Minako plucked a particularly lucky prime number from her garden to ensure the robot exhibition wouldn’t be too crowded. She had been planning the excursion to Universal Studios Japan for months and everything had to be perfect. She made sure Hachi had plenty of dog food and water and then went outside.
Her house stood on the edge of the park containing Mandai Lake. To tell the truth it was more of a glorified pond than a lake, but that didn’t in any way detract from the fact that her house was one of the grandest overlooking the water. Her rival’s house loomed on the opposite shore, a concrete eyesore as ugly as it was large. Exactly halfway between the two houses stood the neighborhood’s most admired residence – a stately compound and garden where Minako’s son and her rival’s worthless daughter lived.
Minako leaned on her walking stick as she walked towards her son’s house. She carried a portable radio and listened to a discussion of the previous day’s baseball games. The sun was shining and the autumn leaves on the trees surrounding the lake were a brilliant red. People from the neighborhood were doing their daily exercise and Minako acknowledged their greetings with a slight incline of her head.
When she had almost reached her son’s house, the front gate opened and Ishikawa Naomi stepped out. She was a thin, imperious woman with hair the color of cigarette ash. Her sweater was decorated with a Mickey Mouse logo.
Minako switched off the radio and placed it in her handbag. “Good morning,” she said, trying to keep the scowl from her face. She wasn’t going to let Ishikawa’s presence interfere with her perfect day with her granddaughter.
She had never been able to determine Ishikawa’s exact age, but her rival was eighty or eight-one years old, at most a year older than Minako. If someone in the neighborhood had a problem the priests at Sumiyoshi Taisha couldn’t solve, they went to see either Minako or Ishikawa. Like her grandmother and mother before her, Minako was the finest practitioner of number magic in all of Osaka. Ishikawa dabbled in the magic of light and shadow, and while admittedly she possessed the ability to perform a handful of fancy tricks with fireworks, she lacked an understanding of how things really added up.
The two of them had once been friends. Ishikawa had been born in Tokyo but moved to the south of Osaka and attended the same junior high school as Minako. They were both smart enough not to fall out over as something as trivial as a boy; instead, their friendship had ended when Ishikawa put a curse on the Hanshin Tigers baseball team. Minako could see no possible way to remain friends with someone who would stoop to that kind of behavior.
Ayumi followed Ishikawa out of the gate. The four-year-old was wearing a bright pink jumpsuit plastered with Disney characters. Her face lit up with joy when she saw Minako. “Obaachan!”
Ishikawa smiled at Minako. “It’s such a lovely day that Ayumi asked me to take her to the park.”
Smiling, damned villain! “We’re going to see the robot exhibition at Universal Studios,” Minako told her granddaughter.
Ayumi squealed in delight. “Robots!”
“The weather is so nice that it would be a shame to spend a day waiting in lines,” Ishikawa said. “We’re going to Nagai Park. You can see the robots another time, sweetheart.”
“Today is my day with Ayumi,” Minako insisted.
“There seems to have been a misunderstanding,” Ishikawa replied. “Last week I arranged with Kentaro for us to swap days. You can see Ayumi tomorrow. Kentaro should have told you that.”
Minako counted to five before replying. It might be true Ishikawa had made a deal with Kentaro – her son was dedicated to his work and absent-minded enough to forget to tell her about a change of plans – but without her own agreement it was a moot point. “I’ve had this day planned for months,” she told Ishikawa. “You can go to the park any time.”
Ever since Ayumi had born, Ishikawa had tried to buy her affection with increasingly extravagant gifts and excursions. Minako wasn’t going to stand for any interference in her plans. She had had to call in a lot of favors to convince the robot exhibit’s chief engineer to agree to give them a personal tour. Admittedly, Ayumi probably didn’t care who was showing her around, just as long as there were shiny and noisy robots.
“We are going to the park,” Ishikawa said.
Minako met Ishikawa’s steely gaze without flinching. “Universal Studios.”
Minako and Ishikawa’s curses and counter-curses had earned them the nicknames of Tiger and Dragon after Takeda Shingen and Uesugi Kenshin, the two great feuding warlords of the Warring States Era. Whether Minako was the tiger or the dragon depended on whom you asked.
Then calamity had struck. Despite their best efforts, their children had fallen in love and married. Ishikawa had given a gift of salt to Minako to seal an uneasy truce. When Kenshin had heard that another opponent had blocked Shingen’s access to much-needed salt, he had sent a gift of salt accompanied by the words, “I do not fight with salt, but with the sword.”
Ishikawa was one of those annoying people who insisted on adding salt to every dish.
After that Minako and Ishikawa had to content themselves by competing to see who could give the more insulting gift. Ishikawa had given Minako an empty bottle of outrageously expensive perfume. Minako had retaliated by purchasing tickets to Tokyo Disneyland and an accommodation voucher for Disney Resort and then waiting a year for the tickets to expire before giving them to Ishikawa.
Ishikawa turned to Ayumi. “Run inside and get your hat, sweetheart. The sun is going to be strong today.”
Ayumi hurried inside.
“Nagai Park.” Shadows fell across the street. Clouds had appeared and hidden the sun.
Minako wasn’t intimidated by Ishikawa’s theatrics. She chanted her personal number. 375. Minako. Mittsu. Nana. Go. Three. Seven. Five. The air around her hummed with infinite possibilities. “Universal Studios.”
“Nagai Park.” The clouds parted and a ray of light streamed into Minako’s eyes, forcing her to look away. Tiger and Dragon. It was time to rid herself of this impudent rival once and for all.
She opened the locket around her neck and removed the golden bead with the number 1260 etched onto it. Vampire numbers had an even number of digits that could be equally divided into two so-called fangs. The fangs were factors of the number and contained all of its digits. 1260 was the smallest of the vampire numbers. Its fangs 21 and 60, when multiplied, equaled 1260.
“Universal Studios.” She flung the bead at Ishikawa.
The vampire number flew towards Ishikawa’s throat.
Particles of pulsating light surrounded Ishikawa and the bead stopped in mid-air. Minako wasn’t going to be thwarted by a tacky light show. She commanded the bead forward.
The lights pulsed even brighter and the bead remained fixed in place.
Minako redoubled her efforts. The bead slid forward, but stopped again. Her knees were weakening and she leaned more heavily on her walking stick.
Unnatural shadows crept along the pathway. The people walking around the lake turned and fled from the dueling women.
Shadowy figures surrounded Minako, clutching at her with claw-like hands. She pressed the button on her walking stick, which extended it to its full height of 153 centimeters. She raised the stick like a staff and called upon the power of 153, a majestic number that was the sum of the first 17 integers, as well as being the smallest 3-digit number, which was the sum of the cubes of its digits.
The shadows broke apart and 153 darts of energy extinguished the pulsating lights surrounding Ishikawa. The vampire number streaked forward.
“Obaachan!” Ayumi screamed.
No! Minako had been so focused on vanquishing her old foe that she had entirely forgotten her granddaughter. She directed the vampire number away from Ishikawa and it exploded into the wall. Light burst from the wall, forcing Minako to shield her eyes.
When she looked back, Ishikawa leaned against the wall, her face wreathed in shadow. Where Ayumi had stood, there was now a shadow in the form of a circle burned into the ground.
Minako’s heart lurched. What had she done?
“Ayumi!” Ishikawa called out, but there was no answer.
Minako leaned against her staff and hobbled over to the shadow. It wasn’t a circle. It was a zero. A shadow number.
At least there was still hope. Perhaps Ayumi was not dead.
In the olden days it had been more common for yokai, demons, and ghosts to walk the earth, but now they had mostly retreated to the shadow realm. One of them had taken Minako’s granddaughter.
Ishikawa turned to face Minako, who braced herself for a barrage of accusations.
“This is our fault,” Ishikawa said. “We endangered the life of our granddaughter.” She reached forward and clasped Minako’s hand. “We caused this problem. We will solve it. We will find our granddaughter.”
Minako was too surprised to do anything except nod. She had forgotten how practical Ishikawa could be. However, she took the chance to examine her hand to see if Ishikawa had laid a curse upon her. She couldn’t detect any enchantment.
“I will find where Ayumi has gone,” she announced. Whenever she was going out with Ayumi, she kept a bag of 131 marbles on hand, in case she was separated from her granddaughter. She took the bag from her handbag and untied it, letting marbles spill onto the shadow zero.
131 was a prime number and the sum of the squares of its digits, 1 + 9 + 1 = 11 was a daughter prime. She scooped up the marbles back into the bag, leaving 11 in the circle. The sum of the squares of 11’s digits, 1 + 1 = 2, a granddaughter prime. She scooped up all but the final two marbles and closed her eyes.
She pictured Ayumi, the sweetest child a grandmother could ever hope for. In response, she saw a vision of Dotombori, one of Osaka’s main pedestrian streets—but this was only a reflection of the real street, filled with shadows. Ayumi stood underneath a seafood restaurant with a giant shadowy mechanical crab on its roof. She wore her pink Disney jumpsuit and her eyes were filled with dark fire. “I am waiting for you,” she cackled in an unearthly voice.
The combination of Ishikawa and Minako’s magic had unleashed a zero shadow, which had taken control of their granddaughter. The sight of Ayumi transformed sickened and enraged Minako. She was going to break some shadowy heads.
“We need to get her back,” Ishikawa said after Minako had explained what she’d seen. “There’s an entrance to the shadow realm at the back of a karaoke bar on Dotombori.”
Trust Ishikawa to know that. When people had problems with the shadow realm, Ishikawa was the one they went to.
“I’ll need to get some numbers from my garden,” Minako said.
Ishikawa nodded. “I’ll prepare some shadow and light.”
“I’ll collect you in thirty minutes,” Minako said. She was tired and her scooter was already loaded into the back of her van. She waved her staff at a terrified neighbor who had been watching the conflict from a distance. “I need to borrow your bicycle, Nakamura-san,” she commanded.
The startled woman reluctantly rode over and presented the bicycle.
“I’ll make sure to triple-bless your next lottery ticket,” Minako said. She climbed onto the bicycle and rode back to her house.
Hachi was always excited when she returned home. The black and brown dachshund barked happily and ran around in circles.
“We’ve unleashed the forces of darkness,” Minako told him.
Hachi replied with a concerned woof.
“It’s worse than that,” Minako continued. “They’ve got Ayumi.”
“I know. This is going to require my special supply.” She hurried to her number garden, located next to her bedroom. The room received plenty of sun and protected the numbers from the elements. She had planted one hundred 100 Yen coins in a long wooden trough she had bought from a pig farmer. Each of the coins had been given to her in payment for a task she had performed and now served as numerical fertilizer.
The numbers themselves grew in the form of different colored petals on a green-stemmed plant covered in thorns. Some of the numbers had taken tens of years to grow, but she plucked them all. Her granddaughter’s abduction counted as a special occasion. She stuffed the numbers into a pouch, which she placed in her handbag.
Hachi gave an approving woof.
“Exactly. It’s best to be prepared. I think it’s best if we don’t let Kentaro know for the moment.” Her son was a math lecturer but didn’t have the same talent with numbers that Minako did. He would only worry.
“You want to come with me? I’m not sure that’s a good idea.”
“Well, yes, I know you love Ayumi, but it will be dangerous.”
“Okay, then.” She hobbled to the garage and opened the van; Hachi jumped onto the back seat. “Wait here,” she said and hurried to the bathroom. It was better to face the horrors of the shadow realm on an empty bladder.
Afterwards she drove to Ishikawa’s house. Ishikawa was waiting out the front, wearing a mirrored samurai helmet on top of her head.
“Take that ridiculous thing off,” Minako insisted. “I won’t be able to see a thing with the light reflecting in my eyes.”
Ishikawa glared at her, but removed the helmet and covered it with a blanket.
The traffic was bad, but Ishikawa used her power to change all of their traffic lights to green. Minako had obviously underestimated the value of Ishikawa’s magic.
Minako drove into one of the back allies connecting to Dotombori. She parked at the back entrance of a three-storey karaoke bar, choosing to ignore the sign informing them that the spaces were strictly reserved for employee parking.
She went around to the back of the van and lowered the ramp for her scooter. She clambered onboard and Hachi jumped into the front basket.
Ishikawa took a piece of black chalk and traced a circle around the van. “This will curse anyone who tries to interfere with the van.” Her powers were indeed more valuable than Minako had suspected.
The back of the karaoke bar had a flickering neon sign that read Karaoke 24 Hours.
“This is the entrance to the shadow realm,” Ishikawa announced.
“How do we get in?”
Ishikawa took the chalk and drew an archway on the wall. “Straight ahead.”
“Are you sure?”
Minako took a deep breath. “Well, you’d better get aboard.”
Ishikawa put on her mirror helmet and hopped on the scooter’s back pedestal. She placed her hands on Minako’s shoulders. If she was planning on cursing her, there was nothing Minako could do about it.
“Are you ready, Ishikawa-san?”
“Please, call me Naomi.”
It had been more than fifty years since Minako had called her former friend by her given name. Naomi was a good name for numbers. Nana, zero, mittsu. 703.
“Please call me Minako. Are you ready, Naomi?”
She revved the scooter and drove through the wall as though it weren’t there. They were on the main street of the shadow Dotombori in front of a karaoke bar whose sign proclaimed, The Dead Sing Free.
There were no clouds or sun visible above them, but the sky was a slate gray. The shadow Dotombori was filled with unappealing restaurants. A vendor at what looked like a takoyaki stand sold blinking eyeballs. Another street stall offered the promise of ghost hair noodles. A shadow was plucking white hairs from a boiled human head and adding them to bowls.
The scooter carried them down the street towards the giant mechanical crab. Along the way they passed a pachinko parlor with shadowy vendors carrying trays of gleaming silver eyeballs. Next up was Shadow Tamade, a discount supermarket filled with goblins screaming, “Shadow Tamade, irasshaimase!”
A ramp allowed Minako to switch to the street’s lower level, away from the blare of the pachinko parlor and supermarket. At least the shadow realm was more accessible than the real street.
They reached the base of the seafood restaurant and stared up at the giant mechanical crab. It waved at them menacingly, but there was no sign of Ayumi.
A street vendor nearby was preparing okonomiyaki. Instead of adding shaved bonito flakes, the vendor reached for a bucket labeled human skin and covered the okonomiyaki with a scattering of skin flakes.
“How do we find Ayumi?” Ishikawa asked.
“I’ll try the marbles again,” Minako said. She poured out the 131 marbles and repeated the process. This time she received an image of Ayumi in a host club.
Minako pointed to the host club on the opposite side of the street. “She’s in there.”
The host club was nine stories high and the sign gave its name as 9413. That was an ill-favored number, recalling the Chinese proverb which roughly translated as 90% percent chance to die, 10% chance to live.
A message above the host club’s main entrance proclaimed in English, “You Don’t Escape From Exciting No More.”
Minako drove the scooter through the club’s doorway into a lobby with walls covered in portraits of spiky-haired shadows. She drove through the lobby into a bar fitted with mirror balls.
Ayumi stood in the center of the room, flanked by four shadows. Darkness filled her eyes. She roared out, “Irasshaimase!” in a deep, booming voice. “I’m glad you could join us. You won’t ever want to leave.”
The shadows slinked forward and surrounded the scooter.
“Get out of my granddaughter,” Minako commanded the zero shadow, but it only laughed in response. The shadows rushed at them.
Naomi’s mirror helmet pulsed with light and beams of energy lanced from her hands, extinguishing two of the shadows.
Minako plucked a special prime number from her handbag. 1,023,456,987,896,543,201 was a palindromic prime, a reflection of itself. She hurled the number at a mirror-ball and it exploded, its fragments cutting the oncoming shadows to pieces.
The zero shadow still held Ayumi in its grasp. Naomi started chanting and particles of light swirled around the zero shadow.
It waved a hand and the lights were extinguished. “You can’t defeat me here,” the shadow crowed.
Minako had dealt with plenty of uppity youngsters and wasn’t going to let some shadow trash-talk her. She reached into her bag for more numbers.
Something grasped her from behind and plucked her out of the scooter. She craned her head and saw that she was held fast in the grasp of a gigantic slug demon. The demon had a man’s torso and the lower half of a slug’s body. It stood almost fifteen feet tall and had left a slime trail across the club’s floor.
Minako still held her handbag, but her hands were held in place so she couldn’t reach inside it.
Hachi gave a ferocious woof, but the demon ignored him.
How could she deal with the slug demon if she couldn’t use her numbers? As much as it annoyed her, she needed Ishikawa’s help. “Naomi!”
Naomi erected a wall of light to protect herself from the zero shadow and turned to face the slug demon. She hurled a beam of energy at the demon, but the energy bounced off the slug’s slimy skin.
Salt could defeat a slug demon. Naomi always carried salt with her, but if Minako told Naomi to use salt, the zero shadow might have a chance to conjure a defense for the slug demon.
“I do not fight with a sword,” she called.
Naomi grinned in understanding. She reached into her bag and brought out a small jar. She wrapped it in light and shadow and hurled it at the creature.
The jar exploded against the demon and it squealed in pain. It released Minako and she dropped to the floor. The slug demon slithered away.
Minako tried to stand up, but it was too much of an effort. Instead she reached into her bag. The zero shadow had blasted away the protective wall of light and wrapped shadows around Naomi’s feet.
Minako took out 9137 and five petals each of 1, 2 and 3.
She multiplied 1 to the 5th power.
2 to the 5th power.
3 to the 5th power.
Summed together equaled 276.
9137 + 276 = 9413, counteracting any special power the shadow derived from the club.
9137 was a left-truncatable prime. Minako tore a strip from the number, changing it to 137, also a prime. Doing so stripped part of the zero shadow away from Ayumi.
She tore again. 37, another prime.
Ayumi’s eyes were now clear. “Obaachan!” she screamed.
The zero shadow was banished from Ayumi. A circular shadow slid across the room towards the exit. Minako readied another number to finish the shadow.
Ayumi rushed over and hugged her.
Minako lowered the final number and the shadow slipped out of the club. She kissed her granddaughter on the cheek. “It’s all right, sweetheart, you’re safe now. Your grandmothers are here.”
Naomi had broken free from the shadows and helped Minako to her feet. They climbed back aboard the scooter. Minako drove out of the club, taking care to avoid the slug’s slime trail.
A dark aura wreathed the mechanical crab on the seafood restaurant roof. The crab flickered and then vanished.Naomi looked worried. “It’s found a way back to our world.”
The scooter took them back to Minako’s van. They climbed aboard and drove back through the portal.
The real Dotombori was in panic. The shadow crab lumbered down the street, threatening terrified bystanders with its giant pincers.
A couple of police officers were trying to take control of the situation, but were failing miserably. Minako sighed. Of all the places that should be prepared for an attack by a giant monster.
She tooted the scooter’s buzzer and called out, “Step aside, young men. Help has arrived.” The police officers looked unsure of what to do, but after a glare from Naomi they made way.
Minako handed Ayumi to a startled officer. “Make yourself useful. Take care of our granddaughter.” Before he had a chance to say anything, she handed Hachi to the other officer. “If anything happens to my dog, I will curse you and your descendants for eternity.”
Minako drove towards the giant crab.
The crab raised its pincers in the air and scuttled forward.
Minako handed her purse to Naomi. “Take coins out that sum to 220 and give them to me. Then take coins out that equal 284.”
Minako swerved the scooter around in a loop and set off in the opposite direction, the crab following behind them.
“Here is 220,” Naomi said, handing her two 100 Yen coins and two 10s.
220 and 284 were the smallest friendly numbers.
The proper divisors of 220 (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110) summed to 284.
The proper divisors of 284 (1, 2, 4, 71, 142) summed to 284.
The numbers were bound together.
The crab was gaining on them. It snapped at the scooter and Minako swerved to the side, barely avoiding the pincer.
“I’ve got 284,” Naomi said.
“Ok, together, we throw them at the crab.”
They held hands and launched the coins at the crab. The coins transformed into shining numbers and struck the crab in a blast of light.
The shadow was banished, leaving behind a lifeless wreckage of robot.
Minako took a moment of quiet repose to regain her strength. “Do you want the crab?” she asked Naomi.
Naomi sighed. “Some of us have more taste than that.”
Minako’s gaze strayed to the Mickey Mouse logo on her friend’s sweater. She was tempted to launch a stinging retort, but instead squeezed Naomi’s hand. “Thank you for your help.”
Minako ordered the police to have the remains of the crab transported to her house. It would make a nice trophy.
She collected Ayumi and Hachi, both of whom were in good health, and ruffled Ayumi’s hair. “We can save Universal Studios for another time,” she told them. “That was enough excitement for today.”
Aidan Doyle is an Australian writer and author of The Writer’s Book of Doubt. He has visited more than 100 countries and his experiences include teaching English in Japan, interviewing ninjas in Bolivia, and going ten-pin bowling in North Korea. http://www.aidandoyle.net @aidan_doyle