Echo Ramsey
Color graphs, advance a proof
Free · Designed for iPad. Not verified for macOS.
Echo is a puzzle game built on real open problems in mathematics.
Each puzzle is a graph: dots joined by red and blue lines. Your job is simple to learn. Tap a line to flip its
color, and break up the "bad cliques" - groups of dots all wired together in a single color. Clear them within your
move budget and the puzzle is solved.
Here is the twist. Every puzzle is a slice of a genuine research graph, and every solution you find is committed to
a shared, growing proof tree. When you solve, your result seeds a brand new puzzle for another player. Their solve
seeds the next. Together, the whole community is searching for colorings no person and no computer has ever found.
You choose which problem to work on from the home screen:
- R(4,4), the practice board - solved long ago, so the search genuinely finishes and you can see the whole engine
work end to end
- R(4,6), the frontier - an open problem where reaching zero would set a real new record
- R(5,5), the legendary one - open for decades, where the prize is the lowest count the network has ever driven it
to
You do not need to know any of that to play. If you can spot a pattern and tap, you are already contributing.
FEATURES
- Quick, tactile puzzles you can finish in a minute
- Three problems to play, from a winnable warm-up to a famous unsolved one, switchable any time
- Five ways to view each graph: circle, spectral, matrix, hub, and density - switch any time to see the structure
differently
- A living progress screen showing the community's best reach and the proof tree growing in real time
- Your lineage: see how many puzzles your solves have seeded for other players
- A built-in glossary that explains every term in plain language
- No ads, no timers, no pressure
Color the graph. Advance the proof. Echo forward.
Ratings & Reviews
- This app hasn’t received enough ratings or reviews to display an overview.
The home "Every solve advances…" hint now opens the correct entry: on R(4,6) it shows "R(4,6) ≥ 37 … 36-dot graph … no red K₄ and no blue K₆," not the old R(5,5) description.
The developer, Sam Rostami, indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .
Data Linked to You
The following data may be collected and linked to your identity:
- Identifiers
Accessibility
The developer has not yet indicated which accessibility features this app supports. Learn More
Information
- Seller
- Sam Rostami
- Size
- 53.8 MB
- Category
- Puzzle
- Compatibility
Requires iOS 14.0 or later.
- iPhone
Requires iOS 14.0 or later. - iPad
Requires iPadOS 14.0 or later. - iPod touch
Requires iOS 14.0 or later. - Mac
Requires macOS 11.0 or later and a Mac with Apple M1 chip or later. - Apple Vision
Requires visionOS 1.0 or later.
- iPhone
- Languages
- English
- Age Rating
4+
- 4+
- Copyright
- © 2026 Sam Rostami