldn't have let him go out without his wool cap, gloves, and scarf, not to mention his heavy winter coat. "It'll get cold after dark," she warned, and she was right. At this time of year it got dark around four-thirty in Bismarck's northerly latitude, and he knew he'd be at the Cafe until its five o'clock closing time.

On Saturdays the Coffee and Cake Checker Club met regularly, but the holidays were approaching and after today there was only one more meeting until the New Year, so the boys would want to make the most of it. But there was more. Today Sal was bringing something extra special to the session.

Sal enjoyed the walk but just the same he was glad to get into the warm interior of the Cafe. He said hello to Deana, the proprietor, and made his way to the big booth at the back. A couple of the boys were already there: Dan, Mike, and Louie were sitting in front of steaming cups of coffee.

"Hey Sal," they all said, "you're late!"

Sal looked up at the clock on the wall. It was three minutes after one. "Just a little," he said, "but you have a point. You're going to need a lot of time today."

The boys--- none of them younger than fifty--- exchanged glances. But just then three more members arrived: Delmer, Larry, and Wayne. It was just about a full house.

When everyone had settled, Sal immediately commandeered one of the checkerboards. "Have I ever got something today," he said. "It's from my pal Brian who said he composed it based on an idea he got from Ed."

Everyone groaned. Brian and Ed composed tough checker problems, but this promised to be really tough.

"Yep," Sal said, "it's a hard one. But it's one of the best ever. You boys will really enjoy it. And I'll enjoy my coffee and cake. You might as well buy it for me right away because you'll never ever get this one."

For years, the idea had been that someone would pose a problem and win or lose coffee and cake depending on whether the others could solve it.

"What've you got today, Deana?" Sal called, looking over to Deana's counter.

"Chocolate chip zucchini bars," Deana replied. "Really good."

Everyone smiled. Who in North Dakota didn't love zucchini bars?

Sal laid out the problem. Then he grabbed two more boards and repeated the position on each of them. "I'll give you an hour," he said, "and you're going to need a lot of coffee."

The boys looked surprised. Usually they only got ten minutes or so to solve a problem. An hour? And Sal was over at the counter buying coffee for everyone? Something was going on, that's for sure.

After about ten minutes, Dan, Mike, and Louie said they thought they had it. But then they changed their minds. "Nope, doesn't work," Dan said, and the others nodded their heads.

An hour passed, then two, then three. Finally Sal interrupted. "It's after four o'clock," he said. "Time to buy me a couple of zucchini bars before it gets too late!"

But there was no reply, just the sound of moves being discussed and pieces being shifted on the checkerboards.

Finally, it was four forty-five. It was dark outside and Deana was saying she was closing in fifteen minutes and couldn't stay late because her boyfriend was picking her up on the dot of five.

"Okay, Sal, show us," Mike said. "We'll buy you a bar to take home."

"Two bars," Sal said. "One for Sylvia."

He paused and after a moment said, "No bars, no solution."

"Aw, c'mon Sal!" Wayne said.

"Bars," Sal repeated.

"Unfair!" Larry said. "This one was too hard and you knew it!"

It was now five to five.

"Everyone OUT!" Deana said, very impatiently.

Now, Deana was not one to trifle with. When she said 'out' then out you went.

"Tell you what," Sal said. "I'll give you until next week to solve it. But if you don't get it, you buy double, okay?"

The boys, not wanting to annoy Deana further, were putting on their coats. "Sure Sal, whatever you say," Mike said, and the others nodded agreement.

"My boyfriend's here," Deana said, shooing everyone toward the door. She turned out the lights and pointedly held the door open.

With the setting of the sun, a wind had sprung up and it was now very, very cold. And Sal had to walk all the way home without a chocolate chip zucchini bar.

"Never mind," he said to himself, "I'll collect double next week."

---

The boys at the Beacon Cafe might have to wait to see the solution, but you don't. However, we suggest that you spend some time on this problem. It's rather difficult, but highly intriguing. Of course you can click on Read More to see the solution whenever you wish, but do really take the time to explore the problem first.

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We say it every year: Thanksgiving is our favorite holiday. It's typically American (and, of course, Canadian), it's relevant to every belief and creed, and it unites us all in giving thanks for the many blessings we have. No matter our station in life, we can all find something to be thankful for.

We like to turn to the great American checkerist and problemist Tom Wiswell for holidays such as these, and what better than a problem, arising from an actual game, that Mr. Wiswell called Dixie.

Mr. Wiswell noted, "We met with the well-known O. J. Tanner, who got us into the following predicament."

White may be a piece up but his options are severely limited by the two Black kings. Can you find your way out of this one? Make yourself a hot turkey sandwich (with mashed potatoes and gravy, of course) as you contemplate this one, and then--- when you've finished the sandwich and come up with a solution--- click on Read More to check your answer.

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Tommy Wagner had girl trouble, and as a ninth grader, he had never expected any such thing.

It all started when he had a match-up against rising star Letitia Wong (see previous Checker Maven story). Tommy won his game, but he was especially impressed with Letitia's sportsmanship and gentle manner. It had given him a funny feeling that was hard to describe.

So a week later, even though he knew maybe he shouldn't be doing it, he gave Letitia a call. Despite his nervousness, Letitia had been very receptive and friendly and even suggested that they go to a movie together on the following Sunday afternoon.

Tommy didn't know how to refuse--- he wasn't even sure that he wanted to refuse--- and that's when the trouble began.

The problem was this: Tommy had been keeping company with another girl, Tina, for quite some little while. They had gone to grade school together and were close friends.

It was just Tommy's luck, or maybe it was fate, that Tina and her older sister went to the very same movie at the very same theater on the very same Sunday. Naturally, she saw Tommy and Letitia together. Tina immediately burst into tears, and after the movie both she and her sister confronted Tommy and Letitia.

It was not a pretty scene and Letitia, mortally embarrassed, was quite angry with Tommy--- although maybe not as angry as Tina was. In the course of not even five minutes, Tommy went from two girlfriends to none, with Tina and her sister stomping off angrily and Letitia telling Tommy pointedly that she'd take the bus home by herself.

Tommy was in a down mood all week, and it persisted into his Saturday morning checker lesson with Uncle Ben, a retired checker professional and Tommy's long-time mentor.

Tommy dragged up the steps to Uncle Ben's front porch and plopped into a chair with only the barest of greetings.

"Something's wrong, isn't it?" Uncle Ben asked. He could have chided Tommy for his lack of manners, but Uncle Ben was too kindly for that.

"No, sir," Tommy muttered. "Everything's fine."

"I don't think so," Uncle Ben said firmly. "Now, you don't have to tell me about it, but please don't deny it. After all these years I think I know you pretty well."

Tommy sat silently while Uncle Ben, wishing to put Tommy at ease, poured out tall glasses of his deservedly famous lemonade.

Tommy couldn't help but relax, if only a little, after his first sip. "Thank you, Uncle Ben," he said, and then, at all once, launched into the story of his girl trouble.

It was Uncle Ben's turn to sit silently for a few moments. Then he said, "Well, Tommy, and who do you suppose is to blame for this situation?"

"If Letitia hadn't asked me to go to the movie with her ..."

"Just a minute! There are three things wrong with that. First, Letitia didn't know anything about Tina. Second, you didn't have to accept her invitation. And third, you're the one who called Letitia to begin with."

"Um ... yeah ... kind of looks like I brought this on myself, doesn't it?"

Uncle Ben didn't need to reply.

"But now, I don't know what to do about it!"

"What does someone do when they've hurt or offended someone else?"

"Er ... well ... they say 'sorry'?"

"Exactly. You've taken the first step by admitting responsibility. That's a big thing. But it's not enough by itself. Now you have to do something even harder. You have to face the people you've hurt and apologize. Do you think you can do that?"

"I don't know ... but I have to, don't I? So I guess ... well, I just have to. Right away."

"Very good, Tommy, the sooner the better. But not quite right away. You can do what you must this afternoon. I think it will go better if you do a little checker study first to clear your head and focus your thoughts."

Uncle Ben pointed to the checkerboard he had set up on a little porch table. "Today we're going to study something known as the back shot. Take a good look at this position. Push everything else out of your mind and focus. Then tell me how to solve it."

Tommy took a deep breath, then another, and began to concentrate. After about five minutes, he said, "I'm ready, Uncle Ben."

---

Do you need a checker problem to clear your head and put you in a frame of mind for doing ... whatever you need to do? We certainly hope you don't have girl (or boy) trouble! Do give the position a good try and then click on Read More to see the solution, notes, and numerous additional examples of the theme.

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It was September and the Coffee and Cake Checker Club was meeting for the first time after the usual summer hiatus. There was a good turnout this week with Sal and five of the boys present: Wayne, Delmer, Louie, Dan, and Mike. The "boys" were all over 50 years old, but nevertheless that's how Sal Westerman, the club's leader, thought of them.

Sal should have been happy. He loved these Saturday afternoon gatherings at The Beacon Cafe more than just about anything else. It was September, 1955, and the weather in Bismarck, North Dakota had started to turn cooler.

So it might surprise you to hear that Sal wasn't the least bit happy.

It all started when Deana, the proprietor of the Beacon Cafe in Bismarck, North Dakota announced that she had fresh lemon bars this afternoon, and Sal frowned. More like scowled, because Sal didn't like lemon bars, not one little bit. He just plain didn't like them, no matter how good they were.

"Lemon bars, Sal!" Dan Kemper, one of the boys, said in a teasing tone. "Don't you want to win one?"

"No, I do not," Sal said emphatically. He looked over to Deana's service counter. "Deana, don't you have anything else?" he asked her.

"Got some fudge brownies left over from yesterday," she said. "Half price for day-old. But the lemon bars are fresh and good." She frowned in turn. Sal had been coming to the Beacon for so long she knew exactly what he liked and didn't like, but she always felt a little insulted when her customers didn't care for her baked products. Everyone said she had the best desserts in town. Even Mayor Lips came here often for coffee and treats with his political pals.

Dan continued, "Come on, Sal, I know you've got a coffee and cake problem for us. Heck, you don't have to eat lemon bars if you don't want to--- more for us!" Dan said, and the rest of the boys added their agreement.

"Okay, okay," Sal said, "it just happens that Ed sent me a nice one." Ed was Sal's checker pen pal in Pennsylvania. "Guess I'll have to settle for those day-old brownies when you boys can't solve it."

They all laughed. "Hey, how about if we do solve it you have to eat a lemon bar?" Dan suggested.

"That's not even funny," Sal replied.

Deana wasn't laughing, either. "You make fun of my food, you leave," she threatened. Deana didn't get upset very often but when she did, you had better watch out.

Sal realized that things were going into the ditch in a hurry. "Okay, okay!" he said. "Let there be peace! I'll show you Ed's problem, and whether you solve it or not, I'll buy lemon bars for everyone and a brownie for myself! How's that sound?"

Everyone now smiled, even Deana. The tension was relieved and Sal set out the following position on one of the checkerboards in the big booth that the club always occupied.

"Ed calls this one Sweet Spot," Sal said. "What do you boys think?"

Five minutes passed with everyone scratching their heads and looking puzzled. "Five more minutes," Sal announced.

---

Can you win this one? And what's your take on lemon bars? (Be careful what you say; Deana might hear you.) When you've given this one a good try, click on Read More to see the solution and the conclusion of today's story.

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