Can You Solve It? How Tall Is The Dog Puzzle

Hey, this is Preshall Walker. Here's a fun puzzle that's gone viral. Based on the information in the diagram, can you figure out the height of the dog? The left side of the figure shows a wooden pole and its height is equal to that of the dog plus 200 cm. The right side of the figure shows the dog sitting on top of the wooden pole and the total height is equal to 300 cm. Can you figure it out? Puzzles like this have gone viral in years past. One of the first was in 2018 from a Chinese elementary school student's homework. You had to figure out the height of the table based on a cat that was sitting and lying down. There was a similar question in the 2018 Math Kangaroo competition. I adapted this question into a turtle and cat problem, which many of you have probably even seen. And lest you think these puzzles are not for serious mathematicians, this very problem was used in the promotional material for the International Congress of Mathematicians in 2022. The Congress is the largest gathering of mathematicians and it only happens once every 4 years. This is where the Fields Medal is awarded. It's often called the Nobel Prize of Mathematics. So with that said, let us solve the original problem. One way to approach it is to use algebra. So let's take a look at our figure and focus on just the left-hand side. We need to figure out the height of the dog, so let's set up a variable so that this unknown quantity is equal to D. We also have another unknown quantity, which is the height of this wooden pole. Let's call that P. We can now figure out the height from the floor to the top of the pole in two different ways. One way is D plus 200 and another way is P. So, these two quantities will be equal to each other. So, we have the equation D plus 200 is equal to P. Now, let's focus on the right-hand side of the diagram. We have the same wooden pole and the same dog. So, those heights will be the same variables. So, the height of the pole will be equal to P. The height of the dog will be equal to D. We can again figure out the distance from the floor to the top of the dog in two different ways. This distance is equal to P plus D, but it is also equal to 300. So, we get the equation P plus D is equal to 300. Let's put it all back together. We now have two equations in two variables, and the beauty of algebra is we can just focus on these two equations. So, how do we solve for the variable D? So, in this particular set of equations, we have already solved for P. So, a good way to eliminate this variable is just to substitute it in. So, in the second equation, we can substitute D plus 200 for P. So, we have D plus 200 plus D is equal to 300. This means 2D plus 200 is equal to 300 or 2D is equal to 100, which means D is equal to 50 cm. And that's the answer. The height of the dog is 50 cm. Now, we didn't need to solve for the height of the pole, but we might as well do that because it'll be easy to do. We just substitute D is equal to 50 into the first equation, and we have that the height of the pole is equal to 250 cm. Now, if all we wanted to do was get to an answer, we would be done. We would just move on with our lives. But, part of the fun of mathematics is figuring out other ways to solve the same problem. So, you think about it from different perspectives. So, I now want to present a couple of visual approaches to this puzzle. As I was preparing the graphics for this video, I was just experimenting with different components of this puzzle. And a thought crossed my mind. We have a dog in the left side of the figure and a right side of the figure. And what would happen if I just overlap the two dogs? Would anything interesting result? So, I went ahead and I placed the left figure right on top of the right figure and overlap the two dogs. And something rather fascinating happened. We now have a distance from the floor to the top of the two figures, and we exactly know that its distance will be 300 + 200. So, this will be 500 cm. But, we can also clearly see that the distance from the bottom to the top is exactly the same length as two poles. So, we have eliminated the variable of the dog by doing this visual operation.

So, we end up with 2p is equal to 500, which means that p is equal to 250. However, we didn't want to solve for the height of the pole. We wanted dog. So, we now need to reverse this operation, and we look at the left side of the figure. We see the distance from the bottom to the top of the pole will be equal to 250. We know that's equal to the height of the dog plus 200, and therefore has to be equal to 50 cm. And we figured it out just from a visual approach. But, it was a two-step process. We had to figure out the height of the pole, and then go back and figure out the height of the dog. But, is there any way to figure out the height of the dog directly in a visual approach. Let's start all over and think about the puzzle visually. By overlapping the dogs, we were able to eliminate the dog variable and just be left with an equation about the poles. So now, we want to eliminate the pole variable and just be left with the height of the dog. So we would need to overlap the poles. So let's translate the left part of the figure. When we shift it over, here's what results. Okay, so this is a little bit messy. Let's clean it up. Let's shift the dog and the distance to the right side of the pole. We now see we have a visual situation where we can figure out the distance from the floor to the top of the dog. On the one hand, it is equal to the height of the dog plus 200 plus the height of the dog, and on the other, it is equal to the 300 cm that was given in the puzzle. So we have the equation D plus 200 plus D is equal to 300. And in fact, we have eliminated the pole variable by overlapping the poles visually. All we need to do is solve this equation for D. We have 200 plus 2D is equal to 300, which means 100, and D is equal to 50 cm. For good measure, we can figure out the height of the pole is equal to the height of the dog plus 200, which means the pole is equal to D plus 200 or 250 cm. And that's the answer to this wonderful puzzle. But I hope this video illustrated that getting the answer is not the only objective of the puzzle. Besides understanding the algebra, we have new techniques to solve puzzles visually. We can eliminate a variable by overlapping those pictures. Wow. 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.