Mind-boggling Maths
Reporting by Evelyn (Year 7)
Did you know that there are over 8 x 10^67 different ways to shuffle a deck of 52 playing cards? The full number has 68 digits! This was the starting point for a mind-boggling lunchtime talk to Year 7 scholars by Maths teacher Mr Astbury-Palmer.
With 52 cards in a pack, there are 52 factorial (52!) different unique ways to arrange the deck, he explained. This means 52 x 51 x 50…all the way until…3 x 2 x 1. This number – 8 x 10^67 – is enormous, but just how big is big?
To illustrate the magnitude of 52!, Mr Astbury-Palmer led us on a thought experiment. Assuming it’s possible to shuffle a new combination of cards every second, how much time would we need to have made 52! shuffles? First, we calculated how many shuffles we could complete in a year (60 x 60 x 24 x 365 = 3.1536 x 10^7). Then we were told we were allowed to take a step forward by 1 metre every 1 billion years. Proceeding in this manner (with one step every 1 billion years and shuffling at a rate of once per second all the while), we worked out that by the time we had circumnavigated the Earth at the equator (a distance of 40.075 x 10^6 m), we would still only have made 1.26 x 10^24 shuffles.
At this point, Mr Astbury-Palmer suggested that for every complete circumnavigation, we would be allowed to remove a single drop of water (0.05 ml) from the Pacific Ocean (a total volume of 710 million cubic kilometres) and were to continue in this manner, one drop at a time after each circumnavigation, until we’d drained it entirely and then instantly refilled it. Then, for every complete emptying and refilling of the Pacific, we would be permitted to place a single sheet of paper 0.1 mm thick on the ground to begin building a tower. By the time the paper stack was tall enough to reach the sun (a distance of 150 billion metres), we would still only have completed 2.68 x 10^64 shuffles. So this entire sequence would have to be repeated just over 3000 times, still shuffling the deck at a rate of once every second to have reached our goal of 52! or 8 x 10^67 shuffles.
With such an extraordinary number of different combinations, it is virtually certain, says Mr Astbury-Palmer, that every single shuffle that has ever been made in human history has never been made before and will never be made again.
