Every four years football gives us the European Championship. At the same time, the Italian company founded by the Panini brothers gives us something far greater. Panini produce a sticker album with hundreds of stickers to collect, representing the players, stadiums and nations at the tournament. In doing so, they give us an intriguing mathematical challenge: how can we collect all 678 stickers as cheaply as possible?

This year there will be, as there always is, several articles claiming to calculate the cost of completing the collection. Most of these articles make the same basic mistakes, and are hugely misleading. Always remember to check the assumptions and logical basis of any of these analyses. This post is based on the facts and assumptions listed below

1) There are 678 stickers to collect

2) Each pack contains 5 stickers and costs £0.90

3) Each sticker is equally likely to occur in any pack

4) Individual stickers can be purchased on the Panini website for £0.28 each

Point 4 is ignored by almost everyone who attempts to calculate the cost of completing the collection – but it makes a huge difference.

You will see articles claiming that it costs £840 to complete the collection. This is wrong, as it doesn’t consider the possibility of buying individual missing stickers from the Panini website.

For those who just want to know how to complete the collection as cheaply as possible – **buy 60 packs, and then buy all the missing stickers online. Total cost: £176**. If you have a friend to swap duplicates with, you should **each buy 85 packs, do your swaps, and then buy the remainder online. Total cost each: £164. **If you have more than one friend in your swapping syndicate, then the price comes down again slightly, but by a smaller amount for each new friend you include.

If you care how to calculate this: read on!

To start – let’s think about the simplest way that someone can collect all the stickers. You could keep buying individual packs until you have them all! But how many packs do you need to buy? We can answer this through simulations. We use random numbers to randomly pick 5 from the 678 available stickers to create one pack of stickers. Then we count how many of those stickers are unique – not duplicates. We then randomly pick another 5 stickers to make another pack, and add those to our simulated collection. We can continue like this until our individual has collected all 678 stickers. The beauty of doing this on a computer is that we can then replicate this a large number of times, to work out how long it would take to complete the collection on average. I did this 10,000 times and have plotted the results below

I found that on average you would have to buy 934 packs to complete the collection! Which would cost a total of £840.60, more than most would be willing to spend on a football sticker album.

Why do we need to buy so many packs to complete the collection? Buying 934 packs would give us 4670 stickers, of which only a meagre 15% would be unique.

The best way of seeing why we have to buy so many packs is another simulation. But this time, instead of simulating the continual buying of packs until the 678 total is reached, we will simulate the buying of a number of packs equal to n, where n is any number between 1 and 1000. We can then record the number of unique stickers we expect to get for each number of packs we buy. This is repeated 10000 times for each number of packs, to get a good estimate of the average number of unique stickers obtained from each number of packs bought.

As you can see, we get a nice curve. And this makes good logical sense. At the start, for each new pack we buy its quite likely that all the stickers we get will be new to us, and contribute to our collection. Towards the end of the curve when we have already got most of the collection, each new pack is likely to contain only stickers that we already have. Note that we already have most of the stickers when we have bought 500 packs, but on average it took an extra 434 packs purchased to complete the collection in our simulations from earlier!

At this point do we give up hope, or start saving up more than £800 and preparing to empty the local Sainsburys of its supply of stickers? Well not quite yet. At any point we can go online and order individual stickers, the ones we need to complete the collection. Why don’t we just do this from the beginning, you may ask. Well aside from that being blatant cheating, individual stickers cost £0.28 each online, compared to the £0.18 they cost when you buy them as part of a pack. However, it seems clear that with this information we no longer have to buy 934 packs. As we have already noted, by 500 packs we already have most of the collection.

So with this new information, the key question becomes ‘at what point should I stop buying packs, and buy individual stickers instead?’. We can easily calculate this by taking our figure above, for the number of unique items we get for each number of packs bought, and converting this into a final cost, but adding the cost of buying that number of packs to the cost of buying the number of stickers required to complete the collection.

The y-axis of this graph is cost, so we can look for the point where this is lowest to find the optimal number of packs, and the cost of buying that number of packs. The answer is 60 packs, for a total cost of £175.93. A much more reasonable answer than 934 packs for a cost of £840

So far we have been using simulations to answer this question, but can we answer the same question with simple maths? Intuitively we know that we are dealing with simple probabilities, the probability of getting any sticker by picking one at random is 1/678. Can we however use this to calculate how many unique stickers we get for each number of packs bought? Yes we can.

This kind of situation is known in maths as the ‘Coupon collectors problem’. Let’s use p to represent the probability of getting a new unique sticker to add to our collection for each new sticker we buy. The chance of getting a new unique sticker from our first one is 1, since we haven’t seen any other stickers so there is 0 chance for a duplicate. The chance of getting a new unique sticker from our second pick is 677/678, since there is one option that we could pick that might be a duplicate. Similarly, for the third pick the chance of adding to our collection is 676/678. This can be formalised to say that the probability of getting a new unique sticker on any draw i = 1 + (n-1)/n + (n-2)/n + … and so on until we have a number of terms equal to i. Can we use this simple maths to work out the time it takes to reach any defined number of unique cards? Yes we can! time and probability can be simple inverses of each other, if defined carefully. For example, if I said that the chance of rolling an even number on a dice is 1/2, we would expect to take 2 rolls to see an even. So we can use our probability equation to calculate the time t to collect any number of unique items = 1 + n/(n-1) + n/(n-2) … and so on. We must be careful to include the fact that we don’t buy individual items, we buy stickers in packs of 5, so we have to divide this time by 5 to convert it to number of packs. Using this we could have arrived at the same answer

So is that the final word, shall we put some of our savings back in the bank and prepare to spend £176? Well actually that’s not the end of it either.

We are a social species, and we love to trade. So I could bring in a partner to collect stickers with me. And we would both benefit because I could give her my swaps for free, and receive her swaps for free!

For me, the easiest way to think about the effect of another player is that it gives us a loads of extra packs for free. For any number of packs we buy, we inevitably get a certain number of duplicates. Our partner has the same. Those duplicates are then packaged up and given to the other player as free packs. This has the effect of shifting the curve for the number of unique cards we get for any number of packs bought

In red here we can see how the blue curve we had before shifts when we take into account all the free cards we get from our partner. So what is the effect on the optimal number of packs we should buy?

In red here we can see the new cost function, how much it costs to complete the collection for any number of packs bought (before doing swaps and buying the rest individually). We have to buy more packs when playing with a partner than we do when alone – 85 packs compared to 60 when playing on our own. The key however is that the total cost to complete the collection is lower; £164. Buying more packs each gives us more swaps to give to the other player, allowing us to obtain a larger collection before we have to buy the comparatively expensive individual stickers online for £0.28 each.

Adding more friends marginally lowers the cost still, but not by much. If you think about it, there might be extra swaps if you have two friends with you, but those swaps also have to be shared amongst more people. You do get a benefit in avoiding a few of those annoying cases where your friends’ swaps are of no use to you, but the benefit is small. To see more, check out my 2018 post on the World Cup sticker collection

If you want to see how your progress compares with the expectation (to see how lucky you are), check out the excel spreadsheet below