# Goldsmiths’ 2014 Mathematics

Tuesday 22nd July

Professor Peter McOwan

The Maths & Computing Magic Show

Any magic trick is part performance and part mathematics.

Because of the magician’s code I won’t give away any how the professor was able to carry out his tricks as he is a member of the Magic Circle.

Torn and restored card trick

Mind control…you will all think of the same card!

You pick a card from a pile and you have to remember it. The pile will be shuffled. A card will be removed by the magician and it will be torn. When you see the rest of the cards you will be able to say that it must be your card that has been torn. But are you sure?

It works because a mixture of colours and numbers puts the brain under pressure and with a short time interval it makes approximates.

The trick dates back to 1905 and is credited to American magician Henry Hardin

Henry Hardin (1849-1929), born Edward A. Parsons, was a musician and music teacher by profession in New Haven, Connecticut.

http://www.geniimagazine.com/magicpedia/Henry_Hardin

What does it have to do with your MP3 player?

Mp3 coding for music

A mobile digital media player (M-DMP), portable media player (PMP), or digital audio player (DAP), is a portable digital consumer electronics device capable of storing and playing digital media such as audio, images, and video files.

http://en.wikipedia.org/wiki/Mobile_digital_media_player

http://en.wikipedia.org/wiki/Data_compression

In computer science and information theory, data compression, source coding, or bit-rate reduction involves encoding information using fewer bits than the original representation. Compression can be either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by identifying unnecessary information and removing it. The process of reducing the size of a data file is popularly referred to as data compression, although its formal name is source coding (coding done at the source of the data before it is stored or transmitted).

The innovation of lossy audio compression was to use psychoacoustics to recognize that not all data in an audio stream can be perceived by the human auditory system. Most lossy compression reduces perceptual redundancy by first identifying perceptually irrelevant sounds, that is, sounds that are very hard to hear. Typical examples include high frequencies or sounds that occur at the same time as louder sounds. Those sounds are coded with decreased accuracy or not at all. In other words loud sounds are kept because the brain would miss them but removing quiet sounds gives more room for data.

Brain Experiments!

The brain breaks down a process. The eye takes in an image and the brain scans for importance. It is possible for a computer to remove part of the image and the brain not notice. This is known as change blindness.

http://en.wikipedia.org/wiki/Change_blindness

Example of images that can be used in a change blindness task

Mathematical models can predict what the brain will notice – neuromophic modelling

http://en.wikipedia.org/wiki/Neuromorphic_engineering

https://www.humanbrainproject.eu/en_GB/neuromorphic-computing-platform

Eyes & seek app
www.qappsonline.com

This fun gaming app gives you a chance to test your own powers of perception against artificial intelligence in a super high tech game of spot the difference. The app is based on state of the art research into the fascinating phenomena of change blindness, where as you will discover your brain often doesn’t notice even big changes in the world around it. – See more at: http://www.qappsonline.com/apps/eyes-seek/#sthash.RY0GQpU6.dpuf

The Piano Trick (aka the invisible palm)…

Enquiry based learning

http://www.cs4fn.org/magic/pianotrick.php

This classic of card magic allows the magician to make a single card change places under very mysterious and possibly musical circumstances.

http://www.mathematicalmagic.com/

A simple algorithm and some misdirection is the key. We are likely to accept a fib if it comes after something that is true. Even clever people can be confused by the maths.

The importance of understanding humans

http://www.chi-med.ac.uk/

CHI+MED (Computer-Human Interaction for Medical Devices, EP/G059063/1) is an EPSRC-funded project to improve the safety of interactive (programmable) medical devices, such as infusion pumps. By understanding more about device design and human factors, medical errors can be reduced thus saving lives.

The perfect shuffle

http://en.wikipedia.org/wiki/Faro_shuffle

The faro shuffle (American), weave shuffle (British), riffle shuffle or dovetail shuffle is a method of shuffling playing cards. Mathematicians use “faro shuffle” for a shuffle in which the deck is split into equal halves of 26 cards which are then interwoven perfectly.

http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle

The Fisher–Yates shuffle (named after Ronald Fisher and Frank Yates), also known as the Knuth shuffle (after Donald Knuth), is an algorithm for generating a random permutation of a finite set—in plain terms, for randomly shuffling the set. A variant of the Fisher–Yates shuffle, known as Sattolo’s algorithm, may be used to generate random cycles of length n instead. The Fisher–Yates shuffle is unbiased, so that every permutation is equally likely. The modern version of the algorithm is also rather efficient, requiring only time proportional to the number of items being shuffled and no additional storage space.

Fisher–Yates shuffling is similar to randomly picking numbered tickets (combinatorics: distinguishable objects) out of a hat without replacement until there are none left.

http://en.wikipedia.org/wiki/Frank_Yates

Frank Yates FRS (May 12, 1902 – June 17, 1994) was one of the pioneers of 20th century statistics

http://en.wikipedia.org/wiki/Ronald_Fisher

Sir Ronald Aylmer Fisher FRS (17 February 1890 – 29 July 1962) was an English statistician, evolutionary biologist, geneticist, and eugenicist. (below left)

S. Brent Morris (above right) is an American author who writes on Freemasonry. He is a Master Mason, a 33° Scottish Rite Mason, and currently the editor of The Scottish Rite Journal, a publication of the Supreme Council of the Scottish Rite Southern Jurisdiction.

Dr. Brent Morris has worked out the mathematics that underlies the mathematics of the perfect shuffle and it became the topic for his doctoral dissertation. He muses that he may have the only doctorate ever awarded in card shuffling.

http://en.wikipedia.org/wiki/S._Brent_Morris

Magic can help with maths. Optimal number of shuffles is connected with binary numbers.

Body language experiment…

http://www.princeton.edu/main/news/archive/S35/82/65G58/index.xml?section=topstories

Princeton University researchers report in the journal Science that facial expressions can be ambiguous and subjective when viewed independently. The researchers asked study participants to determine from photographs if people were experiencing feelings such as loss, victory or pain from facial expressions or body language alone, or from both. In some cases, a facial expression associated with one emotion was paired with a body experiencing the opposite emotion.

Body language and micro expressions can make sure that a magician’s trick always works, but there are people out there who can spot the trickery.

http://www.stat.berkeley.edu/~aldous/Real-World/coin_tosses.html

Persi Diaconis (Born: January 31, 1945 in New York City) is a mathematician, statistician, magician, debunker, MacArthur Foundation award winner and an expert on the mathematics of card shuffling. He has background in magic which has proved useful in another area – catching “psychics” cheating. If a person, even a well-trained scientist, has not had experience with human subjects and with cueing (subtle, body-language hints), then it is extremely difficult to spot what is wrong. Diaconis, however, is an expert at deception and has found cheating, or failure to perform, with every psychic he has been allowed to observe.

http://stat-athens.aueb.gr/~jpan/Persi-Diaconis.pdf

Diaconis is often cited for the simplified proposition that it takes seven shuffles to randomise a deck of cards.

http://en.wikipedia.org/wiki/Persi_Diaconis

Algebra- Lets call them…

http://www.mathematicalmagic.com/docs/mathsmagic_full.pdf

The body language experiment

This is a method of making abstract algebra genuine. Take a complete pack of 52 cards and shuffle them well.

Let’s say the first card that is picked is placed face up (given the symbol B0). The next card is placed face down underneath the black card (given the symbol B1 for now). The third card is looked at and if it is black it will be placed face up on the known black card and the fourth card will be placed on top the face down card. If however the third card is red it will be placed face up on a new pile (given the symbol R0) and the fourth card will be placed underneath this (given the symbol R1) for now. So each odd counted number card will be placed on the correct card pile but each even counted card will simply go on to the matching pile until there is a change in the odd colour.

For instance your first card is black so your second card is placed under it face down. If your third card is black it goes face up on B0 and your fourth card is placed on the card underneath face down. If your firth card is also black it goes on B0 and your sixth card goes on the underneath pile face down. Now if your seventh card is red you start a new pile face up and the eighth card goes face down under it.

So you will end up with four piles of cards. The face up black cards (B0) and the face up red cards (R0), the face cards that go with the B0 (R2 and B2) and the cards that go with R0 (B1 and B1)

http://www.ams.org/samplings/feature-column/fcarc-mulcahy1

You need two volunteers and one is give the R1 + R2 pile and the other is given the B1 + B2 pile. The volunteers are asked vague questions about the quantity of red and black cards they have and they are asked to swap one or two cards (not knowing what they will get) until the magician tells one volunteer that they will have the same number or red cards as the other volunteer has black cards.

This is a final proof that the sub-conscious mind can make us choose random cards to balance those numbers! So you have seen/read about the body language experiment. Is mind control a reality?

Of course it’s not mind control. It’s mathematics, but you knew that didn’t you? But how does this mind-reading miracle work? Well, it’s all down to Abracadabra algebra…

What do we know?

R0 + R1 + R2 = 26 reds

B0 + B1 + B2 = 26 blacks

R0 = R1 + R2

B0 = R2 + B2

(R1 + B1) + R1 + R2 = (R2 + B2) + B1 + B2

2R1 = 2B2 so R1 = B2

Classic of magic… Cutting a person in half

First Public sawing was by P T Selbit 1921 (Stage name of Percy Thomas Tibbles) Finsbury Park

P. T. (Percy) Selbit (1881–1938) was an English magician, inventor and writer who is credited with being the first person to perform the illusion of sawing a woman in half. Among magicians he was known for his inventiveness and entrepreneurial instinct and he is credited with creating a long list of successful stage illusions

http://en.wikipedia.org/wiki/P._T._Selbit

http://www.prestigiazione.it/wp/2012/11/il-17-novembre-del-1881-nasceva-p-t-selbit-linventore-della-donna-segata-in-due/

How can thirteen people become twelve people?

I’m not going to tell you but it is something to do with heights.

Canadian Mel Stover (1912-1999) was a well-respected magician and puzzle creator whose works all seemed to have both impish and amazing qualities.

http://www.puzzles.com/puzzleplayground/Authors/MelStover.htm

Research Questions….

How big a change can we make for it to still be ‘invisible’ to the audience?

Does it work in 2D?

Apply Artificial Intelligence to solve the problem….

Jigsaw picture of spells

http://www.qmul.ac.uk/mathsmagic/

http://www.learningmachines101.com/lm101-001-welcome-big-artificial-intelligence-magic-show-2/

http://en.wikipedia.org/wiki/Artificial_intelligence

http://www.ft.com/cms/s/2/32ab4a22-caee-11e1-8872-00144feabdc0.html#ixzz3BX1ELJx0

Marco Tempest is a Swiss born, New York based cyber illusionist, combining magic and technology (artificial intelligence?) to produce astonishing illusions. He began his performing career as a stage magician and manipulator, winning many awards and establishing an international reputation as one of the world¹s most unique performers. His interest in computer generated imagery led him to incorporating video and digital technology in his work and the development of a new form of contemporary illusion. The expansion of the internet and social media provided more opportunities for digital illusions and ways of interacting with audiences and creating magically augmented realities. Marco is a keen advocate of the open source community, working with artists, writers and technologists to create new experiences and research the practical uses of the technology of illusion. Marco continues to perform around the world, is a media consultant on the subject of magic and illusion and lectures at international conferences on the psychology of deception and creative thinking. Marco is a Directors Fellow at the MIT Media Lab.

http://www.marcotempest.com/

December 3, 1964

http://en.wikipedia.org/wiki/Marco_Tempest

http://sprott.physics.wisc.edu/pickover/esp.html

ESP Zener Cards tricks

The mind match experiment

The ten cards are shuffled by the volunteer and the magician makes two piles of five cards. He intends to influence the volunteer in to making the last card in each pile match.

The magician needs to memorise the order of the cards.

The volunteer is given some poker chips in order to help him make decisions about swapping cards. Each chip represents a swap and there is the same number of chips as cards in each pack. If the volunteer moves, say 2 chips, two cards need to be swapped. Throughout this process the magician is making suggestions. Each time a swap is done two cards end up being paired. Eventually there is only one pair of cards left.

By the magician’s suggestions the last two cards are likely to match.

This is an example of neuro linguistic programming (NLP). Helping the volunteer make a free choice.

Neuro-linguistic programming (NLP) is an approach to communication, personal development, and psychotherapy created by Richard Bandler and John Grinder in California, United States in the 1970s. Its creators claim a connection between the neurological processes (“neuro”), language (“linguistic”) and behavioural patterns learned through experience (“programming”) and that these can be changed to achieve specific goals in life.

http://en.wikipedia.org/wiki/Neuro-linguistic_programming

For books, links, videos and the ISS

Computer science for fun

www.cs4fn.org

http://teachinglondoncomputing.org/resources

http://www.illusioneering.org/

http://en.wikipedia.org/wiki/Modular_arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers “wrap around” upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician, who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, and optics.

http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

A palindromic number or numeral palindrome is a number that remains the same when its digits are reversed. Like 16461, for example, it is “symmetrical”.

http://en.wikipedia.org/wiki/Palindromic_number