# World Cup stickers

28 March 2018

The task of completing this year’s Panini World Cup sticker book will typically set you back around £774, according to a mathematician at Cardiff University.

Eyebrows were raised earlier this month when Panini announced that the cost of a pack of 5 stickers would increase from 50p (EURO 2016) to 80p this year. With 32 squads of players, as well as a special stickers for managers, stadiums and World Cup legends, the task of completing the album is a considerable one.

Professor Paul Harper, from the School of Mathematics, Cardiff University, has explored how much it would cost to collect all 682 stickers from the World Cup 2018 sticker book.

If you were extremely lucky and never had a duplicate in the packets you bought, Professor Harper says that you would need a minimum of 137 packets to complete the entire book, costing you £109.60 (a total of 682 stickers in the entire book, with 5 stickers in each packet and a packet costing 80p: 682/5 = 137 packets x 80p = £109.60).

However, that is an extremely unlikely scenario and all of us have experienced the disappointment of pulling out duplicates or “swapsies” in the hunt for elusive players and “shinies”.

With this in mind, Professor Harper has calculated how many stickers we should expect to buy before getting a new sticker.

“The first sticker you buy is absolutely guaranteed not to be a duplicate,” Professor Harper said. “The second sticker you get has a 681/682 (99.85%) chance of being a new sticker. The third sticker you get has a 680/682 (99.7%) chance of being a new sticker, and so on.”

Professor Harper added up all of the probabilities of getting a new sticker to obtain the following formula in terms of the total number of unique stickers required, n:

n (ln (n) + γ) (where γ = Euler’s constant = 0.557)

It was then necessary to slightly adjust the calculations to account for the fact that the stickers come in packets of 5, which required some more in-depth calculations using conditional probabilities.

The calculations showed that on average you would need to buy 4,832 stickers, or 967 packets, to complete the book, costing £773.60.

Naturally a lot people collecting stickers will be swapping with their friends, which brings the average cost down.

According to Professor Harper, two people collecting and swapping can reduce the number of packets by 30%, five players by 57% and 10 players by 68%.

However, even with 10 friends swapping it might still cost them each £247 on average to complete the album.

“I can still recall the joy of finally completing my first Panini album as a young boy for the 1982 World Cup in Spain. I must have used an awful lot of pocket money to do this, as well as having generous grandparents handing over bundles of packets of stickers, coupled with tense negotiations of swapping duplicates with friends in the school playground!“

Filling an album has become progressively more expensive over the years since then, not just because there are typically more teams competing now, but because Panini have become more creative about allocating spaces,” Professor Harper continued.