1. The First Two Witches

A. Sibyl the Seer

 

“Sometimes I see things,” Sibyl said.

 

“What kinds of things?” I asked her again.

 

“Things that haven’t happened yet,” she explained again.

 

“I see,” I said, although I didn’t really see.

 

“What do you see?” Sibyl asked me, very interested.

 

“I’m not sure,” I said. “It hasn’t happened yet. And what do you see?” I asked her. Again.

 

We’d been going back-and-forth like this for a few minutes. Sibyl had called us, and we came to her house to talk. So far, we’d learned that she sees things. Whatever that meant.

 

“Murder,” Sibyl said. “Although the murder hasn’t happened yet.”

 

Sibyl was very pretty, but in a demure kind of way – in a way that petite women like her could be pretty. She wasn’t beautiful like Carol. Sibyl’s features were plain and understated in the way that the features of some models can be. It was the very “plainness” of her features – the fact that she looked average in every way – that made her pretty. While I would certainly notice her if I passed her on the street, I doubt that I’d quite remember what she looked like a few minutes later. Not at all like Carol.

 

My name is Mickey Maux. My last name is French, and it’s pronounced like the name of the rodent, “mouse.” Sometimes this causes problems for me, especially when I use my actual first name, “Mickey.” That’s why I usually go by “Mick.”

 

Sometimes, people don’t believe me when I tell them my name.

 

By profession, I’m a scientist. I did well for myself, and retired relatively young. Among the many contracts that I’ve worked on were some government ones, which allowed me access to their main computer center. At the time, I held a Top Security clearance. For all I know, I still do, although I’ve not tried to use it in a while.

 

While the government security people don’t know this, I put a “back door” into their computer system which allows me access to it when I need to. This has come in handy in several prior cases.

 

I know that I usually have a different view of the world than most. And that’s where Carol comes in. Carol is my beautiful wife. In addition to being stunningly gorgeous, she brings common sense into our relationship. I tend to be what others call eccentric, and sometimes I miss most of the “bigger picture.” I’m more detail oriented. Carol is much better at facts and real evidence.

 

This is why Carol is essential to our work. And did I mention that she’s beautiful?

 

When I talk about “work,” I’m referring to detective work. While I retired comfortably and we don’t need the money, Carol and I sometimes take on clients who are outside the realm of what the police are willing to do. Sometimes it’s wealthy clients who don’t want the details of their cases to appear in the tabloids. And sometimes it’s cases that are too strange for the police to be either willing, or able to handle.

 

I usually require a substantial payment up front. Again, it’s not because we need the money. It’s because we need to know that the client is serious. In this case, I’d give the client a discount. She wasn’t as wealthy as some of our clients have been. But she was very interesting.

 

This was our first meeting, and I didn’t know whether we’d be taking the case yet. I didn’t know anything about the case. Carol and I had come to meet with Sibyl so that she could explain it to us. It was good that Carol was here. I think that it made Sibyl more comfortable. Some women are uncomfortable if they’re alone with a man that they don’t know.

 

Sibyl had contacted us based on a recommendation that she’d received from a former client of ours. We were here to try to understand what it was that Sibyl needed, and to decide whether we thought that we could help. Sibyl had told us that she was a witch. Neither Carol nor I were exactly sure what that meant. We were about to learn.

 

“Whose murder do you see?” Carol asked.

 

“I don’t know him,” Sibyl said. “But I can see him very clearly.”

 

We were sitting in Sibyl’s family room. She lived in a large house in Connecticut on a heavily wooded lot. It reminded me of our house. Her husband was a contractor who builds houses like this one.

 

Carol and I were sitting on one of the couches. Sibyl was sitting on a mat on the floor, cross-legged. She likely used this mat and position for meditation, but I didn’t ask her.

 

Sibyl had the physique of a vegetarian. She was small-boned and thin. She was wearing very plain clothing: a pair of loose jeans, and a long-sleeved sweater with angular designs on it. She had shoved the sleeves of her sweater up to her elbows so that her forearms were exposed. Like her, her forearms were thin. She wore a simple necklace with a single large glass bead hanging from it, and a bracelet with small charms attached to it. I had no doubt that the glass bead and the charms meant specific things to her.

 

There was an incense burner on the coffee table, and some incense was burning in it. It gave the room a pleasant smell. It was sandalwood incense, which is considered by many to be the best incense for purifying the air, and “for cleansing the body's energy fields,” Sibyl had told us. Whatever that means.

 

While my body didn’t have “energy fields,” Sibyl’s did, or at least she thought so. This was probably why she was sitting “Indian style” with her back rigidly straight, although I’m sure that to her, the posture was relaxing. It wouldn’t have been for me. Being “relaxed” is mostly in the mind, anyway.

 

There was music playing on her sound system. It was playing at a low volume. It was the music of a shakuhachi[1]drifting aimlessly through a Zen meditation in sound by using the tone-color more than the notes themselves as the essential variant in the music.

 

The room had a large fireplace at the end opposite the mudroom. There were about a dozen large candles of different widths, heights, and colors in different kinds of holders for them. Several of them were burning. All of them had been melted down somewhat. It was obvious that she burned these candles regularly.

 

It was a beautiful fireplace, all done in natural stone. Whoever built it had taken time to fit the stones together in a way that made it look symmetric. That’s hard to do with real stone. I assumed that Sibyl’s husband – or his business – had built it.

 

There was some modern art hanging on two of the walls which alternated with the windows. And there were some small, framed, shamanic symbols on the third wall. Most of the third wall opened into the kitchen. I could see two teapots and half-a-dozen large cans of different kinds of tea on one of the counters in the kitchen.

 

Sibyl had made white tea for herself and for Carol. While white tea is mild and should be relatively odorless, I could smell the concoction because she had put spices into it.

 

Instead of tea, I’d asked for a bourbon and water. They had a bar setup in their dining room, so she’d poured me a bourbon before she sat down. She brought me another glass with some ice in it, and a bottle of mineral water from the fridge. It was a quart bottle that had already been opened.

 

While she probably didn’t drink alcohol, her husband probably did. He was in the construction business. And in that business, it’s hard not to. 

 

The fourth wall of the family room had bookcases and a large television set. I noticed a few incense burners that she had besides the one that she was already using. Two were up on the mantel, and a third was in the kitchen next to her cans of tea.

 

In addition to the incense burner that had burning incense in it, the coffee table had several magazines on it. From their covers, I could see that they were about yoga and meditation. There was also a large glass ball in a stand on the coffee table. It was about eight inches in diameter.

 

“Where did you see him?” Carol asked.

 

“In my crystal ball,” Sibyl said, turning slightly, and pointing to the glass ball on the coffee table. Because she was slender, her turn was graceful and effortless; like that of a ballerina.

 

“That ball looks like it’s clear glass,” I observed.

 

“It is,” Sibyl said. “That’s why it works. I don’t always keep it on the table. Sometimes when I meditate, I put it on the floor in front of me, and I stare into it. Sometimes it turns colors. And sometimes I see things.”

 

“Is that where you saw the murder?” Carol asked. “In the ball?”

 

“Yes,” Sibyl said.

 

“What does the man who is being murdered look like?” I asked.

 

“He’s tall and handsome,” Sibyl said.

 

“How tall is tall?” I asked.

 

“I would say that he’s over six feet tall,” Sibyl said. “But not too much more than that. He’s the size of some of the men that work for my husband.”

 

“And what do you mean by handsome?” Carol asked. “Can you describe him?”

 

“Yes,” Sibyl said. “He’s wearing a nice suit, and he has red wavy hair that’s neatly combed. I can tell from his appearance that he’s a neat person.”

 

“Would you recognize his face if you him on the street? Or if you saw a photo of him?” Carol asked.

 

“I’m not sure,” Sibyl said. “When I said that he was handsome, I was talking about the fact that his clothes and grooming made him look ‘handsome’ in a sense. But I didn’t get a clear vision of his face. I was using the word ‘handsome’ in a more generic way. I don’t think that I’d recognize his face. I didn’t really see his face.”

 

“Then we can’t really identify him, can we?” I asked.

 

“I think that I’d recognize him if I could be present with him,” Sibyl said. “I think that I’d remember his aura. He has a strong red aura, and I’d feel it.”

 

“His aura?” Carol asked. “What do you mean by that? What’s his aura?”

 

“He has a red aura,” Sibyl said, as if she had answered the question.

 

“I see,” I said. “And please remind me. What does a red aura mean?”

 

“Among other things, it means that his passions run high,” Sibyl explained. “And that he lives by his emotions and desires. He’s a passionate man.”

 

“That could be a lot of people,” Carol said. “What else does a red aura mean?”

 

“Red is tied to the south. The south is where the sun comes from up here in the northern hemisphere,” Sibyl said. Red connotes the element of fire, which is one of the four earthly elements. Red is linked to career goals, so this man’s ambitions run high. Red both attracts, and it magnetizes, so it indicates lust, passion, and physical desire. While red denotes strength, health, and vigor, it’s also associated with danger, enemies, and war.”

 

“That can be dangerous,” I said. “What else do you remember about him?”

 

“Not much,” Sibyl said. “But I saw him being murdered. Do you think that you can help?”

 

“How is he murdered?” I asked.

 

“He’s shot,” Sibyl said.

 

“Shot?” I asked. “Can you describe the assailant?”

 

“His aura is both black and indigo,” Sibyl said.

 

“What can you tell from that?” Carol asked. “Black and indigo?” she repeated.

 

“A black aura means that he has a dark energy. He’s pessimistic and unkind,” Sibyl said. “A black aura means that the individual could be capable of shape-shifting. On the other hand, an indigo aura means that he is an old soul who has wisdom. It’s very rare that any person has both of those. Usually when you have age and wisdom, you lose your dark energy. The fact that this man has both suggests that he’s basically evil.”

 

“Did you get a good look at him?” I asked. “Can you tell us what he looks like?”

 

I thought about the color black. Black was the darkness out of which God conjured light. It’s both the color of fashion, and the color of mourning. In this sense, despite not being a color, it’s much more expansive than any other color.

 

Renoir had once asserted that white and black were not actually colors. In one sense, he was right. But on the other hand, each is the ultimate expression of presence, and absence, respectively.

 

The notion of death has clung to black as far back as history takes us. Black has long been associated both with mourning, and with witchcraft. But despite its being ubiquitous, black is essential to modern fashion.

 

“No,” Sibyl said. “I won’t be able to see him completely from where I’ll be sitting.”

 

“What do you mean when you say ‘from where I’ll be sitting’?” Carol asked. “Are you going to be there?”

 

“Yes,” Sibyl said. “Of course.”

 

“Where does it happen?” I asked her.

 

“On a street-corner,” Sibyl said.

 

“Can you tell where the street-corner is?” I asked.

 

“No,” Sibyl said. “That’s why I’ll need to hire you two.”

 

“When will this murder take place?” I asked.

 

“I’m not sure,” she said. “Not this week. And not next week. But sometime after that.”

 

“We’d like to help you,” I said. “But frankly, we don’t have much to go on.”

 

“What do you mean?” she asked. “I’ve laid the whole thing out for you.”

 

“Maybe if you went through it once more?” Carol suggested.

 

“Sure,” Sibyl said. “A man will be killed sometime later than two weeks from now. He’s over six-feet tall, has red wavy hair, and he’s nicely dressed. He has a red aura. He’ll be shot on a street-corner by a man who has a black and indigo aura. I can’t actually see the assailant.”

 

“Why will he be murdered?” I asked

 

“I don’t know,” Sibyl said.

 

I took a sip of my bourbon, and watched the thin line of smoke wafting up from the incense. I was pessimistic that we could help her. And if she didn’t know either man, I wondered why she felt that it was her mission to intervene.

 

“And you’ll be there when it happens?” Carol asked.

 

“Yes, but I won’t be in a position where I can see the assailant’s face,” Sibyl explained.

 

“And will do you do when the murder happens?” I asked.

 

“I’ll try to get a look at the assailant but I won’t see him. So I’ll take a ride,” Sibyl said.

 

“And that’s all you see?” I asked.

 

“Yes,” Sibyl said. “But I think you should also talk to my sister. I’m going to call her.”

 

“Who’s your sister?” I asked.

 

“Arabella.”

 

“And why should we talk to Arabella?” I asked.

 

“Because she’s a mystic,” Sibyl said.

 

“I thought that you were a mystic,” Carol said.

 

“No,” Sibyl said. “I’m a seer. My sister is a mystic.”

 

“What’s the difference?” I asked.

 

“I can see into the future,” Sibyl said. “Arabella can’t. But she can see dimensions pertaining to the present that aren’t visible to the rest of us.”

 

“I see,” I said, not sure that I did. “What dimensions are those?” I asked.

 

“The fourth dimension,” Sibyl. “The dimension of dreams and souls.”

 

“Is that why you’re going to call Arabella?” I asked.

 

“Yes. She’ll be able to see who the murderer is,” Sibyl explained. “If we can’t stop the crime, then at least we can bring the killer to justice.”

 

“And what can Carol and I do to prevent this murder?” I asked.

 

“You’ll need to figure out who the killer is, and make sure that he’s not there that day. Or you need to figure out who the tall man with the red aura is, and make sure that he’s not there that day. Either one of those will work.”

 

“What day is it that this will occur?” I asked.

 

“I don’t know,” she said.

 

“Then how can we make sure that either one of them isn’t there on the day that we don’t know at a place that we don’t know?” I asked.

 

“That’s why I called you,” Sibyl said. “I need a good detective to figure that out.”

 

“Normally, I’d say that this sounds like a tough case,” I said. “But Carol and I like tough cases,” I added. “So we’ll take the case.”

 

“Yes, I already knew that,” Sibyl said.

 

“How did you know that?” I asked.

 

“Because I’m a seer,” she said. “Didn’t I mention that?”

 

We sat in silence. I smelled the scent of the incense, and contemplated the shakuhachi music.

 

“And sometimes I see things,” she added.

 

B. Arabella the Mystic

 

Carol and I went to visit Arabella. Sibyl had called her and arranged it.

 

Arabella lived in a house that was similar to Sibyl’s, but it was oriented differently, and decorated very differently.

 

While Sibyl’s house had an eastern entrance, and the family room had a southern exposure, Arabella’s house had a northern entrance, and a western exposure. Normally, I wouldn’t notice these things, but as I was to learn, feng shui was very important to both of these women.

 

Feng shui is the branch of Chinese Metaphysics classified as physiognomy, which is the observation of appearances through formulas and calculations. The feng shui practice comprises invisible forces that bind the universe, the earth, and humanity together. Together, these are called qi. It’s good to know about qi. Especially when you’re playing Scrabble.

 

Arabella’s house had a modern layout. Most of the ground floor was what’s called “open concept.” It was one big space. You could see any of the living areas on the ground floor from any of the other living areas. The kitchen took up about of the Southern wall, and about half of the Western wall. The other half of the Southern wall comprised floor-to-ceiling windows, which were letting the sun shine in right now.

 

The dining space was mostly in the Eastern part of the house, and the living space was mostly in the Northern part of the house. She had some guest children over that were watching television and eating snacks with her children in the living area. I could tell that Arabella loved the children and was a very proud mother. She seemed to enjoy the general din that was coming from the living area.

 

Arabella had let us in, and asked us to sit in the dining area at the center island. She was working in the kitchen, making cakes. We sat opposite her so that we could all talk comfortably. She offered each of us a cupcake.

 

Unlike Sibyl, Arabella looked sturdy, and I could see the strength in her hands when I watched her working with the rolling pin. She was rolling out a fondant. She was wearing a flowery dress tied at the waist with a sash. She had long hair, but had brushed it back and fastened it, presumably because she was baking today.

 

In addition to being a mystic, Arabella is a pastry chef. She operated a small business out of her home, baking cakes and pastries for some of the upscale restaurants in the area. I had always thought that baking pastry required some mysticism.

 

“So you and Sibyl are sisters?” Carol asked, just to start the conversation. “You don’t look alike.”

 

Arabella looked strong and rugged, unlike Sibyl, who was petite. Arabella had pronounced features. She was pretty too, but in a way that was quite unlike the neutral appearance of Sibyl. If I’d passed Arabella on the street, I’d be less likely to notice her, but if I did, I’d remember what she looked like. Her nose was more pronounced than most, and she had thick lips, giving her a more sensuous look.

 

“We’re not actually related, but we are sisters in the same coven,” Arabella explained.

 

“What’s a coven?” I asked.

 

“It is an organization of witches,” Arabella said. “We get together periodically for meetings.”

 

“I see,” Carol said. “I’m glad that now I understand that. When Sibyl had said ‘sisters,’ I’d just assumed that you had the same parents.”

 

“Actually, because we’re in the same coven, we’re probably closer together than most real sisters,” she said.

 

“Really?” I asked.

 

“Yes,” Arabella said. “I do have a sister by blood, but we don’t have much to do with each other. I feel much closer to Sibyl than I do to my blood-sister.”

 

“That’s too bad,” I said.

 

“Not really,” Arabella said. “While there’s a slight animus between me and my sister, that’s not unusual. Sibyl is like my best friend. But my sister can be petty at times.”

 

“I see,” I said. “I had misunderstood. I hate petty people. I keep a list of who they are so that I can avoid them.”

 

After a pause, she wiped her hands clean, and asked: “Can I get you each something to drink?”

 

“If you’re got some cold water and some ice, that would be great,” I said.

 

“I’d like that too,” Carol added.

 

Arabella got us each a glass of water with lots of ice.

 

“And whatever animus I hold toward my sister is her fault,” Arabella said. “She thinks that the fact that I’m a mystic is crazy. I don’t like the negative energy that she sometimes projects. She should be proud that I have a special gift.”

 

“I can see your point,” Carol said. “I think that would upset me too.”

 

“Some things are gifts that we’re born with. They’re absolutes,” Arabella said. “Other things are subjective.”

 

“Yes,” I said. “I’m a strong believer in absolutes. In fact, I’m actually a scientist.”

 

“I try to explain that to people,” Arabella said. “But many of them don’t understand it. It’s just like the concept of twelve inches,” she said.

 

“What about twelve inches?” I asked. “I haven’t heard of that as ‘a concept.’”

 

“Well, how do you know that twelve inches is a foot?” she asked.

 

“Because twelve inches is a foot,” I said.

 

“You’re obviously more of a religious type than a scientist,” she said. “You’re happy just believing that. And you believe it because you were told that when you were growing up. If you really want to know whether twelve inches is a foot, then you need to measure it both ways. That would be scientific.”

 

“You mean that I’d need to measure twelve inches with one ruler, and then measure one foot with a different ruler?” I asked.

 

“No,” she said. “Of course not. You can’t use a different ruler. You’ve got to use the same ruler. With different rulers, the measurements could be different.”

 

“But if I use the same ruler, they’ll be the same!” I exclaimed.

 

“You see?” she said. “You’re more of a religious type than a scientist. I could tell that when you first came into the house. You have an orange aura with some gray.”

 

“What does that tell you about me?” I asked.

 

“Your orange aura tells me that you have lots of creativity, and that the artistry that you employ by using it brings you peace. On the other hand, the gray shows me that you hold skepticism and uncertainty about others. And like I said before, you’re obviously religious. You were quick to believe that a foot had twelve inches in it.”

 

“Actually, I was raised in a religious family, but I stopped attending church once I had a choice,” I said. “Since then, I’ve become a born-again atheist. Are you at all religious?”

 

“I once was,” Arabella said. “But as a girl, I realized that I had the ability to see things that others didn’t. A lot of that made me skeptical of many religious teachings. I gradually stopped going to church.”

 

“I don’t ‘see things,’” I said. “But much of what’s in the Bible doesn’t add up.”

 

“Like what?” Arabella asked.

 

“You said it yourself,” I said. “You asked about twelve inches. Why do you think there were twelve apostles?”

 

Arabella looked puzzled. “I don’t know,” she finally said.

 

“Because Jesus’s people used the United States customary system of measurement,” I said. “Had Jesus really been from Nazareth, he’d have been on the metric system, and there’d only be ten apostles. Jesus was an American. That’s why there were twelve apostles. And that’s why there are twelve inches in a foot.”

 

Arabella thought about this, and looked confused. “But why was his name ‘Jesus’?” she asked.

 

“He was from New Jersey,” I said. “His parents came from Puerto Rico. Let me ask you a question,” I said. “Suppose it was very cold out. Suppose it was 0-degrees out. And suppose that the weatherman said that tomorrow it was going to be twice as cold. How cold is it going to be tomorrow?”

 

There was a long pause. Arabella didn’t know. So finally, I broke the silence. “And what do you mean by you ‘see things’?” I asked. “What does that mean?”

 

“I can see people’s thoughts and feelings,” Arabella said. “I can see relationships that most can’t. And I can see the auras that people give off. That tells me a lot about them.”

 

“You know, I can tell that people think that I’m a very good mind-reader,” I said. “I also have a great memory. Why, I can’t even remember the last time that I forgot something.”

 

Arabella looked uncertain.

 

“And what’s Carol’s aura?” I asked.

 

“She has a yellow aura,” Arabella said. “She’s full of optimism and energy.”

 

“How did you and Sibyl first meet?” I asked.

 

“She joined the coven that I was a member of,” Arabella said. “She has very strong powers. She’s a seer. She sees into the future. I can’t do that.”

 

“How do you know that she sees into the future?” Carol asked.

 

“The first time that I knew it was when our coven had gone up into the mountains for a weekend retreat,” she said. “She did some things that left me amazed and in a state of contemplation. She has strong powers.”

 

“What did you see? What did she do?” Carol asked.

 

“Well first, she dug a hole that wasn’t there,” Arabella said.

 

“When wasn’t it there? Before she dug it, or after?” I asked.

 

“Neither,” Arabella said.

 

“I’m not sure that I understand. Where was the hole?” I asked.

 

“What hole?” Arabella asked.

 

“The one that wasn’t there,” I said.

 

“Well you’ve just said it,” Arabella said. “It wasn’t there.”

 

This time I sat there looking confused.

 

“Then why did she dig it?” Carol asked.

 

“Because it wasn’t there,” Arabella said.

 

“But was it there after she dug it?” I asked.

 

“No,” Arabella said. “That’s how I knew that she had strong powers.”

 

“Did anyone ever tell you that you have a unique way of looking at the world?” I asked.

 

“Of course I do,” she said. “That’s why I’m in a coven. Why did Sibyl want you to see me?”

 

“Sibyl sees a murder that will happen in the future, and she hired us to try to prevent it,” I said.

 

“Who will be murdered?” she asked.

 

“Sibyl doesn’t know,” I said.

 

“Who’s the murderer?” she asked.

 

“Sibyl doesn’t know,” I said.

 

“When will it happen?” she asked.

 

“Sibyl doesn’t know,” I said.

 

“I understand completely,” she said. “Sure, I’d love to help.”

 

“I have a philosophical question for you,” I said. “If you try to fail, but you succeed, which of those have you done?”

 

Arabella looked like she wasn’t sure.

 

It would be good to have Arabella working with us. I wasn’t sure what she’d do for us other than to bake a helluva good cake. But even if that was all that she did, it would be plenty.

 

The children were starting to get loud, and play a little too roughly. Arabella yelled in to them: “Hey kids, let’s keep it down. If you behave, I’ll give you all some cake later.”

 

“OK, Mrs. Lawson,” the oldest looking kid said. “C’mon you guys, let’s behave!” he said to the others.

 

Gee, that boy was quick to comply,” I observed. “And the other kids seem to listen to him. Who’s he?”

 

“His name is Santa,” she said.

 

“Santa?” I asked. “That’s an unusual name. Is he your son?”

 

“No,” Arabella said. “He’s Suzy’s older brother. Their mother dropped them both off. She had some errands to run. Santa is always a big help. And he’s quite precocious. Sometimes he sits her at the counter when I’m baking, and we talk.”

 

“What do you talk about?” Carol asked.

 

“All kinds of things,” Arabella said. “He’s smart beyond his years.”

 

C. Carol and Mick Ponder the Murder

 

Carol and I had our dinner out by the pool.

 

We live at the end of a long driveway that winds through the woods. The woods surround our back yard, where our pool is. Many animals come through when they don’t sense people: squirrels, deer, foxes, raccoons, wildcats, groundhogs, and even the occasional bear. And we have lots of birds. We can hear their songs when we’re quiet.

 

Our pool area is surrounded by a wrought-iron fence. Not only is a fence required by code to keep out small children, but it keeps most of the animals out as well. I wouldn’t want either to drown in our pool.

 

And we’ve grown trumpet vines along one side of the fence. The flowers of the trumpet vine are long and narrow, and they attract hummingbirds. While hummingbirds don’t have a song, they’re always fascinating to watch. They’re only two to three inches long, they’re very colorful, and they can move very quickly.

 

We have a gas grille and an offset smoker out by the pool just for nights like this. I’d put a chicken in the smoker earlier this afternoon, and was taking some sliced vegetables out to cook on the gas grille: eggplant, zucchini, and peppers, all with some olive oil on them. I’d also brought myself some iced tea.

 

I started the grille, and once it was heated, I started grilling the vegetables. They cook very quickly once the grill is hot.

 

Carol came out to join me. She also brought an iced tea for herself, and some coleslaw that she’d made. Carol always puts black mustard seeds in the coleslaw, because she knows that I really appreciate it when she does that. Mustard seeds, a little bit of grated carrot, and some minced onions make a tasty coleslaw. 

 

Carol sat at the table, and I kept one eye on the vegetables that were grilling, and my other eye on Carol. She had put on something nice and summery, and it was refreshing just to look at her. Maybe we’d go for a swim when it started getting dark out. We have a firepit, and sometimes I make a fire, and we go swimming at night.

 

“What do you think of this case?” Carol asked.

 

“It’s our most challenging case yet,” I said.

 

I turned all of the vegetables while thinking in silence.

 

“I’m not sure what we’re even supposed to do. Are you?” she asked.

 

“It’s pretty clear,” I said. “There’s a man who is going to be murdered. He’ll be shot. It won’t be this week or next week, but it will be sometime after that. He has red hair and no face. How hard can that be?” I asked.

 

Carol was obviously enjoying relaxing. Now that we were both inside the fence, some deer emerged from the woods, and started munching on our lawn. What deer “see” is motion. When everything is still, they don’t see you. But deer can smell you if the breeze is headed toward them. So when the deer are out, I try not to move too quickly. I don’t want to scare them off.

 

Their focus on motion is why deer will “freeze” when you step out of the house unexpectedly to find them munching on the lawn. They think that you won’t notice them if they’re still enough. Like people, they think that others view the world the same way that they do. Many have been tripped-up by this notion.

 

I pulled the vegetables off the grille, putting them on a plate with tongs, and I turned the grille off. I put the plate on the table, and then got the chicken. It had smoked for about four hours. I covered it with foil and let it sit for a few minutes so the juices could settle, and then I cut it up. A smoked chicken doesn’t have nearly as much juice as a roasted chicken, so it needn’t sit very long.

 

We served ourselves and started eating.

 

“How do you think we should start this case?” Carol asked.

 

“Maybe we should hold a séance,” I suggested. “We can have Sibyl and Arabella, and ask them to get a few more people from their coven. While I can look into the future, maybe if we get some more witches into the room, things will become clearer. Right now, we have too little to go on.”

 

“That’s a great idea,” Carol said. “I used to have séances when I was a girl. I’d get together with some of my friends. We’d light some candles and turn the lights out. Then all we’d hold hands to form a chain, and we’d chant things. It was a lot of fun.”

 

“Did you learn anything from it?” I asked.

 

“Yes,” Carol said. “That séances can be fun. They’d also give you a chance to focus on a problem. Sometimes, just by focusing, you’ll see things that hadn’t occurred to you before.”

 

“That’s what I always tell students,” I said. “Speaking of which, tomorrow morning I’ll be going to the High School to talk to Pete Fletcher’s class.”

 

Pete Fletcher taught the Advanced Science class at the local High School. He and I had become friends a while back, and I occasionally visit his class to show the students some interesting things that they’ve probably not seen in books. I’d been a professor on-and-off during my working days, and I always liked talking to interested students.

 

“That’s good,” Carol said. “You really enjoy talking to the students, don’t you?”

 

“I’ve always liked talking to people that are interested in learning,” I said. “The problem with many High Schools is that some classes are full of kids that aren’t interested in anything. You can’t really ‘teach’ to classes like that. But it’s fun to talk to the kids that really want to be there.”

 

“What are you going to talk about?” Carol asked.

 

“The stars on the flag,” I said.

 

“Is this about politics?” Carol asked.

 

“No,” I said. “It’s about lattice structures.”

 

Carol’s eyes glazed over.

 

“What else did you do at séances?” I asked.

 

“Sometimes we’d get out a Ouija board,” Carol said.

 

“Those don’t actually work, do they?” I asked.

 

“It depends,” Carol said, laughing. “What you have to do is to get together with a few of your friends ahead of time, and arrange to say certain things. It’s good for lots of laughs.”

 

We’d had a case involving a Ouija board recently.[2]

 

“I think that we need to hold a séance with a real coven of witches,” I said.

 

“That’ll be worth much more than a few laughs,” Carol said.

 

“I hate to bring it up, but witches make lots of my insecurities come to the forefront,” I said.

 

“Why?” Carol asked.

 

“Some bad associations that I’d prefer not to relive now,” I said. “But a few things have been bothering me. Not you, of course. You I love. But a few things have been bothering me.”

 

“Like what?” Carol asked.

 

“Witches,” I said. “I think that I’d like to talk to a shrink to work some of these things out.”

 

“If that will help you, you should,” Carol said. “Where would you find one?”

 

“I know of a few,” I said. “I think I’ll call one of them and pay him a visit. It couldn’t hurt.”

 

“If that would make you more comfortable, you should,” Carol said.

 

D. The Stars on the American Flag 

 

In the morning, I called a shrink and made an appointment. Then I went to the High School to teach a class. I try to do this when I can. Pete Fletcher is a Science teacher there. He and I had struck up a friendship after I’d first come to the High School several years ago to talk to the students.

 

Since I’d been a professor at several universities, Pete liked it when I came and gave the advanced class something to think about. I always tried to show them interesting problems that I knew weren’t in their books.

 

The High School is both large and sprawling: it has several wings, and up to four floors in some places. People who aren’t students or teachers need to check in at the front office where there’s a police officer standing there for security.

 

I parked in a large lot in front of the visitor entrance, and I went in. I knew the drill. You go to the main office where the staff ignores you until you do a loud “Ahem!” Then they look at you feigning surprise, and they ask you to sign in.

 

After I’d done that, I told them that I was visiting Mr. Fletcher’s class. They cleared me to enter. The cop gave me a curious stare.

 

I took an Ell, went up a couple of flights of stairs, and found Pete’s science classroom. It was good to see Pete, and good to see the students. These were the top students in the school. They were interested in learning more about science, and they came to class to learn, not to goof off.

I enjoy working with young eager minds, and sometimes miss the teaching that I used to do at universities. I stopped at the doorway and waved to Pete. Pete waved me in, and we shook hands.

 

Then he said to me, “Professor Maux, we had just been discussing some of the recent stories in the news that could give us, here in the United States of America, some new states. In particular, there’s been talk in California of splitting California into two states, and there was a recent vote in Puerto Rico that shows that many people there want Puerto Rico to formally become a new state. That would give our country two new states. I told them that you though there was a problem with that.”

 

“I’d run into Mr. Fletcher last week in a coffee shop, and he told me that you’d been discussing the implication of adding two states to our country. I wanted to explain why that’s not possible, given the structure of our flag.”

 

“What do you mean by ‘it’s not possible,’ Professor Maux?” Pete Fletcher asked. “Why can’t we add one or two new states to our country?”

 

“We can’t add one or two states because the stars on our flag were placed interstitially. We could add three states, but not just one or two,” I explained. “The flag wouldn’t work.”

 

“What do you mean by ‘interstitially’?” Pete asked. “And what do you mean by ‘The flag wouldn’t work’?”

 

“Generically, an ‘interstice’ is space between objects. For example, a crystalline structure comprises a repeating periodic set of cells that are aligned on what’s called a ‘Bravais lattice.’” 

 

“That lattice determines the periodicity of the crystal, which makes the crystal a filter for specific colors of light. That’s why crystals are pretty. Those cells have simple geometric structures. Each cell comprises atoms at the Bravais points, and usually, interstitial atoms as well. The interstitial atoms are the ones within each of the identical cells of the crystal; they’re the atoms that lie between the Bravais points. The size of the cells defined by the Bravais points determines the crystal’s periodicity, hence the color that we see when we look at it. Its periodicity makes it a filter: only certain colors can pass through it, or can be reflected easily.”

 

I let the class digest that, and then I continued: “Our flag has fifty stars on it. These are not arranged in simple rows and columns. Had it been arranged like this, we’d have five rows of ten stars, or something else like that. But that’s not how our flag is structured.” I went to the whiteboard, and showed the structure of the fifty stars on the American flag.

 

“If we look at this like it’s a crystal, we’d say that there are thirty Bravais points, arranged in five rows of six columns. Notice that 5X6=30. And there are twenty interstitial points, arranged as four rows of five columns. Again, notice that 4X5=20. So, the total number of stars on the flag is 30+20 = 50. We have fifty states, and this works quite well.”

 

“In this case, the Bravais matrix isn’t square, so the structure is wider than it is tall. But it’s nearly square: note that there are six columns and five rows. In general, if the Bravais points are in an n-by-(n+1) structure, then the interstitial points will form an (n-1)-by-n structure, so the number of stars, hence, states, that we can have is n((n+1)+(n-1)) = 2n2.”

 

“In the case of the American flag, n=5, and 2n2 = 50. If we let n=6, then 2n2 = 72, so we’d need to add 22 states to the country to keep the same dimensions. But let’s not enforce the n-by-(n+1) aspect-ratio, and let’s see whether we could place 51 or 52 stars in a way that makes an interstitial structure.”

 

“Instead of using the numbers 5 or 6 to begin with, let’s take a structure of Bravais points that’s square. Let’s say that it’s n-by-n points. Then we’ll generalize it by augmenting that structure.”

I went to the whiteboard and drew a succession of three square structures of Bravais points with their interstices. And then I explained the drawings to the class. (See Figure 2.)

 

“What I’ve drawn is a 2X2 lattice with a 1X1 interstice, a 3X3 lattice with 2X2 interstices, and a 4X4 lattice with 3X3 interstices. Notice that the total number of points in each case is 12+22 = 5, 22+32 = 16, and 32+42 = 25, respectively. So in general, for an n-by-n array of interstitial points, we’ll have an (n+1)(n+1) array of Bravais points. Square lattices can then have only n2+(n+1)2 points in them, for n0.”

 

I allowed the students a minute to study the three drawings, and then I continued.

“If we were restricted to a square lattice, we could have 5, 16, 25, or 61 states. Or even more than that, but it couldn’t be just any number.”

 

“The first possibility after 25 is 61. And if we wanted to have 61 states, we would have to add 11 new states to our country make the American flag a square lattice. This is why the American flag is not a square lattice, but instead starts with a 5X6 set of points. We could not configure 50 states as an interstitial square lattice. The 5X6 lattice is a rectangle.”

 

“Since we are not talking about adding 11 states, let’s see what the least number of states is that we’d need to add to maintain the interstitial structure of the stars, if it were simply rectangular, but not square.”

 

“The question is: ‘How many stars must we add to a square to maintain the rectangular structure of the lattice?’ Let me draw it on the whiteboard so that it will be clear.” (See Figure 3.)

 

Next to the 32+42 lattice, I added two columns to extend the structure, as shown in the figure.

“We can see that for the square, 4X4 set of Bravais points, we started with twenty-five stars, since 32+42 = 52 = 25. To add to the square - to extend it into a rectangle - requires that we add 3 interstitial points and 4 Bravais points. This keeps the lattice complete. Therefore, we can add k sets of 3+4 = 7 points, and still have a complete, rectangular structure.”

 

“In other words, for a rectangle that’s n rows by n+k columns of interstitial points, we’ll have n2+(n+1)2 points – Bravais plus interstitial - in the original n-by-n square, and we’ll add n+(n+1) new points for each of the additional columns added to the square. So, the number of stars allowed on the American flag, regardless of dimension of the rectangle is:

 

n2 + (n+1)2 + k(2n+1) = 2n2 + (2n+1)(k+1), with n

 

Therefore, we can’t just split California into two states, or just add Puerto Rico as a new state, nor can we simply do both. While we can choose n and k to accommodate 50 states (e.g., n=4 with k=1), there’s no set {n,k} that will give us 52 states.”

 

“So how many states would we need?” asked one of the students.

 

“We could make flags like this for 53 states. This would work if we added Puerto Rico, and split California into three states. Or if we added Puerto Rico, split California into two states, and then chose another state to split into two as well.”

 

“What other state could be split into two states?” the teacher, Mr. Fletcher, asked.

 

“Well Texas is pretty big. We could probably split it into two states: Austin, and then the rest of it. That would make a lot of people in Texas happy,” I said.

 

“How would you put 53 states on the flag?” asked another student. “I don’t see it.”

 

“There are actually three ways to make an interstitial grid of 53 points,” I said. “We could have 2 Bravais rows of 18 stars each, with an interstitial row of 17 stars. This would look funny. Or we could have 3 Bravais rows of 11 stars each, with 2 interstitial rows of 10 stars each. This would look kind of funny too. Or finally, we could have 4 Bravais rows of 8 stars each, with 3 interstitial rows of 7 stars each. This would look a lot like the flag looks today.”

 

“Can’t you transpose those to make three other arrangements?” asked Mr. Fletcher.

 

“Yes, of course,” I said. “Good point. What Mr. Fletcher means is that instead of 2 Bravais rows of 18 stars each, we could have 18 Bravais rows of 2 stars each. And the same for the other two arrangements. So, there are six ways to put 53 stars on the flag in an interstitial rectangle.”

 

 “After 53 states, what’s the next number of states that we could have?” he asked.

 

“The next solution is for 56 stars, but this is also long and skinny: 2 Bravais rows of 19 stars each, and an interstitial row of 18 stars. It would look funny. We really don’t get pleasing shapes until we reach 59, 60, and 61 stars, all of which have fairly symmetrical solutions. Therefore, if we added 9, 10, or 11 states, we’d also be in pretty good shape. But we can’t add just 2.”

 

A student raised his hand, and I called on him.

 

“How did Kamala Harris put 51 stars on the flag?” he asked. “You said that the first number that works after 50 is 53. But she did it with 51.”

 

“Ah!” I said. “But she didn’t use a rectangular arrangement.”

 

“She didn’t?” he asked. “What did she use?”

 

“She used a ‘Harris hexagon,’” I said.

 

“What’s a ‘Harris hexagon’?” he asked.

 

“I’m out of time now, but maybe next time I’ll talk about ‘Harris hexagons,’” I said.

 

Finally, I ended the lecture: “I’d like to thank my good friend, Mr. Fletcher, for inviting me to speak to you today, and I’m looking forward to my next visit.”

 

“Let’s thank Professor Maux,” Pete said to the class, and they gave me a round of applause. I waved goodbye, and left. I always enjoy talking to bright young people that are interested in learning new things. Many of us lose this interest as we age, which is unfortunate. Harris Hexagons? It was a new idea.

 

And it made me think about the witches that Carol and I had met. They seemed normal. But they had some extra dimensions to them. I would talk to the students about more dimensions next time.

 

E. I Visit Dr. Shuar

 

I had made an appointment for the morning following my visit to Mr. Fletcher’s class. So I went to see the shrink. His name was Dr. Shuar.

 

The receptionist stood and ushered me into his office. He was sitting in a comfortable-looking lounge chair, and stood to welcome me. His face had a nervous tick in it.

 

“Have a seat,” he said, indicating the lounge chair opposite his. His face ticked. There was a coffee table between the two chairs. “Or if you’d be more comfortable, you can lie in the chaise over there. I’ll pull my chair over.” His face ticked again.

 

“I’ll take the seat facing you,” I said.

 

“Would you like some coffee?” he asked, his face ticking.

 

“No thanks,” I said. “I’m driving.”

 

I was starting to wonder how much this was going to cost.

 

“I see,” he said. “I’m Dr. Shuar. What’s your name?” His face ticked.

 

“I’m Dr. Maux,” I said. I usually don’t bother with formal titles unless the other person starts it.

 

“I’m a psychiatrist,” he said, his face ticking furiously. Then he calmed down.

 

“Really?” I asked. “I thought that the Shuar were an indigenous tribe of Ecuador and Peru that specialized in making shrunken heads[3]. Is that why you became a shrink?”

 

“My name was actually shortened from something longer,” he said, his faced ticking.

 

“It couldn’t have been shortened from something shorter,” I observed.

 

“Quite,” he said, his face ticking. “Now what seems to be the problem?”

 

“Can I have some coffee?” I asked.

 

His face ticked. “Certainly,” he said. “It’s over there.” He pointed to an automatic machine on the counter. “Help yourself.” His face ticked again.

 

I got up and made a cup of coffee. “Would you like one?” I asked him as I was preparing mine.

 

“No, thank you,” he said. “Now what seems to be the problem?” he asked again, his face ticking.

 

“Witches,” I said.

 

“I see,” he said. “And how long have you had this problem.”
 

“It hasn’t started yet,” I said.

 

“I see,” he said, his face ticking.

 

“Of course there was that one Halloween when some witches gave me some candy,” I said.

 

He paused, and looked with interest. His face ticked. He looked like he was anticipating some salacious details.

 

“What kind of witches? And when was that?” he asked, smiling, with his lips twitching.

 

“I was about four,” I said. “Maybe five.”

 

He looked disappointed. “Did they scare you?” he asked, his face ticking.

 

“No,” I said. “They were nice witches. And they gave me some nice candy. Did I mention that they gave me some nice candy?”
 

“Yes,” he said, his face twitching. “Then what’s the problem with witches?” he asked, his face twitching some more.

 

“I have to go meet some,” I said.

 

He looked at me like he thought I was crazy, his face ticking. “Why?” he asked.

 

“Because,” I said. “Maybe I shouldn’t have come here. My wife thinks that I always go to other people to solve my problems. How can I prove her wrong?” I asked.

 

Dr. Shuar looked confused, and his face twitched. “Maybe you can’t prove her wrong,” he said. “And maybe you’re just a little paranoid.”

 

“I know that I’m not paranoid,” I said. “Even though I think that lots of people think that I am.”

 

Dr. Shuar looked confused, and his face twitched some more. “I’m sure that they don’t.”

 

“That’s reassuring,” I said. “I try not to be pessimistic, but whenever I do, it doesn’t work.”

 

Dr. Shuar’s face twitched. “The key is to be happy.”

 

“That’s easy for us to say,” I replied. “But it’s not easy for dwarves to say.”

 

Dr. Shuar’s face twitched, and he looked puzzled. “Why not?” he asked.

 

“Because six out of seven dwarves aren’t ‘Happy,’” I said. “I learned that when I was a kid.”

 

Dr. Shuar’s face twitched some more. “I see,” Dr. Shuar said. “Well I don’t think you should worry about that.” His face twitched some more. “Is there anything else you’d like to discuss?”

 

“Yes!” I said. “You should ask me about my narcissism!!!” I exclaimed, beaming at him.

 

“Do you think that you’re narcissistic?” Dr. Shuar asked.

 

“Sometimes not,” I said. “But at other times, I just don’t know what’s coming over Me, Mmee, Mmmeee, and Mmmmeeee!”

 

“I see,” Dr. Shuar said, twitching. “Do you have any phobias?” he asked.

 

“Like what?” I asked. “What’s a phobia?”

 

“A phobia is a fear. Some people are afraid of the dark,” Dr. Shuar said, his face twitching. “Others are afraid of spiders.” His face twitched dramatically when he mentioned spiders. My hunch was that he was afraid of spiders. I watched his face twitching.

 

“Well, there’s hippopotomonstrosesquippedaliophobia,” I said. “I’ve got that.”

 

Dr. Shuar looked confused. His face twitched. “Hippopoto… I’m sorry, but I didn’t get that. What’s that?” he asked.

 

“Hippopotomonstrosesquippedaliophobia?” I asked. “That’s the fear of long words.”

 

“I see,” Dr. Shuar said.

 

“Well this has been very helpful,” I said. “I think that I’ll save the rest for next time.”

 

Dr. Shuar’s face twitched as I let myself out.

 

 

 

 

[1] An end-blown Japanese bamboo flute.

[2] From the book “The Wuhan RBG Virus.”

[3] True.