KAKURAKU vs Math Surface

KAKURAKU is an educational app that lets you experience probability paradoxes hands-on. Answer with your gut, run the simulation hundreds of times, and compare the result with the mathematical answer. See for yourself where intuition and mathematics part ways. Developer: Fuji Bit Inc. KAKURAKU is the 10th app in the "-raku" series of educational apps, and its theme is probability. It covers 14 famous probability paradoxes, and every one of them unfolds in four phases. Four phases: 1. Gut check: Answer with your intuition first. Most people get these wrong. 2. Simulation: Run one trial at a time or hundreds at once, and see the probabilities as real numbers. 3. The answer: See the mathematical answer and why intuition misses, with diagrams and formulas. 4. Final check: Answer a new question to clear the stage. 14 probability paradoxes in 3 categories: Conditional probability traps - The Door Trap (Monty Hall Problem): Does switching doors win 2/3 of the time? - The Gold Box (Bertrand's Box): Is your 1/2 intuition wrong? - Warden's Verdict (Three Prisoners Problem): Does one word from the warden change A's fate? - Test Trap (False Positive Problem): You test positive, yet the chance you are sick is only 16%? - Sleeping Princess (Sleeping Beauty Problem): 1/2 or 1/3? People still disagree. When statistics fool intuition - Miracle Classroom (Birthday Paradox): With just 23 people, is a shared birthday more likely than not? - Reversed Statistics (Simpson's Paradox): Can the parts and the whole point in opposite directions? - The Endless Bus (Bus Waiting Paradox): Buses come every 10 minutes on average, yet you wait 10 minutes on average? - Red Chain (Gambler's Fallacy): After a run of reds, is black due? - Too Many Ones (Benford's Law) (new): Do about 30% of everyday numbers start with 1? Expected value and decisions - Envelope Dilemma (Two Envelope Problem): Should you keep switching envelopes? - Infinite Gamble (St. Petersburg Paradox): The expected value is infinite, so why would no one pay much to play? - The Predictor's Boxes (Newcomb's Paradox): One box or two? Change the predictor's accuracy and find out. - The Second Mover Wins (Penney's Game) (new): A game of guessing three-coin patterns. Can the second player win up to 7/8 of the time? Features: - Clean, modern design with dark and light themes - Home screen organized by category - Every stage advances in four phases, and you can always see where you are - Graphs and numbers show the results approaching the theoretical value as trials add up - If you go back during the final check, the app asks whether to leave the stage - Available in 73 languages, including English and Japanese (follows your device language) - No sign-in; start right away - Simulations work offline (the network is used only for ads) Great for: - Students learning probability and statistics - Anyone curious about the gap between intuition and mathematics - Teachers looking for classroom material - Anyone who wants to truly understand why the answer is what it is Price and purchases: - Price: Free - In-app purchases: None - Subscriptions: None - Ads: Banner ads by Pangle at the bottom of the screen, and a full-screen (interstitial) ad may be shown when you clear a stage - Ads are personalized only if you allow tracking. You can turn this off in "About this app" > "Ad settings" The "-raku" series of educational apps (selection): - SUUGAKU - Mathematics - BUTURAKU - Mechanics - DENRAKU - Electromagnetism - HARAKU - Waves - NETURAKU - Thermodynamics - KORAKU - Optics - RYURAKU - Fluid dynamics - KAKURAKU - Probability

3D surfaces, 2D curves, the complex plane, probability — all in one app. Type a formula and watch it come alive. Perfect for study, exploration, or just enjoying the beauty of math. Description: MathSurface turns equations into shapes you can hold, spin, and slice. It's a math playground for iPhone and iPad — designed for students, curious minds, and anyone who learns better by seeing. ■ Features • 3D surface plots: visualize z = f(x, y) with smooth rotation and pinch-to-zoom • 2D function plots: explicit y = f(x) or implicit 0 = F(x, y) curves • Complex plane: loci such as |z − α| = r, arithmetic operations, powers, and n-th roots of unity • Probability distributions: binomial and normal, controlled by live sliders • Surface of revolution: spin any 2D curve around the x- or y-axis (rendered with SceneKit) • Cross sections: slice any 3D surface with a plane and see the resulting curve • Compare mode: overlay two functions in the same plot • Preset gallery: paraboloid, monkey saddle, Mexican hat, lemniscate, and many more • Favorites: save your own custom expressions for instant recall ■ Who it's for • High school and university students studying calculus, analysis, or probability • Teachers looking for live visual aids • Anyone curious about how a formula actually looks • People who just enjoy beautiful math A custom Liquid Glass–style formula keyboard makes entering sin, cos, √, π, and friends quick and natural. The dark neon UI keeps the math front and center.