10,958
Piper Pandora Pennington threw her backpack on the faded red seat of the booth and slid in beside it. She crossed her arms and stared at the old man sitting across from her. He didn’t lookup. His nose was buried in a notebook full of mathematical scribbles.
“Well…are you going to talk to me or are you going to sit there like a big lump of doofus?”
Buford Lister slowly lifted up his head. “Lump of doofus? Is that what all the 12-year-old girls are saying this week? Is that the trendy language? Is that the name of a new hit song by some teenybopper that you just can’t stop listening to?”
“I have been listening to The Ramones and Daniel Johnston. And for your information, I don’t spend a lot of time talking to 12-year-old girls.”
“No, you spend your time bothering people who have better things to do than being insulted by a self-proclaimed tween math prodigy.”
“I’m not self-proclaimed, and you know it. I am 100% certified and recognized.”
Buford Lister tried his best to reclaim the smile that wanted to overtake his face. She was right, and she knew it. P-cubed. P to the 3. Present (and hopefully) future math genius.
“Well, did you find anything? I am guessing that you pointed your massive brain at the problem I gave you. So…”
P-cubed reached into her backpack for her notebook. She lifted it up and made sure Buford Lister got a good look at it. “See this,” she said. “I am not some kind of fossil, I don’t need to write things down just so I can remember them.”
“I see.” He wanted to tell her to wait; that time would get her just as it gets everyone else. He decided to keep quiet, she would find that out on her own soon enough.
“You know, Socrates didn’t want anyone to write anything down. I mean, you guys are about the same age, right?”
He took a long drink of water, wiped his mouth with a napkin, and shook his head. “I can tell you what you found, nothing. You got a big zero.”
He turned his notebook to a blank page and held it up in the air.
“Ladies and gentlemen, the contributions of Piper Pandora Pennington, 12-year-old math genius, to the 10,958 problem, are summarized for your reading pleasure on the following page.”
He waved his hand over the blank page as Piper gave him the look, the one that combined sarcasm, raw intelligence, amusement, and the confidence only a girl like Piper could muster. After all, she was too young to realize that there were lots of people smarter than her, and maybe even better looking than her. She just hadn’t met them yet.
“Well, duh? Where’s your big medal? You have done just as much on this problem as I have.”
Buford Lister tried his best to keep a straight face as he put the notebook down.
“You, young lady, are a very hard case.”
*****
Some time ago, I stumbled upon a strange math problem. It is straightforward and yet appears to be unsolvable. Do I have your attention? Thought so.
A Brazilian mathematician named Injer Taneja wrote a paper called “Crazy Sequential Representation,” available for free on the internet. In this paper, he took every whole number from 0 to 11,111 and did something fun and unusual with them. It is easier to show you what he did than try to explain it. Check this out:
10,934 = (12+3+4) x (567+8)+9
10,934 = (9+8x7x65-4) x 3-2+1
Do you see it? He used the numbers 1 thru 9 in ascending and descending order, along with a few mathematical operators to create 10,934. Here are a few more examples:
5 = 12+34+5-67+89
5 = 98-76+5-43+21
87 = 1+2×3+4+5+6+7×8+9
87 = 9+8×7+6+5+4+3×2+1
So, you can use addition, subtraction, multiplication, and division along with exponents (what he calls potentiation). Feel free to put brackets wherever you like, just make sure the numbers are in order. The task is simple; create the target number by that process. What could be easier?
*****
Piper went into a deep think, Buford Lister thought he could see a little steam rising out the top of her head. She was trying to work through something. He sat back and let her go.
She finally spoke. “I set up a Raspberry Pi cluster, and I wrote the code. It has been running 24 hours a day. You know what I found, so I don’t have to tell you.”
“No, you don’t have to tell me. Wasted attempts aside, what do you think about this problem?”
12-year-old math geniuses can sometimes go off the rails, Buford Lister hadn’t seen any indication of that yet from her, but he was continually probing. There are few things worse than a bored genius with vast amounts of computer and math knowledge.
“Primes are the building blocks, I am not sure why we would expect the numbers 1 thru 9 to serve the same purpose. I am still thinking about it.”
“It is a tough problem. I think that maybe there are some big implications, too.”
“Well…duh!”
“By the way, did your little sister help you with the cluster and the coding?”
“Double duh! Susie does all that kind of stuff with me. You should know that by now.”
*****
On page 158 of Taneja’s paper is something curious. Take a look at this:
10,958 = still not available
10,958 = (9+8x7x65+4) x 3-2+1
Still not available? Taneja has a solution for every natural number from 0 to 11,111, except for 10,958 ascending. Not only that, most numbers have multiple solutions. I find this very strange, I am a bit disturbed by it. This shouldn’t be.
Other mathematicians have hopped aboard this train; I have heard rumors that all the numbers up to 30,000 have been solved both ascending and descending…except for 10,958.
*****
Buford Lister started to talk, Piper held up her hand; her universal sign to keep quiet because she was about to monologue. Buford Lister obliged.
“It could be that 10,958 is the first in an infinite list of numbers that cannot be solved by this method. That is one possibility. It doesn’t make any sense at all that 10,958 is the only number that can’t be. I have been thinking a lot about it, and there is nothing special about 10,958.”
“Except that it hasn’t been solved in an ascending fashion.”
“Yeah, and I don’t think there is an answer. My code is good, and I know lots of others have been running programs for a long time. If there were an answer, someone would have found it a long time ago.”
“I think that is right. So, what are you going to do now? Keep running your code? Try to come up with a better algorithm? Give up?”
Piper sat in silence as Rinny, the granddaughter of the owners, approached the table.
“The usual for you, Piper?” Piper nodded. “And you, sir? Need a refill of your Diet Coke? How about that water?”
“Tell your grandparents that I am going to have to find a new diner unless they get a liquor license. I would much prefer a beer to this concoction.”
“Well, for one thing, it is 9:00 in the morning, for another thing, you have been coming in here for decades and telling me and my mom and my grandparents the same thing every day. So, I’ll get right on that for you.”
“Thank you. Much appreciated.”
“Give me your glass! Piper, I’ll be right back with your root beer float.”
Piper nodded her approval and went right back to her monologue.
“If there are an infinite amount of numbers that cannot be constructed, then I would think that there should be more of them this early on in the sequence. This is all very strange. The only thing exceptional or unusual about 10,958 is the fact that it can’t be solved. I don’t know what to think about this.”
“So…one of the reasons I gave you this problem is that it is unlike all the other things you have been doing. It is one thing to open up a calculus book and instantly understand everything in it. When I was younger than you are now, I used to do that, I would get these thick books, and when I read them, it was like I had written them. All that knowledge seemed second nature to me. It was something I had already known, I just hadn’t realized that I already knew it. You…I know you know exactly what I mean.”
Piper nodded.
“Now you are out over your skis. This problem is strange, unusual, and really simple. Maybe, just maybe, it speaks to something deeper about the nature of numbers and the nature of mathematics itself. I just don’t know exactly what that is yet.”
Piper nodded her approval as Rinny approached with her root beer float. She sat in silence as she attacked it with a spoon and a straw.
Buford Lister took out a pen from his backpack and opened his notebook. He jotted down a couple equations as he waited for Piper to eat.
Piper put the straw back in the glass and made a fluttering motion with her left hand. “You know, I did solve it, but I had to use concatenation. I also solved it with a square root.”
“Not allowed. Even so, what is so special about 10,958 that you had to use techniques not needed for all the other numbers?”
“I know. I know. I know. I was just playing around with the code, I wanted to see what would happen.”
“Well, I highly encourage that type of behavior.” He wanted to egg her on a little. “I didn’t know you were that advanced as a coder.”
“Duh…what you don’t know could fill a warehouse!”
*****
One of the great things about this problem is that anyone can work on it. Phones all have calculator apps, notebook paper is cheap, and computer screens are readily available, so all you have to do is pick a method and start concentrating. Feel free to have it; mathematical immortality awaits the person who can solve 10,958 ascending. Good luck, you are going to need it.
