The puzzle has a name. “We call it the coupon collector’s problem,” says Clara. “It is the maths of completing a set when you cannot pick what you get. Football stickers are just the friendly version.” Video-game loot boxes run on the same maths. A close cousin of it shows up when scientists try to predict how a disease spreads.
Getting the full set
Early on, it hardly feels like a problem. “In the beginning, almost every sticker is new, so the album fills up fast,” Clara says. Getting the first half done takes around 680 stickers, only a little more than you would buy in any case.
From there, it slows to a crawl. “When you only need a few stickers, nearly every packet you open is one you already have,” Clara says. Your single last player takes, on average, 980 stickers to turn up, a whole album’s worth for one empty space. By the finish, a typical collector has opened more than 1,000 packets and spent well over 1,500 euros.
What stickers and the internet have in common
The same problem reaches far beyond football. “Underneath, it is always the same question: how long does it take to see everything when you only sample at random?” Clara says. “That comes up in my work.” Her field is enormous networks: the internet and even the wiring of the brain. Both are examples of networks far too big to take in at a glance.
Picture trying to map a social network by hopping at random from one friend to the next. The popular, well-connected people turn up almost straight away. The ones on the edges take far longer to reach, the same way your last sticker hides. Until you know how long reaching everyone really takes, you cannot be sure your map is complete.
This is the grown-up version of the sticker problem. When researchers sample a social network too large to map in full, they need to know how much they have actually covered. Miss a hidden part and the picture is wrong. Clara is drawn to these counterintuitive results. The friendship paradox is another. It explains why your friends have, on average, more friends than you.
What makes an outbreak explode?
The same network thinking carries far higher stakes when the thing spreading is a disease. A virus travels along the contacts people already have. So what happens next depends on the shape of that network. “Whether an outbreak takes off depends far more on how a population is connected than on any single case,” Clara says. In her research she has found that the way people cluster into groups can both speed an outbreak up and hold it back.
The maths even shows where the danger lies. “What surprised us is that the big pattern of groups matters, while the fine detail inside each group barely does,” Clara says. There is also a tipping point: below it an outbreak dies out, while just above it the same disease can reach almost everyone. Reading which side of that line a population sits on is what helps health workers act before a local flare-up turns into an epidemic.
What to do with your doubles
Those doubles are simply the coupon collector’s problem at work, the same puzzle that has frustrated collectors for decades. The way out has not changed either. Find a friend, trade your spares, and let the swaps do what a thousand packets cannot.
