Julia Sets and Orbits of Complex Iterations - Geogebra (Mandelbrot Build Part 1) SparksMaths 2022-01-07 | This is the 'how did' video(s) for the crucial files I used in the @numberphile Mandelbrot video here: youtu.be/FFftmWSzgmkAlso mentioned by Grant Sanderson @3blue1brown in his excellent "Beyond the Mandelbrot Set" here: youtu.be/LqbZpur38nwPart 1 here gets us up to the Julia Sets.Part 2 gets us to the Mandelbrot orbits - and the set itself youtu.be/3BXoNOahFJkThere's a Geogebra file here: geogebra.org/m/uf4AMrQH
GeoGebra: Rotation - find the centre - Transformations in under 1 minute #shorts #geogebra SparksMaths 2024-01-05 | How can you find the centre of rotation if you're not sure where it is?Rotations around a point in 2D are nice when you can see them change dynamically.I'm using Geogebra Classic 5 on a PC, but this will work similarly on the mobile app, or in a browser.
GeoGebra: Rotation - Dynamic Geometry Transformations in under 1 minute #shorts #geogebra SparksMaths 2024-01-01 | Rotations around a point in 2D are nice when you can see them change dynamically. A classroom demo file built from scratch in less than in minute.I'm using Geogebra Classic 5 on a PC, but this will work similarly on the mobile app, or in a browser.
GeoGebra: Alternate Segment Theorem - dynamic Circle Theorems in under 1 min #shorts #geogebra SparksMaths 2023-12-22 | Making an interactive and moving demonstration of circle theorems helps reinforce what the situation is. You can do it very quickly and easily in GeoGebra. This one is for the fact that the angle from a chord to a tangent is equal to the angle off that chord in the alternate (other) segment.ALL CIRCLE THEOREMS LIVE HERE: geogebra.org/m/xkvz2EfF
GeoGebra: Chord Bisectors - dynamic Circle Theorems in under 1 min #shorts #geogebra SparksMaths 2023-12-18 | Making an interactive and moving demonstration of circle theorems helps reinforce what the situation is. You can do it very quickly and easily in GeoGebra. This one is for the fact that a perpendicular bisector of a chord in a circle will always pass through the centre of the circle (and conversely, a line through the centre of a circle perpendicular to a chord will bisect it).ALL CIRCLE THEOREMS LIVE HERE: geogebra.org/m/xkvz2EfF
GeoGebra: Cyclic Quadrilaterals - dynamic Circle Theorems in under 1 min #shorts #geogebra SparksMaths 2023-12-11 | Making an interactive and moving demonstration of circle theorems helps reinforce what the situation is. You can do it very quickly and easily in GeoGebra. This one is for the fact that opposite angles in a cyclic quadrilateral always add up to 180°.ALL CIRCLE THEOREMS LIVE HERE: geogebra.org/m/xkvz2EfF
GeoGebra: Tangents to circles - dynamic Circle Theorems in under 1 min #shorts #geogebra SparksMaths 2023-12-04 | Making an interactive and moving demonstration of circle theorems helps reinforce what the situation is. You can do it very quickly and easily in GeoGebra. This one is for the facts that the two tangents from a point to a circle are the same length, and tangents meet radii at right angles.ALL CIRCLE THEOREMS LIVE HERE: geogebra.org/m/xkvz2EfF
GeoGebra: Angle in a semicircle - dynamic Circle Theorems in under 1 min #shorts #geogebra SparksMaths 2023-11-20 | Making an interactive and moving demonstration of circle theorems helps reinforce what the situation is. You can do it very quickly and easily in GeoGebra. This one is for the fact that the angle in a semicircle is always 90°.ALL CIRCLE THEOREMS LIVE HERE: geogebra.org/m/xkvz2EfF
GeoGebra: Angles at the centre - dynamic Circle Theorems in under 1 min #shorts #geogebra SparksMaths 2023-11-13 | Making an interactive and moving demonstration of circle theorems helps reinforce what the situation is. You can do it very quickly and easily in GeoGebra. This one is for the fact that the angle off chord at the centre is always twice the angle off that chord to the edge.ALL CIRCLE THEOREMS LIVE HERE: geogebra.org/m/xkvz2EfF
GeoGebra: Angles in the same segment - dynamic Circle Theorems in under 1 min #shorts #geogebra SparksMaths 2023-11-06 | Making an interactive and moving demonstration of circle theorems helps reinforce what the situation is. You can do it very quickly and easily in GeoGebra. This one is for the fact that the angle off a chord anywhere in the segment is the same (this is a good starting point for proving many other circle theorems).ALL CIRCLE THEOREMS LIVE HERE: geogebra.org/m/xkvz2EfF
Rolling wheels in Geogebra. Cars, Cycloids, and Easing functions. SparksMaths 2023-11-02 | I make a car with wheels that roll correctly, and discuss cycloids and easing functions along the way.Rolling circle file here: geogebra.org/m/quatamj2Car file I built here: geogebra.org/m/ddyvxhnhInigo Quilez on YouTube @InigoQuilez I recommend watching these two: youtu.be/8--5LwHRhjk?si=IN4VfEa7PTQvSX8Xyoutu.be/BFld4EBO2RE?si=RRruV0-5Iot67Wl8Sections:00:00 Intro01:00 Rolling circle basics11:38 Rolling Car animation26:57 Easing functions37:18 Easing function in the rolling car file41:11 Making the bumpiness happen51:48 Clean final file and outro
GeoGebra: Locus command (make a path from a moving object) in less than 1 minute #shorts #Geogebra SparksMaths 2023-10-30 | I construct a cycloid (the path of a point on a rolling circle) using the Locus command in Geogebra
GeoGebra: Reflection in a circle (circle inversion) #shorts #geogebra SparksMaths 2023-10-23 | You can reflect an object in a circle. This is called circle inversion, and is worthy of a google to find out the details.
GeoGebra: Reflection in a plane #shorts #geogebra SparksMaths 2023-10-23 | You can reflect an object in a flat plane in 3D mode, which is a very familiar thing from looking in a flat mirror!
GeoGebra: Reflection in a point #shorts #Geogebra SparksMaths 2023-10-23 | You can reflect an object in a point. In 2D this is equivalent to rotating by 180 degrees about that point. (Try it in 3D if you're curious)
GeoGebra: Reflection in a line #shorts #Geogebra SparksMaths 2023-10-23 | The reflect command in Geogebra has a few options, but maybe the most familiar to most of us from school is a 2 dimensional reflection in a line.
GeoGebra: 2 Dimensional Sequences (lists of lists), in under 1 minute #Shorts #Geogebra SparksMaths 2023-10-16 | Sequences can hold sequences, and so create 2 dimensional arrays (for example an array of points here).
Spiral of Theodorus - Geogebra Live Build SparksMaths 2023-10-13 | You can construct all possible square roots of the integers, as lengths, by stacking right angled triangles in this way. This is a live build in the @GeoGebraChannel software.I'm using Geogebra Classic 5.The actual file I build is available here: geogebra.org/m/vfnbthfz
GeoGgebra: Basic Sequences, in under 1 minute #Shorts #Geogebra SparksMaths 2023-10-09 | The sequence() command in Geogebra is one of the most powerful tools, but there is no button on the User Interface, you've got to type it in! You don't have to just make sequences of numbers either...wiki.geogebra.org/en/Sequence_Command
GeoGebra: Enlargement - Dynamic Geometry Transformations in under 1 minute #Shorts #Geogebra #maths SparksMaths 2023-10-02 | Enlargement from a centre by a certain scale factor, built from scratch in less than one minute.I'm using Geogebra Classic 5 on a PC, but this will work similarly on the mobile app, or in a browser.In the US this is often called dilation rather than enlargement, and the command in US versions is actually 'dilate': wiki.geogebra.org/en/Dilate_Command
Regression Lines & Deviations demo (and Geogebra lists and the Zip() command) SparksMaths 2023-09-29 | Prompted by a request on Twitter/X I demonstrate how to make an interactive visualisation of the deviations from a line of best fit (and how software can plot a calculated regression line for you).This is really a demo of the Zip() command and how it can be really useful in working with lists in Geogebra. wiki.geogebra.org/en/Zip_CommandThanks to Rob Southern for the prompt: https://x.com/mrsouthernmaths/status/1702642884560781433?s=20
Building the Impossible - A Penrose Triangle in Geogebra 3D SparksMaths 2022-11-06 | I make the (illusion of) the Impossible Triangle, or Tribar, or Penrose Triangle live in Geogebra.Be prepared for a fair amount of me head-scratching and trying to work out what I'm doing!Live build file here: geogebra.org/m/zdjp8fk7I mention Richard Lissaman during the lighting chat, his game SUMAZE is available here: sumaze.mei.org.ukIf you want to skip to the 3D lighting part or any other part, you can do so below:00:00 Intro03:08 Start of build32:52 A momentous moment33:58 Making it spin49:03 Sorting out the lighting (use the dot product)
Circles and Spirals and NOT the Golden Ratio SparksMaths 2022-09-10 | I spot a lovely animation doing the rounds on Twitter, but get upset about it claiming to be something to do with the Golden Ratio. To back up my slightly complainy tweet reply, I live build a demo file in GeoGebra to check my point.My Geogebra file here: geogebra.org/m/hdwkec7wOriginal tweet from Mathladyhazel here: twitter.com/mathladyhazel/status/1568228662784270338?s=20&t=qIv1_aT1R-gjuCYvqGVuCg(which contains link to the original reddit)
SAT Maths Paper Live - alongside Tom Rocks Maths SparksMaths 2022-09-08 | Tom Crawford and I decided to try the maths parts of the US end of high school exam - the SAT paper. We both recorded our attempts live - this is my version. Warts and all.Watch out for the marking video coming up soon!Tom's attempt here: youtube.com/watch?v=u0se7iWdNZw00:00 Intro02:12 Start Section 3 (non calculator)27:10 Start Section 4 (calculator)1:07:09 UPSETTING TYPO SHOCK
Simpsons Rule Geogebra Build Follow up SparksMaths 2022-08-25 | The final demo build mentioned at the end of the previous video (this one: youtu.be/Ca20mFQqZjY)We build a visualisation of Simpson's Rule to approximate a numerical integral with Geogebra.The demo file is here: geogebra.org/m/ucmhkun8
Integration and area functions - Geogebra demos SparksMaths 2022-07-26 | The integral of a function is an Area Function, and this video demonstrates a few ways of showing this in Geogebra, and how to build some other visualisations of the area under curves.00:00 Intro00:52 Part 1: The integral is an Area08:24 Part 2: Rectangle Sums16:59 Part 3: Upper/Lower Rectangle Sums21:05 Part 4: Trapezium Sums24:38 Part 5: Demo File - all sums in one fileHere's the file I demonstrate at the very start and at the end: geogebra.org/m/ucmhkun8Simpson's rule follow up here: youtu.be/6faVbdrgFZo
The Cake and Candles - Geogebra Build as used on Numberphile SparksMaths 2022-05-10 | This is a long one with several parts, showing how I built the files I used in the @numberphile videos here: Part 1: youtu.be/FkVe8qrT0LA and Part 2: youtu.be/l5gUrDg01cQ00:00 Intro02:25 Part 1 - The Visualisation34:45 Part 2 - The Simulations48:27 Part 2b - 2 Dimensions1:00:05 Part 2c - Circle Time1:22:53 Prettification1:34:51 4 AnimationsGeogebra files:3D Visualisation geogebra.org/m/ubwsbw57First 1-D Monte Carlo Simulation geogebra.org/m/usebcewrCircles with options geogebra.org/m/wdzeaer5I have heard (but don't know the origin of) the original puzzle expressed as a stick - with two marks - being broken. Credit to David Bedford and Rob Eastaway for recasting it as a nice cake problem and sending me down this rabbit hole. newscientist.com/article/mg24232361-100-puzzle-09-the-cake-and-the-candlesWatch Grant @3blue1brown and Brady discuss Bertrand's paradox on Numberphile here: youtube.com/watch?v=mZBwsm6B280Extended further discussion between them both here: youtube.com/watch?v=pJyKM-7IgAU&t=0sAlso check @nubdotdev 's lovely video discussing how to generate a uniform random point in a circle, which was entered for the Summer of Maths Exposition (2021): youtube.com/watch?v=4y_nmpv-9lI&t=316s
The Gradient Function (Differentiation using animated graphs) - live Geogebra Build SparksMaths 2022-02-21 | This one's a live build in Geogebra (Classic 5) showing how to make a moving gradient function, to demonstrate differentiation (even if you don't want to talk about it with that vocab yet).00:00 Start00:08 Intro01:51 Start the build07:48 Tidy up the aesthetics13:03 Gradient of sin(x)15:20 Define tangent as limit of a chord approximating the tangent19:10 Labelling and LaTeX
The Chaos Game - Geogebra Build - as seen on Numberphile SparksMaths 2022-01-25 | In this video I build a simple @GeoGebraChannel simulator to generate the fractals seen in the Chaos Game (wikipedia: en.wikipedia.org/wiki/Chaos_game), and as seen in the @numberphile video: youtu.be/kbKtFN71LfsFirst part is short and fairly straightforward.Second part is tinkering with the generalisations in GeoGebra.Third part I get a bit carried away playing with fractal dimension and how the GeoGebra variables affect this. You have been warned.Chapters:00:00 Opening00:31 Part 1 - The Basics09:36 Part 2 - The Upgrade23:18 Part 3 - The MathsThere's a GeoGebra file here: geogebra.org/m/yr2XXPmsI mentioned Ben Orlin - Maths With Bad Drawings (@BenOrlin on Twitter - twitter.com/benorlin)
Mandelbrot Orbits and the Mandelbrot Set - Geogebra (Mandelbrot Build Part 2) SparksMaths 2022-01-07 | This is a continutation of the story from the Julia Set build (youtu.be/ICqj7nJbiRI).We change view to the parameter space, and do some dynamic colouring to get a (crude) view of the Mandelbrot Set itself, before exploring some more ways to see it.The original @numberphile video is here: youtu.be/FFftmWSzgmk (watch this for the full story first?)The video exploring more of the maths of iterations of complex functions from Grant Sanderson @3blue1brown is here: youtu.be/LqbZpur38nwA Geogebra file is available here: geogebra.org/m/XQprvGbW
Animated Cube Present #shorts #GeoGebrAdvent SparksMaths 2021-12-24 | A YouTube #shorts for Alison Kiddle's #GeoGebrAdvent challenge, building an animated present (cube) in @GeoGebraChannel in a few seconds.The Net() command is here: wiki.geogebra.org/en/Net_Command
Xmas Tree Interactive Voxel Lights - GeoGebra Build for Matt Parker @standupmaths SparksMaths 2021-12-24 | Just before Christmas 2020 Matt P @standupmaths managed to turn his Christmas Tree lights into a voxel display, by scanning the coordinates and working out how to light them appropriately to generate 3D patterns. It's great: youtu.be/TvlpIojusBEHe asked me, during the filming back then, to create a GeoGebra file to tinker with the effect, with the real light co-ordinates. This year he's updating the Tree Lights and correcting the light locations (youtu.be/WuMRJf6B5Q4), and I'm releasing a "how-did" video of the GeoGebra file.The one I actually built in this vid is here: geogebra.org/m/bgegzaydThe (slightly more polished one) we used for Matt's stuff is here: geogebra.org/m/p2pdz6v2Thanks to JB Lewis youtube.com/channel/UCNdnjMLe08DV2yK8w1Dcu7A for pointing out the "what-did" or "how-did" vs the "how-to" style... :)
Factorials and the Gamma Function (Geogebra and Python builds) - for Stand Up Maths - Matt Parker SparksMaths 2021-12-10 | Matt Parker @standupmaths asked for my help to make his video about the factorials of non-integers (youtu.be/dGnIJFzkLI4). This video shows how I built the graphics for that video. I used Geogebra (Classic 5) to build the basic mathematical graphics and animations, but then Python for the 3d plots (because of an issue with the Gamma function not accepting complex inputs in Geogebra).The (basic) geogebra file I made is here: geogebra.org/m/ahpucrqfMy website: bensparks.co.ukTwitter: @SparksMaths
Topology (original song Roll to Me - Del Amitri) - for MathsJam Jam 2021 SparksMaths 2021-11-24 | At the Annual MathsJam Gathering in the UK we usually have a bit of fun re-writing lyrics to famous tunes with a mathematical theme, then having a big singalong. It's a MathsJam Jam.In 2021 (like 2020) the gathering happened online, and instead of a live singalong some people made recordings and videos of their re-written songs. This is one of them.Lyrics by Colin Beveridge (and Ben Sparks)Vocals by Ben SparksBacking music (and everything else) Tim SparksColin is a friend I met at the Dorset monthly MathsJam. He's a tutor and author, and has written several excellent popular mathematics books. You can find him at Flying Colours Maths flyingcoloursmaths.co.uk/about and @icecolbeveridge on twitter.Tim is my twin brother, he has his own channel here: youtube.com/c/aquifermusic and the channel for his band here: youtube.com/c/NikkoFirMusicHe's done a proper cover of the original song here: youtu.be/jdzTrxcAgoMFULL LYRICSLook around your twirled piece of paperIt is everything that you hoped you would see?As the end of one side bleeds to the other oneIt’s Sublime TopologyKönigsberg has been driving me crazyIf I crossed each bridge once, I'd be pleasedBut Euler proved that you can't do it, darn itThat's prime topologyAnd I don't think I have ever seenA hole just disappearShapes are porous (like a torus)Or airtight (like a sphere)Your pants are only half-on, there, matey -But your foot's stuck down and you can't get it freeTurn them inside out, with a wee bit of strainingYou'll be dressed fine, topologyAnd I don't think I have ever seenA hole just disappearShapes are porous (like a torus)Or airtight (like a sphere)Get your paper twirled, and then maybeDraw lines to divide it in threeCut the strip along the perforationSublime topology
Normal Distribution Tool Build SparksMaths 2021-11-17 | A quick live build of an interactive normal distribution, with a random sample underlying it.The file I make is here: geogebra.org/m/u7yawge7
Cobweb and staircase diagrams, iterating to solve equations. Live Geogebra Build. SparksMaths 2021-11-07 | In this video I build a demonstration of the cobweb/staircase diagrams that visualise how the fixed point of an iteration can solve an equation.The file I build is available here: geogebra.org/m/xjdpeefyIt's directly relevant to the Bifurcation Diagram I discuss here: youtu.be/rHHlsB5fo1QVeritasium's video (history of cubics and complex numbers) is here: youtu.be/cUzklzVXJwo
Bifurcation Diagram Live Geogebra Build (as seen on Numberphile) SparksMaths 2021-10-24 | Live build of the Logistic Map Bifurcation Diagram in Geogebra.The Numberphile Video is here: youtu.be/ETrYE4MdoLQVeritasium Video here: youtu.be/ovJcsL7vyrkGeogebra website: geogebra.orgTry building it yourself!ERRORAt about 01:00 I mention an accumulation point. The point I'm pointing to is NOT known as the accumulation point - rather an accumulation point is the limit of a sequence (e.g. the period doubling that happens from a=3 onwards, and reaches it's accumulation point at about 3.5699...). Sorry for the slip of the tongue!
GCSE Maths Exam Live - not a competition (ahem) with Tom Rocks Maths SparksMaths 2021-10-16 | I sit a GCSE Maths paper (UK age 16 qualification) live on camera - alongside @TomRocksMaths. We'll switch papers and mark each other's script, then discuss the outcome in another video.Tom and I did not look at the paper in advance and we present to you our 'warts and all' live attempts to do what thousands of teenagers have to do every year. It's definitely not a competition (but it's surprisingly hard not to get competitive at things like this...).Tom's live exam video here: (premiering at 1500 on 16th Oct 2021 - youtube.com/watch?v=hQVcv-T7IiY)Our wash-up discussion here, where you can see what both our final marks were: youtu.be/4DCb5P_uknY.Check out Tom's channel here: youtube.com/channel/UCRfo-DAifrP3lzcxUHtGm_A
The Golden Ratio and Phyllotaxis - Sunflower Spirals - Live Geogebra Build (as seen on Numberphile) SparksMaths 2021-09-19 | In this video I build (from scratch) in Geogebra a way of viewing the spirals in seed heads like sunflowers. It's the resource I used extensively in the Numberphile Video here: The Golden Ratio (why it is so irrational) - youtu.be/sj8Sg8qnjOg (it's probably sensible to watch that first, if you haven't already!)The file I build is here: geogebra.org/m/fh7jp5mqPhyllotaxis on Wikipedia: en.wikipedia.org/wiki/PhyllotaxisYou can get to Geogebra's main website at geogebra.org
Moiré effect/illusion - live Geogebra build SparksMaths 2021-09-12 | I saw this cool Moiré effect done physically by masaya ishikawa @kamone on Twitter (original tweet: twitter.com/kamone/status/1436489417363886080?s=20)They wrote more about it here: hillslife.jp/series/seeing-creates-something/layers-actIn the spirit of trying to record some more live creations (done without preparation to try to model the creative process, problems and all!) I thought I'd have a stab at creating a Geogebra version of Masaya Ishikawa's lovely physical version. Apart from the opening moments this is a live-one-take recording. Apologies for the occasional construction noise in the background.The file I build in the video is here: geogebra.org/m/ejqg6v5nAs Gábor pointed out in the comments, it's easy enough to make the gaps parallel - so there's a version with that implemented here: geogebra.org/m/w5s7yjen (interesting to compare?)You can get to Geogebra's main website at geogebra.org
Circle AEA question - live build no prep SparksMaths 2021-09-12 | A quick response to a glimpse of a discussion on twitter about a geometric maths question (AEA 2005 Q1)Link to original tweet from @colleenyoungtwitter.com/ColleenYoung/status/1436949000935014408?s=20Check out Geogebra at geogebra.orgAutograph at completemaths.com/autographOther excellent dynamic geometry at (for example) desmos.com and mathigon.org/polypad
Building a slope field in Geogebra SparksMaths 2021-07-21 | Quick Geogebra build of a Slope field or Tangent field. Prompted by Robin Houston on Twitter: twitter.com/robinhouston/status/1417781407410237445?s=20I'm using Geogebra Classic 5 geogebra.org/download
906,150,257 and the Pólya conjecture (MegaFavNumbers) SparksMaths 2020-08-19 | #MegaFavNumbers"Most numbers have an odd number of prime factors!"...or do they...?Ben chats about the large counterexample to Polya's conjecture, for Matt Parker and James Grime's MegaFavNumbers project.Ben is @SparksMaths on twitter and at http://www.bensparks.co.uk on the web Polya's conjecture on Wikipedia: en.m.wikipedia.org/wiki/P%C3%B3lya_conjecture#:~:text=In%20number%20theory%2C%20the%20P%C3%B3lya,Brian%20Haselgrove.EDIT:At around 10 mins I suggest that no-one knows whether the summatory Liouville function crosses between negative and positive infinitely often. Tanaka claimed this in 1980 in his paper doi.org/10.3836/tjm/1270216093 - *but he was in fact wrong*. Haselgrove had actually proved this was true, in order to make his 1958 argument. Haselgrove's paper is here: londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300001480 (but you may need to pay to access). Lehman in 1960 clearly recognised this too (ams.org/journals/mcom/1960-14-072/S0025-5718-1960-0120198-5/S0025-5718-1960-0120198-5.pdf) so I'm a bit puzzled why Tanaka claimed it was still an open question.(For full disclosure, Mathworld also made this error on their page on the Polya Conjecture, though it has now been pointed out, and will hopefully be corrected soon).I'm grateful to an eagle eyed viewer for pointing this out to me! Thanks - you know who you are.This video is part of the MegaFavNumbers project. Maths YouTubers have come together to make videos about their favourite numbers bigger than one million, which we are calling #MegaFavNumbers.We want you (yes you humble viewer) to join in! Make your own video about your favourite mega-number. You can think of a cool big number, or think of a cool topic first and hang a mega-number on it. Upload your videos to YouTube with the hashtag #MegaFavNumbers and with MegaFavNumbers in the title, and your video will be added to the megafavnumbers playlist.Submit your videos anytime before Wednesday 2nd September 2020 to be added to the MegaFavNumbers playlist! MegaFavNumbers Playlist: youtube.com/playlist?list=PLar4u0v66vIodqt3KSZPsYyuULD5meoAo
Ben makes Mandelbrot Orbits in under 30 seconds SparksMaths 2019-04-22 | Several people asked me for a way of programming a visualisation of the orbits of the Mandelbrot set in 10 seconds, after the claim I made at the end of the Numberphile Video here: youtube.com/watch?v=FFftmWSzgmkHere's a Geogebra version. To open the spreadsheet in a new Geogebra file click "View - Spreadsheet".I'll let you play with the aesthetics on your own version.You can download Geogebra here: geogebra.orgTo implement this version you'll need to use one of the 'Classic' versions (not the graphing calculator, since it does not include a spreadsheet).Yes - it did take me about 15 seconds. Forgive me. :)
Division controversy: 6÷2×(2+1) SparksMaths 2017-11-02 | The internet was having a meltdown over this calculation:6 ÷ 2 × (2+1) vs 6 ÷ 2 (2+1)Most people report the answer to the first one as 9. Some people report the answer to the second one as 1.This is not helped by some calculators reporting different answers to these two versions and some not.
Utilities problem - Solved! SparksMaths 2016-09-14 | The utilities problem is a mathematical classic - en.wikipedia.org/wiki/Three_utilities_problemIt's impossible to solve it on a flat surface (a plane).However, it is possible on topologically different surfaces - in this case a torus ('doughnut').Ballon from MathsGear.co.uk (http://mathsgear.co.uk/collections/shapes/products/torus-balloons)
