01
The Sum of Exterior Angles
Angle Puzzles
- Q
- Add up a polygon's exterior angles.
- A
- Always 360°, for any polygon
KIKARAKUきからく · Geometry puzzle app
COUNTERINTUITIVE GEOMETRY
Geometry that overturns your intuition.
Feel counterintuitive geometry in 4 steps: guess, play, understand, check. 14 puzzles you can solve with school geometry, yet almost nobody gets right on instinct. Move the shapes with your fingers and turn surprise into understanding.
- Free
- 14 stages
- 74 languages
- No account
- iPhone · iPad · Android
14counterintuitive stages
4steps each
$0no in-app purchases
74languages
4 Steps
Guess → Play → Understand → Check
Every stage follows the same path: answer on instinct, move the shapes yourself, find out why it happens, then finish with a follow-up question.
- STEP 1
Gut check
Answer with your gut first. Don't overthink it — pick what feels right.
- STEP 2
Hands-on
Move the shapes yourself and watch your intuition get overturned.
- STEP 3
Explanation
Use school geometry to work out why it happens.
- STEP 4
Final check
Test your understanding with a new question. Answer it to clear the stage.
Stages
14 "wait, what?" moments
Three categories: shapes in motion, tricks of length and area, and numbers hidden in shapes. Version 4.0 adds 2 new stages: the Reuleaux triangle and Buffon's needle.
Shapes in motion5
02
Napoleon's Theorem
Secrets of Triangles
- Q
- Build equilateral triangles on each side and join their centres.
- A
- Always an equilateral triangle
03
The Coin Rotation Paradox
Symmetry & Motion
- Q
- Roll a coin once around an identical coin.
- A
- It turns twice, not once
04
The Rope Around the Earth
Circle Theorems
- Q
- Lengthen a rope around the Earth by 1 m. How big is the gap?
- A
- About 16 cm, whatever the size of the Earth
05
The Reuleaux Triangle NEW
Constant Width
- Q
- Is the circle the only shape with the same width in every direction?
- A
- No: the Reuleaux triangle rolls just as smoothly
Tricks of length and area5
06
The Missing Square
Pythagoras
- Q
- Rearrange the pieces and a square goes missing?
- A
- Nothing vanished: it hides in a hairline gap
07
Same Area, Different Perimeter
Tangram
- Q
- Same 16 tiles, same perimeter?
- A
- It can more than double, depending on the shape
08
Same Perimeter, Different Area
Area Master
- Q
- Which shape encloses the most area with the same string?
- A
- The circle
09
The Spider and the Fly
Nets
- Q
- What is the spider's shortest path over the box?
- A
- A straight line on the net, across surprising faces
10
Taxicab Geometry
Coordinate Geometry
- Q
- Draw a "circle" using distance along the streets.
- A
- You get a square, tilted 45°
Numbers hidden in shapes4
11
The Square–Cube Law
Similarity & Ratio
- Q
- A giant twice your height weighs how much more?
- A
- 8 times, while bone strength only quadruples
12
The Golden Angle and Sunflowers
The Golden Ratio
- Q
- Which turn per seed packs a sunflower best?
- A
- About 137.5°, the golden angle
13
Gabriel's Horn
Volume & Surface Area
- Q
- Fill it with paint, but never paint its surface?
- A
- Finite volume, infinite surface area
14
Buffon's Needle NEW
Geometry & Chance
- Q
- Drop a needle as long as the line spacing. How often does it cross?
- A
- 2/π ≈ 64%, so dropping needles measures π
Screenshots
Move it, and see for yourself
Drag or slide the shapes and watch lengths, areas and angles update instantly.
Features
Built to make it click
Shapes you can move
Stretch a side, roll a coin, drop a needle. Sliders and drags change the shapes, and the numbers and diagrams update instantly.
Explanations and follow-ups
Diagrams and formulas based on school geometry explain why intuition fails, and a follow-up question checks your understanding.
14 stages, 3 categories
Shapes in motion, tricks of length and area, and numbers hidden in shapes. Cleared stages are marked on the home screen.
74 languages
English, Japanese and 72 more languages. The app follows your device language automatically.
Dark and light
Switch between dark and light themes with one tap. Works on iPhone, iPad and Android, in portrait or landscape.
Free, no account
Every stage is free. No in-app purchases, subscriptions or sign-in, and everything except ads works offline.
FAQ
Frequently asked questions
Is it free?
Yes. Every stage is free to play. There are no in-app purchases, subscriptions or paid unlocks, and no account is needed.
Are there ads?
A Pangle banner ad is shown at the bottom of the screen, and a full-screen ad may be shown when you clear a stage (or leave one during the comprehension check). On iOS, the App Tracking Transparency (ATT) prompt appears after launch; on Android, the app asks once whether ads may be personalized. Either choice can be changed later from "About this app" > "Ad settings" (on iOS, this appears after you have answered the tracking prompt).
What age is it for?
It is designed for ages 13 and up. Every puzzle can be solved with middle- and high-school geometry, so it works as a brain teaser for adults and as material for class or self-study.
Which devices are supported?
iPhone and iPad with iOS 15.0 or later, and Android phones and tablets with Android 7.0 or later.
Is my personal data sent anywhere?
Nothing is sent to us. The app only stores your theme, your ad setting and the stages you have cleared on your device. To deliver ads, the ad SDK sends information such as advertising identifiers to Pangle.
Download
Put your intuition to the test.
Free on the App Store and Google Play.