Can you solve the Big Bang riddle? - James Tanton

It’s moments after the Big Bang and... wow, you’re still reeling. You’re a particle of matter, amidst a chaotic stew of forces, fusion, and annihilation. If you’re lucky and avoid being destroyed by antimatter, you'll be the seed of a future galaxy.

You gravitate towards a ring of particles that have devised a game. A friendly gluon explains: there are an equal number of matter and antimatter particles. At the blow of a cosmic whistle, every particle of matter with a particle of antimatter to its immediate right will merge with that antiparticle to—poof!—both annihilate. The circle will tighten to make a smaller circle, and the process will repeat until just one particle of matter remains. They’d been waiting for someone like you, as with a perfectly balanced setup, no one would be left in the end.

As guest of honor, they’ll let you choose where to insert yourself into the ring. What spot should you choose to make sure you’re the last particle standing? Pause here if you want to figure it out for yourself.

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A general strategy can’t rely on knowing what the circle looks like. But we can rely on how it should end: with you as the last particle standing. Let's start there and work backwards. Just before this, there were also some matter/antimatter pairs. Here’s 3 for the sake of example. This right away suggests a strategy.

You can simulate annihilations in any circle by marking all pairs of adjacent particles, then the newly adjacent pairs, and so on, to reach this final set. If you map them back to the original circle, standing to the immediate left of those final matter particles or the immediate right of final antiparticles means you’ll reach this endgame state and survive. So here are the safe spots in all three given circles.

You find a good position and experience the bittersweet radiance of your new friends pairing off and becoming pure energy. You’re soon left alone. But wait a minute! Today’s the big day. You’re an intelligent creature of matter amidst a chaotic crowd of alien races. Every hundred million years, sentient species of the galaxy send emissaries to the core to pay tribute to the galactic seed you’re all descended from.

Each of the 27,182 attendees has brought a cube of matter and a cube of antimatter. Except you: you’ve been chosen to represent your famed ancestor and reenact its moment of choice. Everyone places their matter and antimatter cubes in a ring. Your job: pick a spot to stand where you won’t be annihilated.

The ring is too large to simulate annihilations like before, but you have a programmable device you can point at spots between cubes. The device can detect the full sequence of cubes around the circle, count cubes of each type, perform arithmetic, and carry out simple logical commands. How do you program your device to identify safe spots?

Pause to figure it out for yourself.

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Let’s go back to our earlier approach: running time backwards starting from you alone, then a sequence of pairs of cubes. A second step back produces more pairs nestled between existing cubes. As does the next, and the next, each adding nestled pairs like a Russian doll of fundamental particles.

A pair never forms around an existing cube because only adjacent cubes annihilate. Your survival depends on the matter particles to your right buffering you from antimatter. We can mathematically model this idea with a sum, starting with your +1. For instance, this would be +1+1-1+1-1. From your perspective, no matter how far you look, this sum will be positive.

Considering how we introduced pairs, this makes sense—each inserted plus was immediately followed by a minus. If we examine the starting position of any other matter cube, its sum will eventually hit zero. The moment that happens marks its exact annihilation partner. Again, our introduction of pairs reflects this; nestled pairs only delay the inevitable for the outer cubes.

We can now program the device. Start a running sum with +1, representing you, and scan around the circle. For each matter cube, add one, and for each antimatter cube, subtract one. If all 54,364 cubes have been scanned without the sum ever hitting 0, that spot is safe. Nothing left to do but enjoy the fireworks.