Numberphile: videos about numbers and stuff
December 29, 2012 11:07 PM Subscribe
Numberphile is a website containing short videos (approx. 5-10 min.) about numbers and stuff. Mathematicians and physicists play around with the tools of their trade and explain things in simple, clear language. Learn things you didn't know you were interested in! Find out why 493-7775 is a pretty cool phone number! What's the significance of 42, anyway? What the heck is a vampire number? Why does Pac-Man have only 255 screens?
Suitable for viewing by everyone from intelligent and curious middle-schoolers to math-impaired adults. Browse their YouTube channel here. (Via)
I like this!
posted by 5_13_23_42_69_666 at 12:48 AM on December 30, 2012 [7 favorites]
posted by 5_13_23_42_69_666 at 12:48 AM on December 30, 2012 [7 favorites]
I don't get what's so interesting about Smith numbers. They don't seem to carry over into other bases; so 27 is 11011 in binary, which is 11*11*11 (or 3*3*3). Adding up the digits in the binary representation yields 100 (4), and adding up the digits in the prime factors yields 110 (6) - at least, if I did that right...
If whether a number is a Smith number depends on which base you choose, his metaphor at the end of the Smith numbers video (linked from 493-7775) about "building blocks of matter" doesn't really connect.
posted by Philosopher Dirtbike at 12:48 AM on December 30, 2012 [1 favorite]
If whether a number is a Smith number depends on which base you choose, his metaphor at the end of the Smith numbers video (linked from 493-7775) about "building blocks of matter" doesn't really connect.
posted by Philosopher Dirtbike at 12:48 AM on December 30, 2012 [1 favorite]
It's no big secret anymore why Pac-Man messes up after 255 screens. But much less well-known is another quirk of many classic arcade games, that has to do with score-based extra life awards at extremely high scores. This happens with many arcade games, and here I'll explain the phenomenon, and why, I think, it happens.
Before Space Invaders, almost all arcade games had a definite length. You played for a certain amount of time, and after that the game was over. You might do well enough to earn one or more time extensions, but these weren't unlimited in number. No matter how well you did, there was a definite time in the future when the game would be over, and it's time for the next player to have a go.
Space Invaders changed that. It had turns of indefinite length. You played a base until it was destroyed, no matter how long that took, and you had a limited number of bases. For this play structure to work in an arcade game, a game's difficulty has to increase until it reaches a place where it's impossible to continue, where losing a life becomes unavoidable. Of course, with many arcade games it doesn't actually become unavoidable, which is why some very skilled players can get extremely high scores, bounded only by their attention and energy.
Space Invaders, instead of awarding a time extension, grants the player a single extra base at 10,000 points. The Pac-Man family of games also use a structure like this, awarding only one extra life no matter how high score gets. Some later games, more sure of themselves, award indefinite extra lives, at periodic score awards, say every 10,000 or 25,000 points. I call these games "of attrition," where you're always losing lives, but you seek to regain them at the same rate or faster. This is an appealing structure to the player because it implies the player can go on forever, even if he dies occasionally. Most of the time, though, these games are quite a bit more difficult to make up for it.
But many of these games have a revealing design flaw. Nearly all arcade games, if you get a score that's high enough to overflow the game's counter, will "wrap around," and your score will appear to be zero again. Well, when you reach the point where the score for the next extra life is at or beyond the rollover point, many games will award an extra life every time the player earns points. It's like a giveaway period. Anything you do, even if it's something that only gives you a minimum score award, like only 10 points, results in an extra life. But these lives are not truly given but kind of loaned; once your score wraps around the counter itself, the giveaway period stops, and for every extra life you earned during it, one normal extra life award will not be granted. If the game rolls over at 1,000,000, extras are earned every 10,000, and you earned 17 extra lives between 990,000 and 1,000,000, then once your score reaches 1,000,000 you will stop earning extras until you reach 1,180,000 points, where extras resume being awarded normally.
Why does this happen in so many games? It's because most of them use a certain algorithm, or something close to it, to determine when the player earns an extra life. Internally the game has a variable, with similar rollover characteristics to the visible score (it may actually be stored in binary-coded decimal to facility display, instead of raw binary), that tracks when the next extra life is awarded. Whenever you earn points the game checks to see if your new score is greater than this value; if it is, the routine that increases the life counter is called, and the next award point is increased by a boost value, in this example 10,000 points.
But when you get near the rollover point, the next-extra-life rolls over before the player's score actually does. At 990,000, the next extra life point is 0. Every time the player scores points, the values are checked, the game sees that the player has more points than the extra life award threshold, awards the extra life, and then increases the score needed for the next award. It's a long way before the score rolls over again so it's unlikely to roll over a second time before the player's score wraps too and ends the phenomenon, but the score boosts from all those free extra lives remains, and must be made up if the player is going to start earning extra lives normally again.
posted by JHarris at 1:13 AM on December 30, 2012 [18 favorites]
Before Space Invaders, almost all arcade games had a definite length. You played for a certain amount of time, and after that the game was over. You might do well enough to earn one or more time extensions, but these weren't unlimited in number. No matter how well you did, there was a definite time in the future when the game would be over, and it's time for the next player to have a go.
Space Invaders changed that. It had turns of indefinite length. You played a base until it was destroyed, no matter how long that took, and you had a limited number of bases. For this play structure to work in an arcade game, a game's difficulty has to increase until it reaches a place where it's impossible to continue, where losing a life becomes unavoidable. Of course, with many arcade games it doesn't actually become unavoidable, which is why some very skilled players can get extremely high scores, bounded only by their attention and energy.
Space Invaders, instead of awarding a time extension, grants the player a single extra base at 10,000 points. The Pac-Man family of games also use a structure like this, awarding only one extra life no matter how high score gets. Some later games, more sure of themselves, award indefinite extra lives, at periodic score awards, say every 10,000 or 25,000 points. I call these games "of attrition," where you're always losing lives, but you seek to regain them at the same rate or faster. This is an appealing structure to the player because it implies the player can go on forever, even if he dies occasionally. Most of the time, though, these games are quite a bit more difficult to make up for it.
But many of these games have a revealing design flaw. Nearly all arcade games, if you get a score that's high enough to overflow the game's counter, will "wrap around," and your score will appear to be zero again. Well, when you reach the point where the score for the next extra life is at or beyond the rollover point, many games will award an extra life every time the player earns points. It's like a giveaway period. Anything you do, even if it's something that only gives you a minimum score award, like only 10 points, results in an extra life. But these lives are not truly given but kind of loaned; once your score wraps around the counter itself, the giveaway period stops, and for every extra life you earned during it, one normal extra life award will not be granted. If the game rolls over at 1,000,000, extras are earned every 10,000, and you earned 17 extra lives between 990,000 and 1,000,000, then once your score reaches 1,000,000 you will stop earning extras until you reach 1,180,000 points, where extras resume being awarded normally.
Why does this happen in so many games? It's because most of them use a certain algorithm, or something close to it, to determine when the player earns an extra life. Internally the game has a variable, with similar rollover characteristics to the visible score (it may actually be stored in binary-coded decimal to facility display, instead of raw binary), that tracks when the next extra life is awarded. Whenever you earn points the game checks to see if your new score is greater than this value; if it is, the routine that increases the life counter is called, and the next award point is increased by a boost value, in this example 10,000 points.
But when you get near the rollover point, the next-extra-life rolls over before the player's score actually does. At 990,000, the next extra life point is 0. Every time the player scores points, the values are checked, the game sees that the player has more points than the extra life award threshold, awards the extra life, and then increases the score needed for the next award. It's a long way before the score rolls over again so it's unlikely to roll over a second time before the player's score wraps too and ends the phenomenon, but the score boosts from all those free extra lives remains, and must be made up if the player is going to start earning extra lives normally again.
posted by JHarris at 1:13 AM on December 30, 2012 [18 favorites]
So I thought this was going to be way too much for my tiny brain..but no
its quite fun!... thanks
posted by quazichimp at 1:50 AM on December 30, 2012
its quite fun!... thanks
posted by quazichimp at 1:50 AM on December 30, 2012
Things I love about numberphile: A viewer paid their neighbours' kids £550 (via an ebay auction) for their appearance in a video and sometimes contributor James Clewett is a genuine tetris maniac. But really.. across the board its a great little youtube series.
As this week between Xmas and New year is traditionally a week of "best of" lists - James Grime, frequent presenter on numberphile, picked 17 as his number of the year for 2012.
posted by samworm at 2:36 AM on December 30, 2012 [2 favorites]
As this week between Xmas and New year is traditionally a week of "best of" lists - James Grime, frequent presenter on numberphile, picked 17 as his number of the year for 2012.
posted by samworm at 2:36 AM on December 30, 2012 [2 favorites]
why 493-7775 is a pretty cool phone number!
Not many people know that 555 is used in Hollywood films as part of a phone number because Hollywood liberals, who are mostly all in league with the devil, picked one less digit in each place instead of 666.
posted by twoleftfeet at 2:47 AM on December 30, 2012 [1 favorite]
Not many people know that 555 is used in Hollywood films as part of a phone number because Hollywood liberals, who are mostly all in league with the devil, picked one less digit in each place instead of 666.
posted by twoleftfeet at 2:47 AM on December 30, 2012 [1 favorite]
Just watched a bundle of these and they're brilliant! Johnny Ball eat your heart out.
posted by comealongpole at 6:13 AM on December 30, 2012
posted by comealongpole at 6:13 AM on December 30, 2012
Vampire numbers are interesting. First thing I wondered is if there are any vampire numbers where the fangs themselves are vampire numbers
posted by 2manyusernames at 7:00 AM on December 30, 2012
posted by 2manyusernames at 7:00 AM on December 30, 2012
That fellow and his Bookie Balls could be a hit on Numberwang!
posted by sneebler at 7:09 AM on December 30, 2012
posted by sneebler at 7:09 AM on December 30, 2012
I'm fond of number pairs (I don't know a special name for them) which do this:
(20+25)2 = 2025
(68320+14336)2 = 6832014336
(2428460+2499481)2 = 24284602499481
...and so on. (You can have fun finding other pairs; it's a good exercise.)
posted by Wolfdog at 8:10 AM on December 30, 2012 [2 favorites]
(20+25)2 = 2025
(68320+14336)2 = 6832014336
(2428460+2499481)2 = 24284602499481
...and so on. (You can have fun finding other pairs; it's a good exercise.)
posted by Wolfdog at 8:10 AM on December 30, 2012 [2 favorites]
In a sort of relation to JHarris's story: back growing up I got really good at playing Sonic: The Hedgehog. This became apparent in Sonic 3 & Knuckles (the lock-on thing you could do with the carts of Sonic 3 and Sonic & Knuckles) when I had accumulated 100+ lives and the life counter, as it's only 2 digits long, begins to display my life total in some glitchy format (I want to say hexadecimal). The save state would always show 99, and I would start with 99 if I wanted to replay levels, but getting one more would start the glitchy display.
Sadly I discovered recently that the save states had finally gone away due to cartridge age (or the CMOS battery finally gave up). :(
posted by JoeXIII007 at 8:40 AM on December 30, 2012
Sadly I discovered recently that the save states had finally gone away due to cartridge age (or the CMOS battery finally gave up). :(
posted by JoeXIII007 at 8:40 AM on December 30, 2012
This is really cool - thanks for sharing! Though I didn't see any reference to the encyclopedia of numbers...
posted by antonymous at 9:19 AM on December 30, 2012
posted by antonymous at 9:19 AM on December 30, 2012
Watching the Pac-Man video now:
I notice in their first clip of the game, they use a home computer version. When they show a clip of the split screen, they use a clip from the arcade version. Weird; why not just use the arcade version for both?
It's also worth noting that game errors like this are all the result of oversight. The programmer figured no one would ever get that far, either because of difficulty or they just wouldn't bother, and so they didn't account for that state. It's not hard to fix the problem if you expect some players to get that far: just watch for level 255 in the code, and artificially advance the player 24 or so levels to get him past the "9th Key" board again, where the game stopped getting harder.
And it's not true that 8-bit computers can only count to to 255. They can count higher, but they have to use more than one byte to do it, which is completely within the ability of most 8-bit processors using something called a carry bit. Memory is in short supply on these machines though (Pac-Man only has 16K of ROM and 4K of RAM, and 2K of that is video memory), so where possible you try to conserve it, using only one byte instead of two. (Of course this is obvious to anyone who thinks about it for a moment; your score is a value much larger than 2. Although in many games the score is stored in BCD, or just as raw digits, to facilitate display. [I don't have proof of this last fact, but it seems evident.])
posted by JHarris at 11:11 AM on December 30, 2012 [1 favorite]
I notice in their first clip of the game, they use a home computer version. When they show a clip of the split screen, they use a clip from the arcade version. Weird; why not just use the arcade version for both?
It's also worth noting that game errors like this are all the result of oversight. The programmer figured no one would ever get that far, either because of difficulty or they just wouldn't bother, and so they didn't account for that state. It's not hard to fix the problem if you expect some players to get that far: just watch for level 255 in the code, and artificially advance the player 24 or so levels to get him past the "9th Key" board again, where the game stopped getting harder.
And it's not true that 8-bit computers can only count to to 255. They can count higher, but they have to use more than one byte to do it, which is completely within the ability of most 8-bit processors using something called a carry bit. Memory is in short supply on these machines though (Pac-Man only has 16K of ROM and 4K of RAM, and 2K of that is video memory), so where possible you try to conserve it, using only one byte instead of two. (Of course this is obvious to anyone who thinks about it for a moment; your score is a value much larger than 2. Although in many games the score is stored in BCD, or just as raw digits, to facilitate display. [I don't have proof of this last fact, but it seems evident.])
posted by JHarris at 11:11 AM on December 30, 2012 [1 favorite]
Thanks for sharing! My son and I will have a great time watching these videos together.
posted by Triplanetary at 11:13 AM on December 30, 2012
posted by Triplanetary at 11:13 AM on December 30, 2012
Your score is a value much later than 255, I should have said.
posted by JHarris at 11:40 AM on December 30, 2012
posted by JHarris at 11:40 AM on December 30, 2012
(-1+9+6) squared = 196 (this is fun, especially if you let yourself use other operators)
posted by Earthtopus at 11:48 AM on December 30, 2012
posted by Earthtopus at 11:48 AM on December 30, 2012
when I had accumulated 100+ lives and the life counter, as it's only 2 digits long, begins to display my life total in some glitchy format
The original Super Mario Bros would famously do this as well, and it was relatively easy to reproduce if you knew the trick.
SMB rewarded you for consecutive enemy stomps/bounces without touching the ground, so if you bounced off a goomba you'd get 100 points, but if you then landed on and bounced off another goomba that'd be an additional 200 points, 400 for the next, and so on until on the eighth or ninth bounce you were rewarded not with points but with a 1up. So too for every bounce after that. Done right you could rack up multiple extra guys in short order.
In practice it would be very hard to string together that many bounces; enemies were just not that thick on the ground and Mario didn't have a lot of tricks back then to work with. Certainly it was a lot easier to pull off in SMB3 with the raccoon tail and the pipes that would spit out goombas indefinitely, as you'd let a few spawn and then start bouncing off their heads and tail-floating slowly down to the next one; you even got a bit of a slow-mo assist in that context, since the NES couldn't handle manipulating that many character sprites on screen at full speed.
But in the original SMB there were a couple of spots in the game (the one I remember specifically was in World 3-1) where with a bit of careful jumping and luck you could get into a very fast, stable cycle of bouncing off a turtle shell on a "staircase" (just a diagonally-ascending stack of blocks) and have your wee bounce land back on that shell just as it rebounded the wee distance from the staircase block adjacent. So you'd get your eight or so scoreful bounces and then start racking up the extra lives.
And if you let that keep going past 99, the numbers would start getting replaced with odd glyphs, which was neat; and if you pushed it too far (maybe past 128, enough to get a signed 8-bit int to wrap around into negative numbers), you got the game to crash, which was somewhat less neat if you were hoping to have a stress-free time exploring the game's tougher later levels, though if that wasn't a concern then it was kind of a holy of holies as far as crazy shit to make your Nintendo do.
posted by cortex at 12:54 PM on December 30, 2012 [2 favorites]
The original Super Mario Bros would famously do this as well, and it was relatively easy to reproduce if you knew the trick.
SMB rewarded you for consecutive enemy stomps/bounces without touching the ground, so if you bounced off a goomba you'd get 100 points, but if you then landed on and bounced off another goomba that'd be an additional 200 points, 400 for the next, and so on until on the eighth or ninth bounce you were rewarded not with points but with a 1up. So too for every bounce after that. Done right you could rack up multiple extra guys in short order.
In practice it would be very hard to string together that many bounces; enemies were just not that thick on the ground and Mario didn't have a lot of tricks back then to work with. Certainly it was a lot easier to pull off in SMB3 with the raccoon tail and the pipes that would spit out goombas indefinitely, as you'd let a few spawn and then start bouncing off their heads and tail-floating slowly down to the next one; you even got a bit of a slow-mo assist in that context, since the NES couldn't handle manipulating that many character sprites on screen at full speed.
But in the original SMB there were a couple of spots in the game (the one I remember specifically was in World 3-1) where with a bit of careful jumping and luck you could get into a very fast, stable cycle of bouncing off a turtle shell on a "staircase" (just a diagonally-ascending stack of blocks) and have your wee bounce land back on that shell just as it rebounded the wee distance from the staircase block adjacent. So you'd get your eight or so scoreful bounces and then start racking up the extra lives.
And if you let that keep going past 99, the numbers would start getting replaced with odd glyphs, which was neat; and if you pushed it too far (maybe past 128, enough to get a signed 8-bit int to wrap around into negative numbers), you got the game to crash, which was somewhat less neat if you were hoping to have a stress-free time exploring the game's tougher later levels, though if that wasn't a concern then it was kind of a holy of holies as far as crazy shit to make your Nintendo do.
posted by cortex at 12:54 PM on December 30, 2012 [2 favorites]
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