George and the Unbreakable Code

by Stephen Hawking

  “So here’s the thing,” said Merak. “You’re trying to do a deal with me on a BOGOF with your friend. I keep you … I get the spelling-bee-challenged friend. But you don’t seem to understand that you’re not in a strong position. You don’t get to set the rules—this is my spaceship, my robot army, my quantum computer; and when it comes down to it, it’s pretty much my planet down there as well. Given that I am in control, here’s how it works—either you stay and help me, or I throw both of you out into space, where you face certain death. Deal? Or no deal?”

  As Merak was speaking, George noticed Ebot moving around so that instead of floating upright like Annie, George, and the robots, he was now upside down.

  “No deal,” he said firmly.

  “No deal?” Merak sounded astonished. “Why? Why would you do this? Why wouldn’t you want to stay with me and break every code on Earth? Why don’t you want to be the winner?”

  “Because there’s one code you’ll never break. Or understand,” said George fiercely. No portal had emerged. He figured that, in a matter of minutes, either a missile from Earth would blow up the whole ship, or he and Annie would get hurled out into space. Either way, his final moments had come. A great calm stole over him.

  “It’s the code of friendship,” said George. “Between people—real people who like each other, stand up for each other, and care what happens to each other. You’ll never crack that code. You can’t decrypt it because it would make no sense to you. I don’t know anything about you, but I do know that happy people don’t behave like you behave. They don’t go round bribing and bullying people into obeying them by intercepting their secret messages and then lording it over them and hurting them. You’ll never be able to decrypt friendship. It’s the code you can’t break. Friends.”

  “Yeah!” said Annie. “Friends! That’s what we are—George, you’re brilliant.” She floated over to give him a hug. “If this is our last minute, at least we’re together.”

  As George hugged her back, he spotted something. “Look down—I think Ebot’s finally come to the rescue,” he whispered.

  He realized that upside-down Ebot’s eyes had brightened. Two beams of light shot out of them as he started to outline the doorway, unnoticed by Merak and his robot army, who were still focusing on the kids.

  “Oh, look!” said Annie loudly. “Over there …” She hurled the tiger tail as far as she could. As she did so, all the robots and Alioth Merak leaped after it, away from Ebot and the doorway.

  While their attention was elsewhere, Annie and George dived for the space portal. As they jumped through, Annie reached back, grabbed Ebot, and dragged him with her and George: instantly they were transported away from the doomed space station—toward Cosmos, Earth, and home once more.

  Traveling from microgravity into normal gravity isn’t a pleasant feeling, but Annie and George didn’t care. They landed, the three of them, in a heap on the floor of Old Cosmos’s basement.

  “Gerroff me!” Annie pushed George and Ebot off her and rolled over. “Phew!” she said. “That was horrible.”

  George lay on his back beside her. Both of them still had their eyes closed as they listened to Old Cosmos’s teletype, which had resumed its mechanical chatter, producing page after page of slowly unfurling paper.

  When George finally opened his eyes, he saw that on the paper spooling by him on the floor were dense blocks of text lines of complicated math, presented like newspaper headlines (peppered with funny little numbers offset a little above and below), and also several diagrams drawn entirely out of characters—including one that looked like the space station from which they had so narrowly escaped.

  The printer made a different sound as it produced these pictures, skimming quickly over large blank areas, then stopping to hammer a tiny part of the image, then shooting off to do the same thing a bit farther on, and finally flying back to start the next line.

  But when George looked up, past the reams of emerging paper, a horrible sight greeted him. Standing upright, holding his tail, as well as some fresh printouts from the supercomputer, was none other than Alioth Merak. He was smirking horribly.

  George’s heart sank. He squeezed Annie’s hand and she opened her eyes in shock.

  They thought they’d managed to escape from terrible danger, and at the same time saved the Earth from a power-hungry man in a onesie—only to find that they had brought their deadly foe back with them. The disappointment was shattering. To see Merak here when they thought they had vanquished him … it was even more frightening than their first encounter with him. They had guided him to the home of Old Cosmos, and who knew what damage he might be able to do if he got control of the supercomputer? It looked as though their happy ending had gone terribly wrong.

  Merak himself looked immune to fear or anxiety. He seemed perfectly relaxed. “Nanorobotic single qubit element,” he mused, reading aloud from the pages. “Operating temperature range 140 to 250 Kelvin. Accounts for space location—need to guarantee consistently low temperature range … Oh, so that’s why I built it in space, is it?” said Merak. “Well, that’s good to know! So glad two kids and a technological dinosaur turned up to tell me. Did you know”—he addressed the two children, who were still lying on the floor—“that 250 Kelvin is actually 9.67 degrees below zero in Fahrenheit?”

  He carried on reading while the kids lay there, petrified. “Oh, and apparently I built a nanorobotic solar array to prevent my quantum computer from overheating. Well, aren’t I the clever one! Look at this—so sweet! Your ancient and outmoded friend has printed the code of an entire program written in the programming language C. What’s this … ?”

  Old Cosmos began typing once more. Merak read out: “Current time: 04:31:18. No malicious activity detected for 153 seconds. Oh dear! Is he trying to tell me that my space station has been destroyed? Obviously that’s a pain, but it’s hardly the end for me. Being a positive sort, I choose to see it as a beginning.”

  “How did you get here?” George got to his feet and pulled Annie up with him. He wasn’t going to let this nasty piece of work stand over them any longer.

  “I followed you through the portal,” said Merak. “I just had time to dive after you—you were so eager to escape that you forgot to look behind you! A schoolboy error, one might say … so appropriate.”

  “It’s the end for you if you don’t have a quantum computer anymore.” Annie wasn’t going to let him get the better of them. She and George had come through so many challenges together—they weren’t going to let this madman bring them down.

  “I can rebuild it,” said Merak casually, dusting down his onesie. “Wow—that portal thing is old-school. I didn’t think they made them like that anymore! So slow to shut down as well—that’s how I was able to jump through after you. Hey, Grandpa!” He kicked Old Cosmos. “How does it feel to be entirely irrelevant in the modern age?”

  “Don’t kick Cosmos!” said Annie angrily.

  Merak smiled at her and kicked him again.

  Annie threw herself at him, pummeling her fists against his onesie. “I hate you!” she burst out. “You’re evil and rude and nasty, and you want everyone to do as you say rather than let them choose what they actually want.”

  Merak pushed her away and she fell at George’s feet.

  “There’s nothing more you can do,” George said bravely, stepping in front of his friend. “Eric knows we are here.” He hoped this was true. “He’s sending people to meet us. You won’t get away with it! You’ve got nothing left to bully us with!”

  “So sad,” tutted Merak. “I wish you wouldn’t keep harping on about bullying! It’s just a show of superior strength, which you silly little kids find demoralizing and overwhelming.”

  “I thought you said George was your heir!” protested Annie, getting up again.

  “I was wrong… . Not something you will often hear me say.” He smiled. “I have now given the command to my global network of 3-D printers to start replicating my robot arm
y. A few of them are already on the planet, as you know. I have now summoned them here. They weren’t far away.”

  “Are you replicating the Boltzmann?” asked Annie.

  “I have decided to discontinue that model,” Merak told her. “They are incredibly hard to pull off and not at all reliable when you succeed. I hoped I could be nice when I came to save the Earth, but your negative reception has made me change my plans. I have now decided to implement a regime of punishment instead. You will have to watch while I rip the world as you know it to pieces, and know that it is entirely your fault. Fun, huh?” He swished the tail around like a rope charm. “Well, for me, anyway.”

  A great pounding noise resonated from outside the basement room. The two friends gulped. As they watched helplessly, the door was relentlessly pummeled until it gave way. For a moment George hoped they’d see Eric and his fellow scientists, but it wasn’t to be. Two of Merak’s identical robots forced their way into the basement.

  George turned to look at Ebot: the android that had survived the kidnapping, the space station, and the portal, returning to Earth with his hair only slightly ruffled, seemed to have lost power and was now uselessly slumped in the corner.

  The magnificent ancient machine, Old Cosmos, offered no obvious protection either. In any case, George realized, to reach their old friend they would have been forced to go around Merak. Had they really survived all their adventures only to end up the prisoners of an evil robot army with a madman in a planetary onesie who sought world domination? Was this really the end of everything for them?

  Once more, George reached out for Annie’s hand, and the pair clung on to each other, determined to face the threat together, united.

  But just as they’d given up even a tiny glimmer of hope, they realized that the robots hadn’t actually seized them. Instead, they were grabbing hold of Alioth Merak, who was struggling in their pincer grips.

  “Let go of me, I command you!” he snapped. “You’ve misunderstood my orders… . It’s not me, you pathetic piles of metallic junk! It’s them!” He tried to free his hands to point, but they were pinned tightly behind his back.

  As the robots held their former leader, Annie and George heard Cosmos tapping away once more.

  George ran over to the computer and ripped off the sheet.

  “He’s not the only person who knows how to break a code,” he read out loud, “or intercept a message and change the contents. ROFL!”

  He burst out laughing. “Annie!” he said gleefully. “Cosmos has taken control of the robot army. Isn’t that right, old fella?”

  Cosmos twinkled. Yes it is, he replied. I shall direct them all to be extremely helpful, wherever they are. Except for these two, which I will use to keep this man in custody until Eric arrives.

  “Wowzers, Cosmos!” exclaimed Annie, who ran over as if to give him a hug, but then stopped, realizing that this was kind of impossible. “You saved the day!”

  Next time, said Cosmos, remember that there is wisdom in age as well as innovation in novelty.

  “Okay!” said George. “We’ll do that, you marvelous machine! You”—he turned to Merak, who was still vainly struggling to get free—“can stay here until Eric arrives! I expect he’ll have some Quantum Error Detection to do with you!”

  “Where are you going?” Alioth Merak asked sulkily. “You can’t just leave me here in this basement, with these robots! That’s not fair. I’ve got nothing to eat, nothing to drink, nothing to do. This is against the International Convention of Robotic Activity. I’ll get my lawyers on to you! You’ll pay for this!”

  “Typical,” said Annie as she and George turned to leave the room. “Now that he’s lost, it’s unfair. When he was winning, he didn’t care what rules he broke.”

  “Let’s go home, Annie …” George picked up his skateboard and headed toward the stairs. “I don’t know about you, but I’m hungry!”

  WHAT CAN’T A COMPUTER DO?

  All known computer designs (including quantum computer designs) can compute no more than a Turing Machine could compute if given enough time and memory. However, Turing was able to prove that some problems in mathematics are uncomputable, that is to say, they cannot be solved by a Turing Machine—and hence not by any known computer today! He demonstrated this with a problem concerning Turing Machines themselves, known as the Halting Problem.

  The Halting Problem

  When will a Turing Machine halt? If it only has one state (state 0), then only two rules are needed—what to do if the machine reads 0, or 1. There are varied ways these rules could lead to different results, depending on how the 1 rule is formulated:

  • The 0 rule says leave 0 and march right, continuing until it finds the input of a number 1, then halt. The machine halts and outputs the answer.

  • But a Turing Machine could find itself in an endless loop: choosing “if 1 is read, write 1 and move left” would make the machine move back to the previous 0, then move back to the 1 at the next tick of the clock (following the 0 rule), and then repeat these two moves forever.

  • It is also easy to make a Turing Machine that will not ever halt. Changing the 1 rule to “if 1 is read, write 0 and move left” will cause the machine to move back to the previous 0, then return, but this time it sees 0 and continues past until the next 1. The machine will turn all the 1s into 0s and then disappear off to the right forever.

  Machine H

  Alan Turing himself posed the question: is there an algorithm that, when fed with the program of any Turing Machine and some extra input, will output the answer 0 if that machine with that input doesn’t ever halt and output an answer?

  Suppose for the moment that such an algorithm existed—then there would be a Turing Machine to perform it. Let’s call this machine H and input data such that H halts if, and only if, its input is the program of a Turing Machine that doesn’t halt when input with its own program.

  So what happens if we feed H with its own program?

  If it does halt, then it is an example of a Turing Machine that does halt when input with its own program—but then H was designed not to halt when fed with the program of such a machine!

  If it doesn’t halt, then H is a machine that doesn’t halt when input with its own program, but that means that H fed with the H program should halt, because it was designed specifically to detect such machines.

  Either way, this is a contradiction! A nonsensical situation like this tells a mathematician that what they were assuming is true was wrong. Constructing the imaginary Turing Machine H—which cannot exist—was therefore very clever. It proved there cannot be a Turing Machine able to compute whether any Turing Machine with any input will halt. And if this question cannot be settled by a Turing Machine, therefore it is uncomputable on any computer we can currently imagine building.

  Put simply, a computer can’t solve this problem!

  Infinite numbers

  The number of possible programs and Turing Machines is infinite, but because every computer program can be turned into one big binary number, a mathematician would describe the set of all programs or machines as countably infinite, which is not quite the same thing—this just means that we can list them in order of size.

  But there are much bigger infinities, for example the infinity of decimals with infinite decimal places—these are called the “real numbers”: numbers like 0.33 recurring for 1/3. This means that there are real numbers whose digits cannot be generated by a computer.

  For example, the real number pi (which you use in working out the circumference of a circle, for instance, and probably know as standing for 3.142) can be written out to any number of decimal places by a computer. The first few are 3.1415926535, and a computer has done this to trillions of decimal places. Most real numbers, though, cannot be generated like this: they are fundamentally uncomputable—a computer can’t do it!

  The future?

  Some theorists speculate that new types of computer, relying on as yet unknown physics, will be disco
vered in the future that can compute more than a Turing Machine can compute, and that the human brain (the original “computer”) may even turn out to be one of these.

  “Wait!” said Annie. “We have to say good-bye to Old Cosmos!”

  “Thank you.” George looked at the massive computer and smiled. “Thank you, Cosmos. You’ve saved not just us but the whole world.”

  It was my pleasure, Cosmos said, and his lights seemed to glow. It is nice to be useful. Please make sure you tell Eric all about it, just in case he was thinking of decommissioning me.

  “We won’t let that happen,” Annie promised. “You’re our friend forever now!”

  • • •

  Outside on the street, a silvery dawn was just breaking over the quiet streets of the beautiful university town; the stone facades of the ancient buildings reflected the newly minted sunlight. In doorways and under arches, people slumbered as the two friends rolled past on their skateboards, discussing what they most wanted to eat for breakfast.

  “Pancakes,” said George, his mouth watering at the thought. “A huge tower of them with maple syrup.”

  “Bacon!” said Annie. “Hot crispy bacon!”

  “Bacon?” George was thinking of his old pig, Freddy.

  “You don’t have to eat it,” Annie told him. “Anyway, it’s not like we’ve actually got any food at home.”

  George thought of the empty kitchens in their two houses. “Wow, it’s going to be hard to put everything back together,” he said as they rolled along.

  “I wonder what will happen now… . I mean, how will Dad and the others explain this to the world?” Annie wondered as they approached Little Saint Mary’s Lane.

  “What do you mean?”

  “They can’t just say, “Oh, by the way, people of Earth, we were attacked by a madman in a onesie who pretended he wanted you all to have free money and food but in fact wanted to rule the world.”’

  “I don’t know,” said George thoughtfully. “Perhaps it would be best just to tell everyone the truth?”