puzzle 94 curious village

Puzzle 94: The Vanishing Square (from Professor Layton and the Curious Village) is a classic geometric dissection puzzle that plays with perception. Here's how it works: The Puzzle Setup: You're shown two arrangements of the same four shapes (a right triangle, a trapezoid, and two smaller irregular quadrilaterals) that form a larger triangle. In the first arrangement, the shapes form a 13x5 triangle with no gaps. In the second arrangement, the same shapes are rearranged to form an identical-looking trianglebut now theres a small empty square (1x1 unit) in the middle. Question: Where does the extra square come from? Solution: The "extra" square is a result of visual deception. The two larger triangles (13x5) arent actually perfect trianglestheir hypotenuses arent straight lines. First Arrangement: - The hypotenuse has a subtle concave bend (inward), but the shapes fit snugly, hiding the imperfection. - Total area covered: (13 5)/2 = 32.5 square units. Second Arrangement: - The same shapes are rearranged so the hypotenuse now has a convex bend (outward), creating the illusion of a perfect triangle. - The "missing" 1x1 square (area = 1) accounts for the difference: 32.5 (original) + 1 = 33.5, but the new triangle would require 33 units (if perfect). Key Insight: The combined shapes dont form a true triangle in either casethe slopes of the smaller pieces (2/5 vs. 3/8) dont match the larger triangles slope (5/13). The "extra" space comes from this mismatch. Visual Proof: Calculate the slopes: - Red triangle: 2/5 = 0.4 - Blue triangle: 3/8 = 0.375 - The larger "triangle": 5/13 0.385 The slopes are not equal, so the hypotenuse isnt straight. The rearrangement exploits this to create the illusion of a missing square. Why Its Clever: The puzzle teaches you to question assumptions (like "this is a perfect triangle") and look closely at angles and proportions. Its a classic example of the missing square paradox. Let me know if you'd like further clarification!