Can you solve this 4th grade problem?

hey this is press tow Walker here's a puzzle that was popular on Twitter from Dr Katherine young I may have a PhD but I'm also apparently failing fourth grade math here's the problem the square below is made up of four small squares can you shade half of it so that the unshaded part is also a square I love sharing puzzles on online and I think it's something we should all do I myself share puzzles that I can't solve online and I love getting help and I love getting feedback no matter how smart you are even if you have a PhD or you've done a math degree there's still going to be problems you can't solve that's fine I love that we can share this on the internet and we can learn something new so here's a puzzle for kids that is stumped adults can you figure it out pause the video if you'd like to give this problem a try and when you're ready keep watching to learn how to solve this [Music] problem so here's a grid of 2x two squares making up a larger square and we need to shade half of it so the unshaded part is also a square now at first this puzzle seems impossible shading half would suggest suggest that you shade two of the four squares so if you were to shade these two squares then the remaining portion would be two squares that are unshaded and this forms a rectangle now we have four squares and we want to shade two of them so there are six different ways we can do this here's one of the ways this is not a solution here's another way here's a third way which will leave two squares here's a fourth way which leaves a rectangle a fifth way which leaves a rectangle and finally a sixth way which leaves two squares so none of these methods work so the question is how else can we shade the diagram in half so let's think about this as an adult let's imagine that one of the small squares has a side length equal to one so the other Square also has a side length equal to one and the entire large square has a side length equal to two the total area of the large Square will be 2^ 2 or 2 * 2 and that equals 4 now we want half of the area to be shaded so half of the total will be 4 over 2 which equals 2 and we want this remaining half which is unshaded to be a square so we need the unshaded squares area to be equal to two so if we let the unshaded square have a side length equal to S we need S2 to be equal to 2 which means that s is equal to < tk2 so how are we going to get a square with a side length equal to < tk2 let's think about this each square has a side length equal to one so now the length of the diagonal of the square will be equal to square < TK of two so let's shade this portion we have a right triangle with each of the legs equal to one so it's hypotenuse will be equal to < TK of 1 + 1 which equals the < TK of 2 so let's repeat the shading in each of the other three small squares we're going to end up creating a picture frame and in the middle we have a quadrilateral which looks like a square now let's just go ahead and show this is in fact a square each of the Shaded right triangles is an isos right triangle that bisects each small square so bisecting a 90° angle we'll mean we have a 45° angle the other acute angle will also be equal to 45° this is true for all of these unshaded triangles going around now in this unshaded quadrilateral 45° + 45° is equal to 90° so we have four right angles furthermore the Shaded Isles right triangles are congruent to each other so all of the hypotenuse lengths will be equal to each other so we have four side lines that are equal to each other and four right angles and therefore this is a square we've shaded half of each of the small squares which means we've shaded half of the total diagram and we have left an unshaded Square so this is in fact a solution we have shaded half of the diagram and the unshaded portion is also half of the diagram and also a square we know it's half because we've shaded each of these small squares in half so this is a solution that was probably intended and most people came up with but you might be wondering are there any other Solutions so how can we think about this well here's one way let's imagine we rotate this Square 45° so is this a solution notice that the square that is unshaded has exactly the same area as before it was rotated that square was 50% of the total area so this Square will also be 50% or half of the total area so that means the Shaded portion has to be the other area which will also be half of the total area so in fact we have shaded half of the large square and the resulting unshaded portion is also a square this is a solution but now we can think about bending the rules to try to find more solutions there's nothing specified that each of the four small squares have to be shaded each in half so we can in fact move this unshaded portion what if we put it to one of the corners this would also be another solution we can do this for any of the Four Corners so we can move this to any of the four corners and we're going to get other Solutions as well the part that's shaded will be half the square and the parts that's unshaded will be also half the square people also thought about putting it to the middle of each side so this will also be a solution you can do this for each of the four sides of the square so there are a lot more solutions than at first one might think so let's characterize them we have this rotated Square this is the Tex book answer we also have this Square in the center then we have solutions that don't involve shading each of the small squares exactly in half these are solutions where the total large Square will be shaded in half and the unshaded portion will be half of that area so these are types of solutions but are these the only solutions to the puzzle let's think about it let's go back to this diagram where where we have an unshaded Square in the center what would happen if we were to arbitrarily rotate it and move it around do we have another solution right here let's think about it the Shaded area plus the unshaded area is the total area of the large square but we know the unshaded area is half of the total because we've taken a known solution and just move the square around therefore the Shaded area is half of the total as well in other words we've shaded half of the large Square leaving a square that is unshaded in fact we can move this unshaded square around in any arbitrary way and we can get other Solutions as well this will be a solution we can just move it over here this will be a solution we can really move it any way we want and each of these will be a solution into the puzzle so I really hope that if a student was approaching this problem they wouldn't be limited to just thinking inside the box and thinking about those symmetric Solutions I would really hope that you can share one of these Creative Solutions because that's what mathematics is about it's not about just sharing the same tired old narrative but it's actually about thinking about a problem differently and coming up with a solution that people didn't even think was possible what a very interesting in problem thanks for making us one of the best communities on YouTube see you next episode of mind your decisions where we solve The World's problems one video at a time

Channel: MindYourDecisions

Views: 1,184,631

Keywords: mathematics, math, maths, riddle, brain teaser, puzzle, math puzzle

Id: TB__Og3XnR8

Length: 9min 23sec (563 seconds)

Published: Sun Apr 21 2024