Maths / Problem
Roots All the Way Down
Algebra / Algebra and Limits
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · 5-10 min
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Maths and Physics problems, with every Modulo and Physicle edition.
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Started includes problems you have answered or opened help for, and unfinished games. Previous solves are kept when you practise again.
What Do the Difficulty Tiers Mean?
- Tier 1 - Element
- Focused around one main idea or observation.
- Tier 2 - Relation
- Requires a few ideas, quantities or steps to be connected.
- Tier 3 - Dimension
- Requires several pieces of reasoning to be organised into a solution.
- Tier 4 - Tensor
- A layered challenge requiring substantial reasoning or technical control.
- Tier 5 - Singularity
- The highest challenge tier, usually requiring deep or non-obvious reasoning.
Difficulty describes the challenge, not the solver. It is a Society guide rather than a formal academic level.
Physics / Problem
A Rock, a Boat, a Lake
Fluids / Buoyancy
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · 5 min
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Maths / Problem
Hello, Everyone
Combinatorics / Counting Pairs
Difficulty Tier 1, Element. Focused around one main idea or observation. · 2 min
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Physics / Problem
On the Rocks
Fluids / Buoyancy
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 3 min
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Maths / Problem
A Pair in the Dark
Combinatorics / Pigeonhole Principle
Difficulty Tier 1, Element. Focused around one main idea or observation. · 3 min
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Maths / Problem
One Six Changes the Odds
Probability / Conditional Probability
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · 5-8 min
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Maths / Problem
The Missing Corners
Logic & Proof / Invariants
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · 5 min
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Maths / Problem
Building a Square
Number Theory / Patterns and Proof
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 4 min
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Physics / Problem
Two Paths, Less Resistance
Electricity & Magnetism / Electric Circuits
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 4 min
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Physics / Problem
Twice the Rhythm
Waves & Optics / Waves
Difficulty Tier 1, Element. Focused around one main idea or observation. · 3 min
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Physics / Problem
A Fair Fall
Mechanics / Forces and Motion
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 3 min
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Physics / Problem
Same Heat, Different Rise
Thermal Physics / Thermal Physics
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 4 min
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Maths / Problem
Clockwork
Geometry / Angles and Clocks
Difficulty Tier 1, Element. Focused around one main idea or observation. · 3-4 min
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Maths / Problem
A Tangent at One Point
Calculus / Differentiation
Difficulty Tier 1, Element. Focused around one main idea or observation. · 2-3 min
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Maths / Problem
Four Numbers in a Box
Linear Algebra / Determinants
Difficulty Tier 1, Element. Focused around one main idea or observation. · 2-3 min
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Maths / Problem
The Shortest Route
Combinatorics / Lattice Paths
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 4-6 min
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Maths / Problem
Say the Opposite
Logic & Proof / Quantifiers and Negation
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 5-7 min
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Maths / Problem
A Repeating Last Digit
Number Theory / Modular Cycles
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 3-5 min
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Maths / Problem
Distance on the Number Line
Algebra / Absolute Value Equations
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · 6-8 min
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Maths / Problem
Which Box Did It Come From?
Probability / Bayes' Theorem
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · 5-8 min
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Maths / Problem
Every Letter in the Wrong Envelope
Discrete Mathematics / Derangements
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · 7-10 min
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Maths / Problem
The Largest Rectangle
Calculus / Optimisation
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · 8-12 min
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Maths / Problem
One Integral, Surprisingly Simple
Analysis / Integration by Parts
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · 8-10 min
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Maths / Problem
Waiting for Two Heads
Probability / Expected Waiting Time
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · 15-20 min
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Physics / Problem
Waiting for the Echo
Waves & Optics / Sound
Difficulty Tier 1, Element. Focused around one main idea or observation. · 3 min
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Physics / Problem
Twice the Frequency, Twice the Photon?
Quantum Physics / Photon Energy
Difficulty Tier 1, Element. Focused around one main idea or observation. · 2 min
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Physics / Problem
Three Half-Lives Later
Modern Physics / Radioactive Decay
Difficulty Tier 1, Element. Focused around one main idea or observation. · 3 min
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Physics / Problem
The Elevator Scale
Mechanics / Newton's Second Law
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 5 min
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Physics / Problem
Double the Resistance
Electricity & Magnetism / Electric Power
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 4 min
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Physics / Problem
How Much of It Sinks?
Fluids / Floating Equilibrium
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · 5 min
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Physics / Problem
Where Does the Image Form?
Waves & Optics / Thin Lenses
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · 6 min
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Physics / Problem
Mixing Hot and Cold
Thermal Physics / Calorimetry
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · 7 min
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Physics / Problem
A Quantum Jump
Quantum Physics / Particle in an Infinite Square Well
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · 5-7 min
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Physics / Problem
Half Charged
Electricity & Magnetism / RC Charging
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · 8-12 min
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Physics / Problem
Staying on the Loop
Mechanics / Circular Motion and Energy
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · 10-15 min
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Physics / Problem
How Fast Must the Muon Travel?
Relativity / Time Dilation
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · 15-20 min
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Maths / Modulo
A Remainder of Curiosity
Number Theory / Integer Deduction
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 001
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Maths / Modulo
Evenly Narrowed
Number Theory / Integer Deduction
Difficulty Tier 1, Element. Focused around one main idea or observation. · Edition 002
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Maths / Modulo
A Square to Uncover
Number Theory / Integer Deduction
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 003
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Maths / Modulo
Prime Suspects
Number Theory / Integer Deduction
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 004
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Maths / Modulo
Sevens and Sums
Number Theory / Integer Deduction
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 005
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Maths / Modulo
The Final Residue
Number Theory / Integer Deduction
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 006
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Maths / Modulo
First Steps
Number Theory / Digits and Parity
Difficulty Tier 1, Element. Focused around one main idea or observation. · Edition 007
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Maths / Modulo
Two Digits
Number Theory / Digits and Parity
Difficulty Tier 1, Element. Focused around one main idea or observation. · Edition 008
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Maths / Modulo
Step by Five
Number Theory / Divisibility
Difficulty Tier 1, Element. Focused around one main idea or observation. · Edition 009
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Maths / Modulo
Between the Markers
Number Theory / Perfect Squares
Difficulty Tier 1, Element. Focused around one main idea or observation. · Edition 010
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Maths / Modulo
The Reversed Pair
Number Theory / Digit Reversal
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 011
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Maths / Modulo
The Factor Trail
Number Theory / Divisibility
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 012
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Maths / Modulo
Remainder Relay
Number Theory / Congruences
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 013
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Maths / Modulo
Three Moduli
Number Theory / Congruences
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 014
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Maths / Modulo
A Shared Factor
Number Theory / Common Multiples
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 015
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Maths / Modulo
Three Steps Ahead
Number Theory / Perfect Cubes
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 016
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Maths / Modulo
A Product Mystery
Number Theory / Prime Factorisation
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 017
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Maths / Modulo
The Triple Lock
Number Theory / Congruences and Primes
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 018
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Maths / Modulo
The Long Way Around
Number Theory / Perfect Squares
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 019
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Maths / Modulo
Residue Intersection
Number Theory / Common Multiples
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 020
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Maths / Modulo
Doubling Back
Number Theory / Powers of Two
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 021
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Maths / Modulo
The Last Digit
Number Theory / Divisibility
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 022
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Maths / Modulo
The Shortlist
Number Theory / Primes and Digits
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · Edition 023
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Maths / Modulo
A Rare Structure
Number Theory / Divisor Counts
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · Edition 024
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Maths / Modulo
The Final Intersection
Number Theory / Congruences
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · Edition 025
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Physics / Physicle
Something Worth Conserving
Mechanics / Mechanics
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 001
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Physics / Physicle
Keeping the Rhythm
Waves & Optics / Waves
Difficulty Tier 1, Element. Focused around one main idea or observation. · Edition 002
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Physics / Physicle
An Electrical Obstacle
Electricity & Magnetism / Electricity
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 003
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Physics / Physicle
A Question of Warmth
Thermal Physics / Thermal Physics
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 004
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Physics / Physicle
Light between the Crests
Waves & Optics / Optics
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 005
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Physics / Physicle
What Changes Motion?
Mechanics / Mechanics
Difficulty Tier 1, Element. Focused around one main idea or observation. · Edition 006
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Physics / Physicle
Opposing Forces
Mechanics / Forces
Difficulty Tier 1, Element. Focused around one main idea or observation. · Edition 007
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Physics / Physicle
A Measured Wave
Waves & Optics / Waves
Difficulty Tier 1, Element. Focused around one main idea or observation. · Edition 008
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Physics / Physicle
One Path for Current
Electricity & Magnetism / Circuits
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 009
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Physics / Physicle
Two Half-Lives
Modern Physics / Radioactive Decay
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 010
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Physics / Physicle
The Same Push
Mechanics / Forces
Difficulty Tier 2, Relation. Requires a few ideas, quantities or steps to be connected. · Edition 011
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Physics / Physicle
How Much Sinks?
Fluids / Buoyancy
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 012
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Physics / Physicle
A Fixed-Volume Gas
Thermal Physics / Thermal Physics
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 013
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Physics / Physicle
The Returning Sound
Waves & Optics / Sound
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 014
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Physics / Physicle
Crossing the Boundary
Waves & Optics / Refraction
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 015
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Physics / Physicle
Light and an Electron
Quantum Physics / Quantum Physics
Difficulty Tier 3, Dimension. Requires several pieces of reasoning to be organised into a solution. · Edition 016
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Physics / Physicle
Two Ways Through a Circuit
Electricity & Magnetism / Circuits
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 017
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Physics / Physicle
Another Circular Orbit
Mechanics / Circular Motion
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 018
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Physics / Physicle
The Bright-Fringe Spacing
Waves & Optics / Interference
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 019
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Physics / Physicle
A Pendulum Elsewhere
Mechanics / Oscillations
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 020
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Physics / Physicle
Power in One Resistor
Electricity & Magnetism / Circuits
Difficulty Tier 4, Tensor. A layered challenge requiring substantial reasoning or technical control. · Edition 021
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Physics / Physicle
Reversing the Collision
Mechanics / Collisions
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · Edition 022
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Physics / Physicle
The Other Ship's Clock
Relativity / Special Relativity
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · Edition 023
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Physics / Physicle
The Lift and the Floating Object
Fluids / Buoyancy
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · Edition 024
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Physics / Physicle
The Third Polariser
Waves & Optics / Polarisation
Difficulty Tier 5, Singularity. The highest challenge tier, usually requiring deep or non-obvious reasoning. · Edition 025
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