The maths behind the largest ever World Cup sticker book

Collecting stickers of players ahead of the World Cup is a much-loved tradition. This tournament the sticker book is the largest ever – this is what it takes to complete it.

At first, it is hard to see what the fuss is about. They're just stickers, after all. 

Yet, for decades, children (and some adults) have excitedly bought, traded and coveted stacks of stickers featuring pictures of the best football players in the world.

I remember, as a child growing up in the 1990s, the thrill of shuffling through a friend's "swaps" pile in the hope of finding one of the few remaining stickers I needed to complete an album. The words "Got. Got. Got. Need!" would ring out across the playground.

The fervour around them can be compared to collecting baseball cards or Pokémon Trading Cards. As can the frustration of those stickers that seem to come up again and again.

Some stickers – the rarest of the most famous players – become so sought after that they sell for huge sums of money. One 1979 sticker of Argentinian footballing legend Diego Maradona sold at auction in 2021 for £413,000 ($555,000).

And ahead of the 2026 World Cup, demand is strong.

More than 8,000 fans collecting for Panini's 2026 World Cup sticker album recently gathered at a massive "swap meeting" in Santiago, Chile.

Their aim? To fill the pages of an album with icons of all of the players from teams competing in the World Cup. With this year's album being the biggest World Cup sticker book to ever be released, it will be quite some challenge.

So what does it actually take to complete a World Cup sticker album? Fortunately, we can turn to maths to help us work it out.

Much of it depends on your strategy.

The 2026 Panini World Cup sticker album comes complete with 980 different blank spaces to fill (by comparison there were just 270 stickers to collect in the 1970 World Cup in Mexico and 670 in the 2022 World Cup in Qatar). If you are planning to simply buy packet after packet to complete your set then you're going to have to have deep pockets.

Stickers can be bought in packs of seven, costing $2 per pack in the US and £1.25 in the UK. This would mean you are looking at a minimum outlay of $280 or £175.

But this would require each sticker you find to be a new one, with no swaps. While this is a statistical possibility, it's so astronomically unlikely, it practically impossible.

In reality, for each gap you want to fill, you'll end up buying multiple stickers. After you get your first sticker, the probability that the next one will be any of the 979 stickers you haven't already got is 979/980. This means that on average we will need to buy 980/979 (just over one) stickers to fill this next slot. And so the trend continues with the next spot needing 980/978 stickers and the one after that requiring 980/977 stickers on average.

As the probability of finding different stickers early on is close to one, it means it is unlikely that you'll get any repeats for a while. However, as you draw closer to completing the album, the expected number of stickers you need to buy to fill the next gap balloons massively. To obtain the penultimate sticker you would face a probability 2/980 of getting it for each sticker you buy. This means you'd need to buy 490 stickers on average to fill that penultimate gap. The last sticker is even worse. On average you'd need to buy 980 stickers to find that elusive one.

You can see the formula for calculating the number of stickers you'd need to buy to fill an album below.

In total we would expect to have to buy 7,316 stickers, bought in packs of seven. That adds up to a lot of money – around $2,100 if you are in the US or £1,300 in the UK.

The diminishing set of fractions inside the brackets are a well-known mathematical object called the "Harmonic Series". The name comes from the concept of "overtones" or "harmonics" in music.

The formula for completing a sticker album crops up in a huge variety of different areas because it relates to the cost or effort required to fully explore an unknown set. It can tell you how many people you may need to survey to get a full range of consumer preferences in market research. In ecology, scientists use the harmonic series to estimate the number of species in a particular habitat.

But heading back to our sticker collection problem, the harmonic series formula not only tells us how much it will cost to fill a sticker album, but also explains why it is so much more satisfying (not to mention cheaper) to fill the first half of the album than the second. To fill the first 490 gaps of a 980-sticker album we would expect to have to buy just 679 stickers. But to fill the second half we would need to buy another 6,637 stickers.

But much of the fun in sticker collecting is the ability to swap duplicates with your friends.

There is another mathematical formula relating to the harmonic series which can tell you how many stickers you and your friends would need to buy to complete all your albums. It's complicated, but let's imagine you've got 10 friends who are all collecting.

This would mean that together, you are hoping to build up 11 complete albums – one for each of you. Together you'd need to buy roughly 26,230 stickers, which works out at roughly 2,385 stickers each. That is going to cost each of you roughly $680 or £430 – a substantial saving compared to doing it alone.

Even with just four other friends, the cost would come down about $850 or £530 each – still a substantial saving.

But there is another way – head to the Panini website and order any missing stickers. They typically cost you twice as much as a normal sticker and there's a strict limit on the number you can purchase.

The most cost-effective way to complete the 2026 World Cup album would be to collect the first 490 stickers by buying packets – this would cost you about $200 or £120. Then if you order the rest from the Panini website, it will cost you about $280 or £175. This would set you back about $480 or £300 in total.

Whether Panini let you order 490 missing stickers directly is a different matter.

More like this:

• These scientists have spent decades perfecting World Cup grass

• The mathematics of the perfect penalty shootout

• A bluffers guide to the World Cup

Of course, in this modern age, you can collect your "stickers" in an online album. There's no need to go to the shops to buy your sticker packets, no call for spending time swapping with your friends and no requirement to painstakingly stick the stickers into the album in the right place.

But where's the fun in that?

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