place rooms of any size and watch a simulated cell explore. when the rooms are small enough, the shape alone determines how long it dwells; physics doesn't matter

dial a delay up to 48 hours and the fraction of devices affected; watch grid frequency shift; toggle the paper's mitigation and see it clamp the hazard

slide into chaos, toggle control on, and watch the bursts collapse as the parameter self-adjusts in real time

draw a graph of who wants which goods and hit split; the algorithm allocates everything fairly, but only when your graph avoids short cycles

place nodes on a canvas with one marked root. adjust an epsilon slider to set how much longer each node's path to root can be. hand-draw edges connecting all nodes to root without breaking any stretch cap. your score: tree weight divided by minimum spanning tree weight. chase the lowest ratio

set sliders for division asymmetry and feedback strength. randomize a population of cell phases around a circle. press run and watch them snap into clusters — mothers and daughters self-organize into a repeating p:q rhythm with no external pacemaker. the population syncs itself

build a graph by hand. drag to add vertices, click to connect edges. cap each vertex at degree d. make every pair of vertices reachable in 2 hops or less. a score tracks your count versus the moore bound: the theoretical max for these constraints. it's barely achievable even at small d

drag a wavy landscape to set how fast it is at each point. adjust the delay in how particles respond to it. watch a swarm crawl along it. push the delay slider and the entire swarm reverses direction — the response lag alone is enough to pump particles one way or the other

pick a cricket career. shuffle its innings in random order and watch the gaps between personal bests shift — real careers hold a shape that scrambled innings lose

four objects that match on every count — identical, indistinguishable — yet each shuffles four labeled points in its own way. pick one and watch it shuffle

hold a value in a small recurrent network's memory by sliding a dial. add noise, then toggle how neurons inhibit each other. watch memory either drift smoothly or snap between discrete points

run a repeated auction where each bidder follows a simple budget rule. slide the population from a few to thousands and watch the system's convergence go from chaotic to a perfect straight line

drag any payoff in a game and watch it split into a cooperative part and a conflict part that sum back to the original. every game decomposes this way

pick a spanning tree as your favorite in a weighted network. what's the minimum total cost adjustment to make it beat all rivals? the tool computes it exactly and shows you which tree you're racing against

drop a quantum wavepacket on a hard floor and watch it bounce. heat marks where the local energy exceeds the highest energy in the packet—superenergy, hotter than its parts. hunt for the peak

play iterated prisoner's dilemma where each move flips to its opposite with a hidden probability. find the critical noise level where cooperation always collapses, no strategy survives

tune an auction rule with sliders: lean on a 'give it to the lower bidder' penalty and watch your worst-case revenue guarantee climb. sometimes losing is winning

add seats one at a time to nested groups, keeping each between its fair share rounded down and up. errors grow like log(depth) no matter what you do. try it by hand and watch monotonicity break

quantum walk on an 11-vertex path bounces forever. pick any way to average its motion over time: point samples, a uniform window, exponential decay. no matter what you choose, it never becomes uniform. watch a certificate prove it cannot

build a bipartite graph by toggling edges between two columns. two numbers below: dominating sets in your graph, dominating sets in the opposite, their sum. the paper sets a bound on that sum. can you make it equal exactly? some bipartite graphs do

place points on a canvas. draw edges between them, color each edge red or blue, and pack them dense. but no edge can be crossed by more than one edge of its own color. how many edges fit? the paper sets an exact limit for any board

grow a spherical mesh by attaching triangle to triangle, each triangle sharing edges with the last. stop when you've closed it. no vertex can touch more than six triangles. now fold the net into 3d—you get exactly one equilateral-triangle solid, always

draw two line lengths with compass and straightedge, pick a number k, then construct the third length solving 1/c^k = 1/a^k + 1/b^k. works only when k is a power of two. try k=3 or k=5? algebra forbids them, every time

pick n, then find the shortest run of consecutive integers where each of 1..n appears as one distinct multiple inside. drag them onto a number line and verify. the bounds on how short are absurdly far apart—nobody knows which is closer

slide ion diffusivity asymmetry and external field strength. a propulsion arrow on the particle updates live from the formula. cross zero on the asymmetry slider and the arrow flips—which ion species escapes fastest can reverse the entire direction of motion

pack disjoint rooted trees into a graph or find a small set of vertices that blocks them all. one always works. the paper proves that adding the root requirement keeps the bound linear—no quadratic penalty, just as tight as the classical theorem

mark a set on a clock. use translates to tile each hour exactly once. this same set also satisfies a completely separate mathematical condition. the paper is the first to prove both always work together for an infinite family of cyclic groups

draw arrows between dots. each vertex glows green when its neighborhood contains a cycle. the sparsest strongly connected graph satisfying this needs exactly 7n/3 edges—a density bound the paper proves is tight. race it on screen

run a four-stage supply chain: factory, distributor, wholesaler, retailer. each gets its own autonomy dial. watch orders and inventory move as you tune them. the finding: raise autonomy upstream and chain costs fall. raise it downstream and they spike. same dial, opposite effects by position

build a planar graph hitting every degree from 3 to n, as small as possible, while a live counter watches for symmetry — rotations, reflections, permutations that map it to itself. chase the degree checklist. at degree 14, the symmetries vanish. every minimal graph breaks perfectly lopsided