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What’s real fun is that you can just draw this “board” on any piece of paper and immediately produce a calculator. Add more rows and columns for bigger numbers
Since you will always have a checkered sheet of of paper you can build your own checkerboard during a math test. Cut some small paper chunks and paint them with your pencil and you have your "coins"
If someone is genius enough to come up with the idea of using a checkerboard as a calculator, then they should be allowed to do so because they have damn well earned that right. If I were a maths teacher, I would be more impressed with someone who was smart enough to convert an innocent board into a makeshift calculator than someone who is scribbling calculations on a rough sheet of paper.
The actual techniques you use on this chessboard mirror the same techniques used on 8-bit processors before multiply and divide instructions were added.
Yep this is a representation of a computer. Save this to teach it to are descendants so They can defeat their machine overloads without the need of a machine fifth column. 😂
Teacher: "You can bring a 3x5 reference card for the exam" Me: **frantically tracing graph paper grids onto a 3x5 card and snapping off bits of pencil lead to use as counters**
@@vishu1334 For large numbers, and especially if you're willing to take "good enough" approximations, yes It's basically a slightly more sophisticated but much more complicated slide rule
@@bluerendar2194Would it even be 'good enough approximations'? Since this is binary, as long as you have enough 'bits', you can represent any number you want. So, you should be able to get the exact answer every time without having to approximate anything.
I wonder if you might even be able to double the bits in the same grid size by doubling the pennies. So, instead of 1 penny, you could have 3 or 7 pennies to triple the amount of bits you can represent per grid spot. That'd be 1, 3, and 7 pennies total for each square if all in the same spot. 1, 2, and 4 for being in separate spots. You should be able to separate them out for the final result based upon that. Or, you could have different colored pieces or different coin types just to simplify everything a bit.
i mean it seems likely that you could get used to it after a bit of practice, even if it’s not quite as intuitive as multiplication on the checkerboard at first glance. division just always has to suck i guess lol
@@_rr_-qm5og For mental calculaton, I've once and for all learned the three first decimals of the inverse of 1 to 50 by heart. I recommend it! Then it's just multiplication to get any rational number, within reasonable error margin for many practical purposes. Division is much more useful than multiplication. Division allows you to put quantities in relation to each other. It should be mandatory in elementary school, together with the "times table". And it's not 50 unique numbers, most are multiples or have other nice patterns. 1/9=0.111... 1/11=0.0909... See how 1 and 9 go together? It has to do with the base 10 system. Just looking at the inverses of integers makes one realize that there's something to number theory. At least it helps memorizing some inversed integers. The inverse of a prime has a repeating sequence of decimals the length one shorter than the prime. And the first half of those decimals plus the second half results in all 9s. "Go figure!" is a fitting exclaim here. 1/7=0.142857 so 142+857=999 or 14+28+57=99 if you prefer. 1/13=0.0769230 so 769+230=999 but 076+923=999 or 07+69+23=99 also works. 1/17=0.05882352941176470 so 58823529+41176470=99999999. 1/13 only has 6 repeating digits instead of 12, or well, I suppose one could say that 1/13=0.076923076923 with a repetition within the repetition, as if by chance. And by the way, the inverses of primes have an iterated digit sum of 9. But don't go insane, there are solid logical explanations.
You describe lark as an obsolete word but where I live it's in everyday use. "We did it for a lark", "the kids were just larking about". Hopefully it'll catch on on your side of the pond! Solid word!
Absolutely! I'd like to bring back hootenanny, hullabaloo, swanky, lark, and cattywampus
หลายเดือนก่อน +38
@@WrathofMathYou could also learn from Indian English and enjoy preponing your meetings. (I haven't convinced my Indian friends yet of the brilliance of 'poning' our appointments, ie having them exactly on time. But everything is better than just boring postponing.)
How to do 1 variable algebra on chessboard: 1) Do a bit of eliminating (subtraction) 2) place numbers on 1 place while x on the other (reverse operations) 3) do addition & subtraction on both sides 4) do division 5) Now you have the value of x, yay!
This is such a good idea, if the collatz conjecture works with multiples of 3, wouldn't it make more sense to work with it using a base 3? Why has no one thought of this.
This uses the fact that humans can at a glance find the intersection of horizontal and vertical grid lines. Because while the human mind cares not at all about doing advanced mathematics for what appears to be no practical reason, it does still use advanced mathematics for things that do have benefit, like intersections which help traverse the real world. The trick is always setting up the problem in a way that the human mind can understand as useful, once done to a sufficient degree the brain is perfectly capable of outpacing a calculator easily.
@@Jeff-ss6qt yes, some of the first people who got research-purpose neural implants in the 2010s can output text as fast as they think it. imagine what someone who grows up with one of those could do. i'm not to interested, personally.
For lambda calculus you can proof existence of fixed point combinator for every lambda function but this is not true for all algebraic functions. So you shouldn't use lambda calculus for everything because it's quite restricted
20 วันที่ผ่านมา
@@lpi3 You can stimulate everything else in (untyped) lambda calculus, if you wanted to badly enough.
You could use heads and tails on your pennies for ternary, too. Or mix different size coins. But that would probably slow you down more than it's worth.
Using a higher number base doesn't make the math faster. It'd actually probably be a lot slower because it makes it more complicated and you have more rules to follow.
I mean, to be fair, you could do the same thing with a chess board and just use multiple types of coins, too. You could even do base-4 with pennies, nickels, and dimes, for example. (But I agree with @Andoxico that that would probably actually be a lot harder to do math with, in most situations.) A Go board could also be used just as well for this same binary-based system, though, and it would allow you to do larger numbers in a more compact way...
Honestly i think that introducing this to schools might be quite good, this is both useful as a tool to make some calculations easier as well as being a good showcase for some mathematical principles, which should also make it easier to do these steps in your head. Additionally, unlike with real calculators, there is a high likelihood that over time, the students will do more and more steps, only including the more difficult parts, since that takes less effort. The only real problem is that it can't be used for the more difficult calculations involving fractions and irrationals.
The Egyptians managed by using binary as the foundation for their mathematical algorithm/procedures (including fractions). Today this can still be seen in the Ethiopian method of doing arithmetic.
That’s actually really devious, you can distract your classmates with the sounds of coins sliding on wood. This would only be a good strategy when you knew you were going to fail and just wanted to lower the class average, but still
That just made the following scene occur to me Diligent Student: Mrs. Teacher, can I borrow your calculator? Teacher: Sorry, I don't have any more to lend Devious Student: You can borrow mine (reaches into backpack for chessboard 😈)
to be fair, I think you could convince a fairly large amount of professors to substitute your calculator for this for fun (except when there's not enough space)
Yeah I don't know how a professor would feel, but I think a high school teacher would say no way - you're not gonna use some dumb useless old calculating method and tank your grade, get a TI-84 out the bin.
Unless i'm mistaken, you don't have to label the axis in base 2. You could use base 4, base 10, whatever, to get a much bigger range with just an 8x8 board.
@@mt.penguinmonster4144 I have no problem with binary, as I started programming in BASIC at school in 1977 (when 16K was a LOT of RAM!), and I soon found the 'front panel' of the Research Machines 380-Z, which got me into Z80 coding, and I got my own computer (a Nascom 1 with 1K RAM) in 1979, so I could ONLY code in Z80!
@@mt.penguinmonster4144 Computers do the exact same thing so i see no problem. The 1/2, 1/4 are in the decimal system so you don't have to convert binary to decimal, it's already done.
Even better, use the rectangle paper they give you and tear off the longer side to make it a square. Tear the scrap into ‘penny placeholders’ and you can do it with just paper and pencil.
@@michaldzurik535Just gotta check, in case the universe changed. Remember to do it twice in case a cosmic ray flipped a bit in the calculator and it messes the calculation up. (I do this every single math exam even though I have A+ in all of my math courses)
5:09 But you do have a bigger chessboard! Ol' Johnny Napes labeled all 15 diagonals, so he had labels on the left and top as well, so 256 all the way up to 2^(15-1).
This is actually similar to a factoring method I developed using Matrices, except you have a Chessboard instead of a matrix and the factoring only works for small numbers as the cost increases exponentially.. 😄
That’s a really interesting thing, you’re doing binary operations without converting to base 2, and the same idea it’s behind the computation of the curta computers, using gears instead of squares
If you think about it in a bit more detail, there are some additional rules to doing the square-root calculation that can make it a lot easier to know what moves to try to make. First, the whole thing must start with a single penny on a white square (this is the only way to make the necessary top-left corner of any solution). Therefore, if your number starts with a black square, you immediately know that you need to substitute downward, and then you also know you will only use one penny from that and then substitute the second one downward again. Also, the number of pennies starting from a black position must always be even. Therefore, if you have an odd number of pennies on a black starting position, you know you will need to substitute down at least one of them. More importantly, you can't just move a penny to any square you feel like. There are basically only two forms of valid "move" after the first one: 1) You can move _two_ pennies to fill an unoccupied spot on the first column of the result, and also the corresponding spot on the top row (a bottom-left and top-right corner combo). 2) You can move a penny to some other column _but only if that position already also has a corresponding penny in both the first column and the top row._ (basically, move #1 is the only way to create new columns/rows in the answer, and move #2 is the only correct way to fill in empty spaces in existing columns/rows.) Therefore, based on these rules, it becomes pretty obvious that at 16:34 and 17:05, due to the restrictions on rule #2, those moves you considered and discarded are both going to be invalid moves, so it's not really worth even considering them.
I am so amused by you setting both pennies aside and getting a new one at 4:50, like they're permanently 64 markers now, you can't simply put one on 128
Wow, so simple and elegant. Chess board really is a tool for the mind. Great video and great presentation in on point, keep up the great work and have fun 😁
You know what... I've been using that method on math exams multiple times. Never used a chessboard, though, just a strip of paper where I write the binary sequence of whatever number I'm calculating
Come to think of it, you're usually permitted all the scrap paper you want, you could easily rip some up, use it as counters and use more paper to make the board. Check mate.
@@QSBraWQ I've never actually been that good at math. I've always been faking it.. counting on fingers; using scrap paper to write binary calculations...
If I were a math teacher, I'd 100% allow this calculator because you HAVE to know math to use this calculator. The multiplication technique is literally a representation of polynomial multiplication in action.
Question #1: If you have an 8x8x8 3D chess board, can you do cube roots? Question #2: If you label the vertical as 0, 1, 2, 3, ...7, can you do logarithms?
The division method makes me feel iffy. I was expecting you to complete the rectangle by columns, but sometimes you randomly decide to move a penny up to the top row before moving the next penny to finish the column. Which is weird because once you set a penny to a square, one in its column _must_ have a penny unless you backtrack. But you prioritize filling in the rows first.
Very nice video! Nice to see another Napier device for facilitating calculations (I was only familiar with the Napier's bones and this other little thing called logarithms) Great that you looked on the original source, and there is a lot on number theory embbeded there. Great job!!
I feel like remembering the rules of operating this calculator is harder than remembering the formulas needed to solve the problems by hand (the exception maybe being square roots as those are a pain"
I like the graphical nature of the basic arithmetic. Often times we forget what we're doing with symbols and this brings it back to a more graphical representation. I wonder if it's possible to compute logarithms this way.
This is fascinating!! Genuinely gonna go play around with a chessboard and try this myself, it seems like a fine new puzzle. I think I’ll try to fill out a times table using this method.
@@darinpringle5611listen again to what he said. I understand I can be the "🤓" type of guy but he clearly said "bigger chess board". I still understood that he wanted to have more squares don't worry.
I get to say something technically wrong because everyone will know what I mean and it is a short way to say it, and you get to tell a funny joke, it's a win win
Reminds me of the time I found a way to factor numbers using only increment and subtraction. Trivially simple and useful, even if you have to contrive your situation somewhat.
i love this! it's fast, and doesn't have any limits; PLUS I can learn to do it in my head! i think i might try to memorize the binary for all of the numbers upto 1024, and then start practicing this on the chessboard in my head
@LordDIO-z4w yeah, it gives you the remainder 128/5 = 16+8+1 + (2+1)/5 = 25 + (3/5) You could multiply the remainder by 10 and then divide by 5 again if you want the next digit: = 25.6 (3*10)/5 = (2+1)*(8+2)/5 = (16+8+4+2)/5 = 4+2 = 6 So you know it's 25 + 0.6
so, if an infinite chessboard is Turing complete, and we can emulate any computational device with any turing complete computer, and if neurology can be emulated with a binary computer, that means any chordate brain, and surrounding biological systems, can be emulated with an infinite chessboard. Which means that given the right chessboard and program, it really *is* a Lark!
The chess board was used as a counter board in many civilisations. That is why Napier (as a young boy) started with it. He was only 16 when he realised the power of using multiples of 2. But it can be used with powers of any number.
It's not that difficult to generalize it to any integer base k. One issue is that you need to remember (or have it written on paper) the multiplication table up to k-1 by k-1 so it's no longer as simple to operate. Anyway it boils down to long multiplication just with a physical instrument rather than paper so it's more convenient. Especially for computing square roots.
" Why You Can't Bring Checkerboards to Math Exams ", wtf are you talking about, you get as much scrap paper as you want and can draw chessboards onto that.
I really wish my college professors taught binary arithmetic in this way back when I was still in college (Computer Engineering), as a visual learner this clicks for me so much easier than trying to understand why it works from a purely mathematical perspective.
If I see a person using a checkerboard during a test, I will allow him to do so. He is actually using the binary system. He is actually using his mind.
23 วันที่ผ่านมา
The way you describe how this works does very much sound like the way I learned to divide in school.
Maximum Ammount Of Numbers To Be Inserted In This Checkerboard: 128 × 128 = 1024 + 2560 + 12800 = 3584 + 12800 = 16384. So this COULD be OP, but really limited, without using more than 128.
But a square root isn't a unary operator. There is an implied 2 as the root that is operated on the radicand. That's 2 numbers required for the operator.
its wild how many easier hacks for math there are, like weighted numbers for averages blew my mind when i learned it a few days ago (im 23 and been out of highschool for like 5 years xD)
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That looks to be a portion of A Go Board, not a checker board
Texas Instruments probably already knows.
@antstone2898 better watch the skies. Texas Instruments is also an international arms manufacturer
"There is then nothing more
that these pennies on 64
can accomplish"
Is not a sentence I thought I'd ever hear
They also kinda have the monopoly on rocket guiding systems so if they find out you might be fricked.
What’s real fun is that you can just draw this “board” on any piece of paper and immediately produce a calculator. Add more rows and columns for bigger numbers
Any calculating power you desire can be yours
What's even funnier is that you can just do long division on paper
@@matejnovosad9152 but that's not as fun
If in the hypothetical scenario of a desert or island (without a calculator,) just draw in the sand?
@@1994AustinSmiththat works i guess
So when my calculator runs out of battery during an exam I can't rely on my backup chessboard?
Unfortunately, no.
But maybe you can use your back up Rubix Cube I heard that can also be a calculator also haha.
It would be a lark!
@@ZDTF 0:58
Just don't stalemate yourself
90’s teachers, “you won’t always have a calculator in your pocket in the future.”
Jokes on you.
*pulls out checkerboard*
😂
Me: literally always has a calculator app in his pocket.
Since you will always have a checkered sheet of of paper you can build your own checkerboard during a math test.
Cut some small paper chunks and paint them with your pencil and you have your "coins"
Me who gets a whole paper table and starts making the setup as I pull out pennies
If someone is genius enough to come up with the idea of using a checkerboard as a calculator, then they should be allowed to do so because they have damn well earned that right. If I were a maths teacher, I would be more impressed with someone who was smart enough to convert an innocent board into a makeshift calculator than someone who is scribbling calculations on a rough sheet of paper.
The actual techniques you use on this chessboard mirror the same techniques used on 8-bit processors before multiply and divide instructions were added.
FYI, the instructions do this in the microcode instead! Binary math is binary math!
Somehow, this is the only explanation that has ever actually gotten me to understand 8-bit multiplication lmao
Yep this is a representation of a computer. Save this to teach it to are descendants so They can defeat their machine overloads without the need of a machine fifth column. 😂
@@GodzillaGoesGaga Unless CPU has HW multiplier in form of combination circuit.
or you can just use full adders and then do a repetitive addition for multiplication and repetitive subtraction for division
Teacher: "You can bring a 3x5 reference card for the exam"
Me: **frantically tracing graph paper grids onto a 3x5 card and snapping off bits of pencil lead to use as counters**
😂😂
Not a bad idea, I wonder if this method is faster than just doing the math
@@vishu1334 For large numbers, and especially if you're willing to take "good enough" approximations, yes
It's basically a slightly more sophisticated but much more complicated slide rule
@@bluerendar2194Would it even be 'good enough approximations'? Since this is binary, as long as you have enough 'bits', you can represent any number you want. So, you should be able to get the exact answer every time without having to approximate anything.
I wonder if you might even be able to double the bits in the same grid size by doubling the pennies.
So, instead of 1 penny, you could have 3 or 7 pennies to triple the amount of bits you can represent per grid spot. That'd be 1, 3, and 7 pennies total for each square if all in the same spot. 1, 2, and 4 for being in separate spots. You should be able to separate them out for the final result based upon that.
Or, you could have different colored pieces or different coin types just to simplify everything a bit.
When you got to root 2 I was really hoping you'd pull out a second chessboard and move into the 4th quadrant.
my camera is usually very zoomed in, so that would have been a good time for a dramatic cut to zoom out
The division procedure made it obvious why this invention of the a bit too brilliant Napier did not catch on very much.
yes 😂
i mean it seems likely that you could get used to it after a bit of practice, even if it’s not quite as intuitive as multiplication on the checkerboard at first glance. division just always has to suck i guess lol
@@_rr_-qm5og For mental calculaton, I've once and for all learned the three first decimals of the inverse of 1 to 50 by heart. I recommend it! Then it's just multiplication to get any rational number, within reasonable error margin for many practical purposes.
Division is much more useful than multiplication. Division allows you to put quantities in relation to each other. It should be mandatory in elementary school, together with the "times table".
And it's not 50 unique numbers, most are multiples or have other nice patterns. 1/9=0.111... 1/11=0.0909... See how 1 and 9 go together? It has to do with the base 10 system. Just looking at the inverses of integers makes one realize that there's something to number theory. At least it helps memorizing some inversed integers.
The inverse of a prime has a repeating sequence of decimals the length one shorter than the prime. And the first half of those decimals plus the second half results in all 9s. "Go figure!" is a fitting exclaim here.
1/7=0.142857 so 142+857=999 or 14+28+57=99 if you prefer.
1/13=0.0769230 so 769+230=999 but 076+923=999 or 07+69+23=99 also works.
1/17=0.05882352941176470 so 58823529+41176470=99999999.
1/13 only has 6 repeating digits instead of 12, or well, I suppose one could say that 1/13=0.076923076923 with a repetition within the repetition, as if by chance. And by the way, the inverses of primes have an iterated digit sum of 9. But don't go insane, there are solid logical explanations.
@@_rr_-qm5og If you stick to a more granular log scale and are willing to accept approximations, then eh not really
its pretty cool how it uses binary
I find it irresponsile to unleash such a powerful tool on humanity
what can i say, I am a daring and mischievous lark-fiend
@@WrathofMath dude, you taught me what lark means, and i will never look at my friend's 1961 studebaker the same again.
You describe lark as an obsolete word but where I live it's in everyday use. "We did it for a lark", "the kids were just larking about". Hopefully it'll catch on on your side of the pond! Solid word!
Absolutely! I'd like to bring back hootenanny, hullabaloo, swanky, lark, and cattywampus
@@WrathofMathYou could also learn from Indian English and enjoy preponing your meetings. (I haven't convinced my Indian friends yet of the brilliance of 'poning' our appointments, ie having them exactly on time. But everything is better than just boring postponing.)
Why be a daredevil when you could be a larkfiend
@@WrathofMath cattywompus is still used
@@WrathofMath Such chobblesome words!
I'd allow checkerboards in my math exams. Good luck trying to do formal algebra on it :-)
That... actually sounds kind of interesting to try.
Charles Babbage and Ada Lovelace have entered the chat...
You're on.
How to do 1 variable algebra on chessboard:
1) Do a bit of eliminating (subtraction)
2) place numbers on 1 place while x on the other (reverse operations)
3) do addition & subtraction on both sides
4) do division
5) Now you have the value of x, yay!
You only need a 3d board for that
Prophecy: a board like this, except with one of the axes labelled with powers of 3, will one day be used to settle the Collatz conjecture.
Don't give me any ideas, Collatz has already usurped far more of my time than it ought to have 😭
This is such a good idea, if the collatz conjecture works with multiples of 3, wouldn't it make more sense to work with it using a base 3? Why has no one thought of this.
@@MilkGlue-xg5vjDividing a number in base 3 by 2 can be annoying. Perhaps base 6 could be somewhat insightful instead.
@@Opisek Yes, I relized that after I wrote my comment.
@@Opisek But how will you multiply by 3 without 2 then?
when bro claims he grew up poor but never used a chessboard and old pennies as a calculator
This uses the fact that humans can at a glance find the intersection of horizontal and vertical grid lines. Because while the human mind cares not at all about doing advanced mathematics for what appears to be no practical reason, it does still use advanced mathematics for things that do have benefit, like intersections which help traverse the real world. The trick is always setting up the problem in a way that the human mind can understand as useful, once done to a sufficient degree the brain is perfectly capable of outpacing a calculator easily.
this is one of the best comments I've seen on youtube
Can the brain outpace a calculator when you remove the bottleneck of typing everything in manually?
@@Jeff-ss6qt Yes.
@@Jeff-ss6qt yes, some of the first people who got research-purpose neural implants in the 2010s can output text as fast as they think it. imagine what someone who grows up with one of those could do. i'm not to interested, personally.
It makes me feel so stupid that there is no way I could even figure this out on my own but seeing it explained seems so intuitive
Same
There is something very appealing about squares being used to find squares
if you break math down like in lambda calculus you find that all operations can be unary, not just square roots and some other operations.
You could use SKI calculus.
For lambda calculus you can proof existence of fixed point combinator for every lambda function but this is not true for all algebraic functions. So you shouldn't use lambda calculus for everything because it's quite restricted
@@lpi3 You can stimulate everything else in (untyped) lambda calculus, if you wanted to badly enough.
Division by zero?
One more: define f x = f (x+2). Find f 1 with lambda calculus
you could make a ternary calculator this way with a Go board. Empty for 0, white and black for 1 and 2
We're gonna push this calculator to the limit
You could use heads and tails on your pennies for ternary, too. Or mix different size coins. But that would probably slow you down more than it's worth.
Using a higher number base doesn't make the math faster. It'd actually probably be a lot slower because it makes it more complicated and you have more rules to follow.
I mean, to be fair, you could do the same thing with a chess board and just use multiple types of coins, too. You could even do base-4 with pennies, nickels, and dimes, for example. (But I agree with @Andoxico that that would probably actually be a lot harder to do math with, in most situations.)
A Go board could also be used just as well for this same binary-based system, though, and it would allow you to do larger numbers in a more compact way...
@@foogod4237 in the limiting case, you can use a marker to write directly onto the board
I'm just gonna draw a calculator in class
Just seeing it may be enough to inspire your mind's calculations. I know how inspired I feel when I see a TI-108
I mean you could draw a checkerboard and just remember where the pennys should be
@@andrewcherry3058 i just draw a blackboard in my head. sometimes the numbers get smudgy but I make do
@@WrathofMath What is a TI-108?
@spin4team4096 th-cam.com/video/xrmqoKchspo/w-d-xo.html
Honestly i think that introducing this to schools might be quite good, this is both useful as a tool to make some calculations easier as well as being a good showcase for some mathematical principles, which should also make it easier to do these steps in your head. Additionally, unlike with real calculators, there is a high likelihood that over time, the students will do more and more steps, only including the more difficult parts, since that takes less effort. The only real problem is that it can't be used for the more difficult calculations involving fractions and irrationals.
Just extend the checkerboard on the right and bottom, adding 1/2, 1/4, 1/8, etc
The Egyptians managed by using binary as the foundation for their mathematical algorithm/procedures (including fractions). Today this can still be seen in the Ethiopian method of doing arithmetic.
This is seriously what I'm looking for in TH-cam - new content that *actually* blows my mind
Directly followed, thank you for your effort!
That’s actually really devious, you can distract your classmates with the sounds of coins sliding on wood. This would only be a good strategy when you knew you were going to fail and just wanted to lower the class average, but still
That just made the following scene occur to me
Diligent Student: Mrs. Teacher, can I borrow your calculator?
Teacher: Sorry, I don't have any more to lend
Devious Student: You can borrow mine (reaches into backpack for chessboard 😈)
to be fair, I think you could convince a fairly large amount of professors to substitute your calculator for this for fun (except when there's not enough space)
Yeah I don't know how a professor would feel, but I think a high school teacher would say no way - you're not gonna use some dumb useless old calculating method and tank your grade, get a TI-84 out the bin.
@@WrathofMath Very true
As a professor, I’d happily let my students use this!
Unless i'm mistaken, you don't have to label the axis in base 2. You could use base 4, base 10, whatever, to get a much bigger range with just an 8x8 board.
@@mieperb6579you cannot represent all possible numbers in it
Finally, a use for all 6 of my chess boards
This is their true purpose!
@@ZDTF Alright man, relax
Wow, I thought I was the only girl on the internet /s
@@Ethorai_Astraliswdym
girls aren't real
i can confirm because i am one myself and i clearly don't exist
@@Ethorai_AstralisYou're a liar, there are no girls in the internet
For square root of 2, you could also relabel the rows and columns: 2, 1, 1/2, 1/4, ...
Yes, but that would give you a binary approximation. Multiplying by 100 gives a decimal approximation
@@mt.penguinmonster4144 I have no problem with binary, as I started programming in BASIC at school in 1977 (when 16K was a LOT of RAM!), and I soon found the 'front panel' of the Research Machines 380-Z, which got me into Z80 coding, and I got my own computer (a Nascom 1 with 1K RAM) in 1979, so I could ONLY code in Z80!
@@mt.penguinmonster4144 Computers do the exact same thing so i see no problem. The 1/2, 1/4 are in the decimal system so you don't have to convert binary to decimal, it's already done.
Step 1: have some sort of flat candies
Step 2: have a piexe of squared paper
Step 3: calculate all square roots >:j
Computers are sweating; we're coming for them!
Even better, use the rectangle paper they give you and tear off the longer side to make it a square. Tear the scrap into ‘penny placeholders’ and you can do it with just paper and pencil.
0:13 do they have to be slightly sticky or can I use extra sticky ones?
I wouldn't risk using extra sticky ones, it might get really...sticky 🤭
Repent from your sins brother. @BadxManners
What????
...
it’s getting sticky in this batch
Calculator ❌
Abacus ✔️
13:27 I was like "Yep, I also don't know the square root of 9" 🤦♂️
That's what calculators did to us. Couple of years ago at a test I saw my friend inserting 1+1 into the calculator. That shit scares me to this day.
@@michaldzurik535Just gotta check, in case the universe changed. Remember to do it twice in case a cosmic ray flipped a bit in the calculator and it messes the calculation up.
(I do this every single math exam even though I have A+ in all of my math courses)
5:09 But you do have a bigger chessboard! Ol' Johnny Napes labeled all 15 diagonals, so he had labels on the left and top as well, so 256 all the way up to 2^(15-1).
They start trembling when i do 100 moves in one hour
😂
This is actually similar to a factoring method I developed using Matrices, except you have a Chessboard instead of a matrix and the factoring only works for small numbers as the cost increases exponentially.. 😄
That’s a really interesting thing, you’re doing binary operations without converting to base 2, and the same idea it’s behind the computation of the curta computers, using gears instead of squares
Me on my way to creating the biggest checkerboard ever to do calculations without electricity using only pennies. 💀
"a lark" is very much still in common use in british english.
This brings a whole new meaning to the term "calculating moves"
If you think about it in a bit more detail, there are some additional rules to doing the square-root calculation that can make it a lot easier to know what moves to try to make.
First, the whole thing must start with a single penny on a white square (this is the only way to make the necessary top-left corner of any solution). Therefore, if your number starts with a black square, you immediately know that you need to substitute downward, and then you also know you will only use one penny from that and then substitute the second one downward again.
Also, the number of pennies starting from a black position must always be even. Therefore, if you have an odd number of pennies on a black starting position, you know you will need to substitute down at least one of them.
More importantly, you can't just move a penny to any square you feel like. There are basically only two forms of valid "move" after the first one:
1) You can move _two_ pennies to fill an unoccupied spot on the first column of the result, and also the corresponding spot on the top row (a bottom-left and top-right corner combo).
2) You can move a penny to some other column _but only if that position already also has a corresponding penny in both the first column and the top row._
(basically, move #1 is the only way to create new columns/rows in the answer, and move #2 is the only correct way to fill in empty spaces in existing columns/rows.)
Therefore, based on these rules, it becomes pretty obvious that at 16:34 and 17:05, due to the restrictions on rule #2, those moves you considered and discarded are both going to be invalid moves, so it's not really worth even considering them.
I am so amused by you setting both pennies aside and getting a new one at 4:50, like they're permanently 64 markers now, you can't simply put one on 128
Yes! You got that right!
Now I'm picturing tiny gnomes or fairies inside my old 8 bit home computer rapidly sliding coins on tiny electric chessboards.
Some annoying people:"Binary isnt useful"
Binary(It works with powers of two):
Wow, so simple and elegant. Chess board really is a tool for the mind. Great video and great presentation in on point, keep up the great work and have fun 😁
You know what... I've been using that method on math exams multiple times. Never used a chessboard, though, just a strip of paper where I write the binary sequence of whatever number I'm calculating
Is it faster than normal?
Come to think of it, you're usually permitted all the scrap paper you want, you could easily rip some up, use it as counters and use more paper to make the board. Check mate.
@@QSBraWQ I don't know
@@QSBraWQ I've never actually been that good at math. I've always been faking it.. counting on fingers; using scrap paper to write binary calculations...
@@koppadasaoWait, you're not supposed to do that!?
Next, why not take the step up and discuss the Slide Rule?
I need to get one. I have an addometer, an old mechanical calculator, but no slide rule! I have a lot of old vintage electronic calculators as well
If I were a math teacher, I'd 100% allow this calculator because you HAVE to know math to use this calculator. The multiplication technique is literally a representation of polynomial multiplication in action.
Question #1: If you have an 8x8x8 3D chess board, can you do cube roots?
Question #2: If you label the vertical as 0, 1, 2, 3, ...7, can you do logarithms?
The division method makes me feel iffy. I was expecting you to complete the rectangle by columns, but sometimes you randomly decide to move a penny up to the top row before moving the next penny to finish the column. Which is weird because once you set a penny to a square, one in its column _must_ have a penny unless you backtrack. But you prioritize filling in the rows first.
Very nice video!
Nice to see another Napier device for facilitating calculations (I was only familiar with the Napier's bones and this other little thing called logarithms)
Great that you looked on the original source, and there is a lot on number theory embbeded there.
Great job!!
This method is pretty versatile; it works with cents, too!
WELL NAH 🤯
It works with any object that can fill a tile
the username checks out
makes sense
@spin4team4096 I believe you would likely struggle a lot to make the method work with piles of cinnamon, chunks of white phosphorus, or a spray can.
this just made math fun again and really helped me visualize value relations a lot easier. Thank you
You can make this use signed bytes instead of unsigned bytes (2's complement) by replacing 128 by -128.
I think this is my favourite video from you so far
Thanks so much! I love talking about these weird calculating methods!
This makes calculatieons feel like a game!
This is a pretty useful way of visualizing math in your head. I’ll keep this is mind for when I’m in class to speed things up.
0:25 i see powers of 2, this is going to be a video on bitwise arithmetic isn't it
Edit: 1:58 Argh he got me
A bit too obvious?
I've always struggled with maths but this visual representation makes it mindblowingly so much easier for me!!!
I feel like remembering the rules of operating this calculator is harder than remembering the formulas needed to solve the problems by hand (the exception maybe being square roots as those are a pain"
It is AS HARD AS the formulas you want to remember, because the only difference is the base
It does seem mostly useful for optimizing operations on computers rather than for humans, yah. Since they only see in base 2 to begin with.
I like the graphical nature of the basic arithmetic. Often times we forget what we're doing with symbols and this brings it back to a more graphical representation.
I wonder if it's possible to compute logarithms this way.
So when i forgot my calculator then i'll just get my chesssboard with dirty dirty coins
Yup!
This is fascinating!! Genuinely gonna go play around with a chessboard and try this myself, it seems like a fine new puzzle. I think I’ll try to fill out a times table using this method.
4:58 "and if we had a bigger chess board"
It doesnt work like that...
A bigger chess board would still be 8x8
Nope: en.wikipedia.org/wiki/Chess_on_a_really_big_board
Chess on a "bigger board" not on a bigger CHESS board.
@@ClarkCox Well... yea he didn't specify if it was a "standard" chess board or not but who uses 16x16 chess boards anyways?
@@darinpringle5611listen again to what he said. I understand I can be the "🤓" type of guy but he clearly said "bigger chess board".
I still understood that he wanted to have more squares don't worry.
I get to say something technically wrong because everyone will know what I mean and it is a short way to say it, and you get to tell a funny joke, it's a win win
Reminds me of the time I found a way to factor numbers using only increment and subtraction. Trivially simple and useful, even if you have to contrive your situation somewhat.
It's okay, Texas Instruments already knows about this. Their chips are miniaturized chess boards.
Very creative, you did a good job explaning and making this video. Keep it up! Have a wonderful day.
Gotta say, I don't hear "oldfangled" nearly enough
One day we'll all be oldfangled, might as well embrace it!
i wish i learned this when i was younger.. this is suprisingly intuitive
i love this! it's fast, and doesn't have any limits; PLUS I can learn to do it in my head!
i think i might try to memorize the binary for all of the numbers upto 1024, and then start practicing this on the chessboard in my head
You can make binary really fast using subtraction by powers of 2, pick the largest power of 2 smaller then your number and keep subtracting it.
It doesn't work well if the result is not a whole number.
Try to get 128÷5 to work.
@LordDIO-z4w yeah, it gives you the remainder
128/5 = 16+8+1 + (2+1)/5
= 25 + (3/5)
You could multiply the remainder by 10 and then divide by 5 again if you want the next digit:
= 25.6
(3*10)/5 = (2+1)*(8+2)/5
= (16+8+4+2)/5
= 4+2
= 6
So you know it's 25 + 0.6
“fast”
We used to play with pennies for chess and checkers, because we lost the pieces as kids.
so, if an infinite chessboard is Turing complete, and we can emulate any computational device with any turing complete computer, and if neurology can be emulated with a binary computer, that means any chordate brain, and surrounding biological systems, can be emulated with an infinite chessboard.
Which means that given the right chessboard and program, it really *is* a Lark!
If they kick me out of an exam for using a chessboard instead of my TI-84, someone's getting a sharp #2 pencil in the eye.
This cool and all, but how do i divide by 0?🧐
You explode the chessboard with dynamites
Learned about Mr Napier with his quite simplistic multiplication method, now known as Napiers Bones, quite glad to see he was somewhat of a madman!
Great, now how do I integrate?
The chess board was used as a counter board in many civilisations. That is why Napier (as a young boy) started with it. He was only 16 when he realised the power of using multiples of 2. But it can be used with powers of any number.
@ 11:26 - Just use 7-zip.
No
That's what your mom said last night
"You have been playing checkers, whilst I've been doing math"
-me
8:58 what about 7 divided by 7
Anything divided by itself IS ONE. It's a very basic rule in maths
@spin4team4096 uh... wanna run that by me again?
@spin4team4096do you mean to say 1 or what?
@@AdjustedMaple Yes, my bad. I had 4 exams right after another for 5 days so I'm burnt out and my brain isn't braining anymore 😭
@@AdjustedMaple what did he say before?
I've done multiplication with the Montessori checkerboard, but never did it with a real checkerboard! This is incredible!
5:21 ‘Highly advanced Calculation Device’
*shows a checkerboard
🤣🤣🤣🤣🤣
It's not that difficult to generalize it to any integer base k. One issue is that you need to remember (or have it written on paper) the multiplication table up to k-1 by k-1 so it's no longer as simple to operate.
Anyway it boils down to long multiplication just with a physical instrument rather than paper so it's more convenient. Especially for computing square roots.
1:13 as soon as he said kicked out of the room TH-cam kicked me out and crashed
I read ur comment right as he said that phrase
Neat demonstration, thanks for sharing!
The square root is a unary operator in the same way that "dividing by two" is a unary operator
" Why You Can't Bring Checkerboards to Math Exams ", wtf are you talking about, you get as much scrap paper as you want and can draw chessboards onto that.
This was a fantastic video. Very well done.
Thank you!
Lovely little visual rephrasing of the classic binary arithmetic operators.
I really wish my college professors taught binary arithmetic in this way back when I was still in college (Computer Engineering), as a visual learner this clicks for me so much easier than trying to understand why it works from a purely mathematical perspective.
you can use the middle of the squares yes, but also the intersections for even more lines
If I see a person using a checkerboard during a test, I will allow him to do so.
He is actually using the binary system.
He is actually using his mind.
The way you describe how this works does very much sound like the way I learned to divide in school.
the chess update be hittin different.
Maximum Ammount Of Numbers To Be Inserted In This Checkerboard: 128 × 128 = 1024 + 2560 + 12800 = 3584 + 12800 = 16384. So this COULD be OP, but really limited, without using more than 128.
you can have more than 1 coin on 128 you know
i've never really thought of bringing a checkerboard to a math exam but thanks for the extra info
np
The thing is that you can easyly recreate the system with a paper and pencil using small paper ball as marker
If you consider +/- then base 3 also works.
It is good to explore new areas of calculations.
I love this so much and I can't believe I had to exist on this spinning hellball for 32+4+2 years before learning about it!!
But a square root isn't a unary operator. There is an implied 2 as the root that is operated on the radicand. That's 2 numbers required for the operator.
That's so cool. My jaw fell when you showed each new thing!!!!!!!!!!!
This is the new common core instead of just stacking and multiplying like a normal person.
its wild how many easier hacks for math there are, like weighted numbers for averages blew my mind when i learned it a few days ago (im 23 and been out of highschool for like 5 years xD)