Seven Shuffles- the amount of shuffles said by Diaconis and Bayer to completely randomise a deck of cards. In particular, a set of cards which have been perfectly riffle shuffled (Eastaway, 2008, p44).
According to Eastaway (2008, p41) ‘a riffle shuffle involves cutting the pack into two roughly equal piles, then bending the piles up with the thumbs, bringing them together’. There are many variations of riffle shuffle, but for the purpose of this blog post, I plan to relate to what is called a ‘perfect riffle shuffle’ (Eastaway, 2008, p41).
Within a ‘perfect riffle shuffle’ (Eastaway, 2008, p41), one whole deck of cards is divided equally into two piles of 26:
This can be carried out in two ways:
The first, is a ‘Perfect Out Shuffle’ (Eastaway, 2008, p42). This shuffle occurs from dropping the ‘bottom-half’ of the pack of cards first to create a mixed set of cards. This shuffle allows both the top and bottom card to stay in the same place (Eastaway, 2008, p42). This could be said to give a magician a strategical advantage for working out the location of other cards in the pack, and therefore the cards which you choose when playing.
The second is called a ‘Perfect In Shuffle’ (Eastaway, 2008, p42). It follows the same concept as previously, however it places ‘the card that was originally at the top of the pack’(Eastaway, 2008, p42) directly underneath the top one. The original bottom card is placed ‘second from the bottom’ (Eastaway, 2008, p42). This means that ‘the cards that were originally on the outside of the pack are now tucked inside’ (Eastaway, 2008, p42).
‘If you take a complete pack of 52 cards and out shuffle them (perfectly) exactly eight times, you will discover you have restored the pack exactly to the order that it was in when it started’ (Eastaway, 2008, p43). If you do it one less time, it could be said that the pack will end up completely randomised. However, it is important to be critical of this, as hardly anyone is able to achieve a perfect shuffle, giving variation to the results.
How does this knowledge link to maths?
Whilst knowing the perfect amount of times to shuffle a pack of cards may not be useful unless you are a magician, there is a distinct pattern which can be shown and predicted throughout the shuffling process. Eastaway (2008, p40) states that ‘in every game there is an element of finding patterns…but even more importantly there is an element of chance’, and this is why the example of playing cards resonates with me. When watching card tricks, I have always been in awe of how the magicians are able to guess your cards. I had always put this down to a lucky guess, but in fact there is a mathematical element of chance and probability behind the magic, and it comes in the form of shuffling a pack of cards and being able to predict which set of cards will end up where.
Mathematician and magician Persi Diaconis created a formula to define and prove the theory that if you shuffle a pack of cards using a basic riffle shuffle, then it will only need shuffled 7 times to have completely randomised the deck. This is the main formula used to describe the theory:
Whilst this formula is resonant of an advanced understanding of mathematics, one which I don’t quite have a grasp of yet! (There is a paper in the references below if you fancy having a go at it yourself), it has highlighted to me in a rather unique way, why it may be useful to children to learn about the concept of chance and probability.
In a real world context, the theory can have negative implications for casinos as Persi Diaconis and Dave Bayer have proven (Eastaway, 2008, p44). This is because ‘after one, two, three shuffles, the original card pattern does begin to mix, but traces of the original order can still be spotted’ (Eastaway, 2008, p44). This provides people who are highly skilled in card play, the opportunity to track and more successfully win a game (Eastaway, 2008, p44).
Chance and probability play a large part in people’s everyday lives. It goes beyond a mathematical concept and trickles into everything that we do. This is why I believe that teaching children chance and probability is of key importance. Life is not one linear predictable process it changes constantly. It is important for children to learn this, and on a very basic level, it could be argued that learning this in school begins to build the foundations of children’s understanding of the world. According to Askew and Eastaway (2013, p260) ‘we are surrounded by uncertainty, and being able to cope with events that are not entirely predictable is one of the most important life skills’. This highlights the importance of learning chance and probability in order to navigate the world around us.
References:
Askew, M and Eastaway, R (2013) Maths for Mums and Dads; Take the pain out of maths homework. London: Square Peg.
Eastaway, R (2008) How many socks make a pair? Suprisingly Intersesting Everyday Maths. London: CPI.
Persi Diaconis (N.D) Mathematical Developments from the analysis of riffle shuffling. Available at: http://statweb.stanford.edu/~cgates/PERSI/papers/Riffle.pdf
(Accessed 15th November 2015)
