The Missing Piece Puzzle: Why This 'Triangle' Doesn't Add Up скачать в хорошем качестве

The Missing Piece Puzzle: Why This 'Triangle' Doesn't Add Up

What is going on here as we rearrange these four different pieces? It seems like we are magically conjuring up a missing block out of nowhere. But is that really the case? This classic math puzzle—known as the "missing piece" puzzle—challenges us to question our assumptions and reveals some fascinating geometry along the way. Let’s dive in. Check out the main channel! ‪@polymathematic‬ Whenever you encounter a puzzle like this, the first question to ask is: What are my assumptions, and how might those assumptions be failing me? In this case, the initial assumption is pretty straightforward: we believe that this is a 5-by-13 triangle. If that’s true, we can calculate the area using the formula for the area of a triangle: (base × height) ÷ 2. Here, the base is 13 and the height is 5, so the area would be: (13 × 5) ÷ 2 = 32.5 square units. Simple enough. But here’s where things get weird. When we rearrange the four pieces, they seem to form that same 5-by-13 triangle, yet there’s clearly a missing square block. How is that possible? Can this triangle have an area of both 32.5 square units and 31.5 square units at the same time? Of course not. Something is off here. To figure out what’s going on, we need to go back to our assumptions. Specifically, let’s re-examine whether this figure is actually a triangle at all. To do that, we’ll break it down piece by piece. Start with the two Tetris-like pieces. These are straightforward because they’re just made up of blocks. The orangish piece consists of 7 blocks, and the greenish piece consists of 8 blocks, for a total of 15 blocks. Next, look at the red triangle. It has a base of 8 units and a height of 3 units, so its area is: (8 × 3) ÷ 2 = 12 square units. Finally, the blue triangle has a base of 5 units and a height of 2 units, so its area is: (5 × 2) ÷ 2 = 5 square units. When we add these areas together, we get: 15 (Tetris pieces) + 12 (red triangle) + 5 (blue triangle) = 32 square units. Wait a second. That’s not 32.5. It’s also not 31.5. Instead, it’s a number right in between. What’s going on? It turns out that the problem lies in our original assumption that the figure is a triangle. Let’s examine the slopes of the two triangular pieces. The red triangle’s hypotenuse has a slope of 3/8 (rise over run). The blue triangle’s hypotenuse has a slope of 2/5. At a glance, these two slopes look very similar. But they’re not identical. This tiny difference means the hypotenuses of the red and blue triangles don’t actually line up. Instead of forming a single straight line, the pieces create a subtle kink. In other words, what looks like a 5-by-13 triangle is actually a slightly distorted quadrilateral. That’s why the area calculations don’t add up and why a "missing" block appears when you rearrange the pieces. So, what’s the big takeaway here? This puzzle is a great reminder to always question your assumptions. What seems obvious at first glance might not hold up under closer scrutiny. And beyond that, it’s a perfect example of how mathematical concepts like slope and area can help us solve puzzles and uncover hidden truths. #MathPuzzles #CriticalThinking #GeometryFun Follow Tim Ricchuiti: TikTok:   / polymathematic   Instagram:   / polymathematicnet   Mathstodon: https://mathstodon.xyz/@polymathematic Reddit:   / polymath-matic   Facebook:   / polymathematic   Watch more Math Videos: Math Minis:    • Math Mini   Math Minutes:    • Math Minutes   Number Sense:    • Number Sense (UIL / PSIA)   MATHCOUNTS:    • MATHCOUNTS