On The Swords of Truth

'The Swords of Truth' trick is an ingenious and intriguing little mathemagic trick based on the work of magician and computer programmer Alex Elmsley, notable for his mathematical card tricks, and for publishing on the mathematics of playing card shuffling. The trick has been described by Michael Stevens (@Vsauce) and Professor Hannah Fry (@fryrsquared) in their 'The Rest is Science' podcast as 'one of the most deceptively tricky logic puzzles of the last century... [a] this seemingly innocent riddle [that has] fascinated mathematicians, programmers and puzzle-solvers for decades'.

Specifically, the 'Swords of Truth' trick — designed by Dave Janelle with a wonderful wooden keychain version manufactured and sold by Creative Crafthouse — utilises a binary grouping logic Alex Elmsley applied to a card trick using 16 cards. Elmsley originally published this routine in the September 1958 issue of the magic journal 'Ibidem (Issue 14)', under the title  'Mathematics and Mentliasm', which can also be viewed in Stephen Minch's 1991 'Collected works of Alex Elmsley, Volume 1', pp378-380.

For a maths teacher, 'The Swords of Truth' is a potent conceit that piques students' interest and that — when used carefully of course (see 'When Magic Fails in Mathematics,' by Junaid Mubeen, for example) — levers and encourages strong, mathematically mature thinking. After playing the trick out on/with students, it is demystified carefully, and collectively, before students use their newfound knowledge and understanding creatively in devising a different trick, based on the same principles. (Downloads of the 'swords' can be accessed here, in pdf, png and ppt.)

The Trick

• Ask someone to secretly choose any one of the 'swords'.

• Ask them to secretly choose any one of the numbers on that 'sword'.

• Ask them to read aloud the shapes surrounding the numbers on that 'sword' in order, but they must lie about the shape surrounding their chosen number, i.e., if a circle surrounds their chose number they must read it as a square, and vice-versa. For example, if they chose number 12 on the 5th sword, they will say 'Circle Circle Square Circle* Square Circle', lying about the square surrounding the number 12 and saying it was a circle. 

The Reveal

• Convert the shapes into binary numbers, where 'circle' is zero and 'square' is one. 

• Split the six digits you now have into two groups of three.

• Convert the first set of three from binary to decimal to give you the first digit of their chosen number.

• Convert the second set of three from binary to decimal to give you the second digit of their chosen number.

 

The How / Why?

• The number 1 in binary (1)₂ = (1 x 2⁰) = (1 x 1) = (1)₁₀ → Hence in any group of 3 shapes, should number 1 be the chosen digit, the sequence of shapes will be 001, i.e., Circle Circle Square.

• The number 2 in binary (10)₂ = (1 x 2¹) + (0 x 2⁰) = (1 x 2) + (0 x 1) = (2)₁₀ → Hence in any group of 3 shapes, should number 2 be the chosen digit, the sequence of shapes will be 010, i.e., Circle Square Circle.

• The number 3 in binary (11)₂ = (1 x 2¹) + (1 x 2⁰) = (1 x 2) + (1 x 1) = (3)₁₀ → Hence in any group of 3 shapes, should number 3 be the chosen digit, the sequence of shapes will be 011 , i.e., Circle Square Square.

• The number 4 in binary (100)₂ = (1 x 2²) + (0 x 2¹) + (0 x 2⁰) = (1 x 4) + (0 x 2) + (0 x 1) = (4)₁₀ → Hence in any group of 3 shapes, should number 4 be the chosen digit, the sequence of shapes will be 100, i.e., Square Circle Circle.

• The number 5 in binary (101)₂ = (1 x 2²) + (0 x 2¹) + (1 x 2⁰) = (1 x 4) + (0 x 2) + (1 x 1) = (5)₁₀ → Hence in any group of 3 shapes, should number 5 be the chosen digit, the sequence of shapes will be 101, i.e., Square Circle Square.

• The number 6 in binary (110)₂ = (1 x 2²) + (1 x 2¹) + (0 x 2⁰) = (1 x 4) + (1 x 2) + (0 x 1) = (6)₁₀ → Hence in any group of 3 shapes, should number 6 be the chosen digit, the sequence of shapes will be 110, i.e., Square Square Circle.

• The number 7 in binary (111)₂ = (1 x 2²) + (1 x 2¹) + (1 x 2⁰) = (1 x 4) + (1 x 2) + (1 x 1) = (7)₁₀ → Hence in any group of 3 shapes, should number 6 be the chosen digit, the sequence of shapes will be 111, i.e., Square Square Square.

Examples

For example, if the number 12 is chosen on the 5th sword: 

• Circle Circle Square Circle* Square Circle  →  (001)₂ | (010)₂
• (001)₂ = (1 x 2⁰) + (0 x 2¹) + (0 x 2²) = 1 + 0 + 0 = (1)₁₀ → So the first digit is 1.
• (010)₂ = (0 x 2⁰) + (2 x 2¹) + (0 x 2²) = 0 + 2 + 0 = (2)₁₀ → So the second digit is 2.
• Hence their chosen number is 12.

For example, if the number 54 is chosen on the 4th sword: 

• Square Circle Square Square Circle* Circle → (101)₂ | (100)₂
• (101)₂ = (1 x 2⁰) + (0 x 2¹) + (1 x 2²) = 1 + 0 + 4 = (5)₁₀ → So the first digit is 5.
• (100)₂ = (0 x 2⁰) + (0 x 2¹) + (1 x 2²) = 0 + 0 + 4 = (4)₁₀ → So the second digit is 4.
• Hence their chosen number is 54.

For example, if the number 2 is chosen on the 1st sword: 

• Circle Circle* Circle Circle Square Circle → (000)₂ | (010)₂
• (000)₂ = (0 x 2⁰) + (0 x 2¹) + (0 x 2²) = 0 + 0 + 0 = (0)₁₀ → So the first digit is 0.
• (010)₂ = (0 x 2⁰) + (1 x 2¹) + (0 x 2²) = 0 + 2 + 0 = (2)₁₀ → So the second digit is 2.
• Hence their chosen number is 2.

 

More Discussion and Explanation

Michael Stevens (@Vsauce) and Professor Hannah Fry (@fryrsquared) play with the wonderful wooden 'Swords of Truth' keychain produced by Creative Crafthouse in their 'The Rest is Science' podcast 'The Magic Trick that Fools Everyone'. They give the binary code explanation and move onto discuss error correcting codes.

Possible Teaching Approaches

Thinking through some possible starting questions for students to consider (or to consider with students), before moving on perhaps with students creating their own 'Swords of Truth' trick. You may also want to follow with 'On The 'Mystery Calculator' Trick', another and more well-known mind reading mathemagic trick that utilises binary grouping.

• What do you notice about the numbers on each 'sword'?
• Can we have a number larger than 76 on any of the swords?
• What would be the largest possible number if we had fewer or more 'swords'?
• Would we need more 'swords' if we wanted there to be higher numbers?  
• Are there any constraints to the numbers we place on the cards?
• How did the puzzle designer make the 'lie' work?
• Would a similar trick in another base, maybe base-3, be more magical?
• How high can we count in binary on our fingers? (Watch this Ted-Ed video from James Tanton.)

Super Curricular Activities in Mathematics

Super-Curricular Activities are activities that lie outside the classroom, beyond the school curriculum and beyond schools' extra-curricular offers. And whilst Super-Curricular participation is undoubtedly of benefit with University applications [1], the one-page of 'Super-Curricular Activities in Mathematics' collated and suggested here are done so purely for the 'more interested' young secondary school mathematician to explore, play and engage with outside of school. 

The 'Super Curricular Activities in Mathematics' to download below is a one-page collation of links to free resources and activities, from pre-school reading to University Admissions Tests. (Visits to museums, attendance at seminars and lectures, either online or in person, are not included in this collection, but of course are very much encouraged in super-curricular terms.) It includes suggested reading lists, suggested films and documentaries to watch, interviews and talks and explanatory animations to watch, podcasts and radio programmes to listen to, online courses to follow and engage with, and more. 

Once downloaded you can of course edit and adapt as you wish. The collation may complement schools' 'more able' programmes, but it is in no way designed nor intended to be restricted for exclusive use with students identified as 'more able'. Indeed, the intention is to get more students more interested in mathematics, expose more students to mathematics that would not ordinarily be visited in school curricula, and in doing so, expose more students to the power and beauty of the discipline.

Download:

• Super Curricular Activities in Mathematics (.pdf format)
• Super Curricular Activities in Mathematics (.doc format)
• Super Curricular Activities in Mathematics (.png format)

Notes, References & Links:

The First Equation

The 'First Equation' is a story about the evolution of the notation we use for addition (+), subtraction (–), and equality (=), which led to the first time an equation was written using the modern notation we use today, 14x + 15 = 71. Interestingly, the equals sign was introduced some two centuries after the introduction of the addition and subtraction signs, but some 50 years before the multiplication sign and a century before the division sign. 

The 'First Equation' is a story that will help students appreciate how the invention of mathematical notation fuelled the development of mathematics. Moreover, it is a story that may help students appreciate the diversity inherent in the development of mathematics; let them frame their own learning not just a function of that diversity, but also as an intrinsic part to the continuing story of mathematics. It can be used by teachers to support their teaching of setting up and solving algebraic equations to solve problems.

Downloads:

• Lesson slides (.ppt format)
• Lesson slides (.pdf format)

The introduction of the + and – signs

In her 2000 paper A Brief History of Algebraic Notation, Lucy Stallings identifies 'one of the earliest known symbols for addition and subtraction' to come from ancient Egypt, as seen in the Rhind papyrus (c. 1650 BC). The symbols used were 'a pair of legs walking forward [for] addition and a pair of legs walking away for subtraction' (p233). Howard Eves, in his 1983 book An introduction to the history of mathematics, describes a symbol for subtraction being used by the Greek mathematician Diophantus in the 3rd century, and the use by Hindu mathematicians of yu (यु ) as an abbreviation of yuta, meaning 'added', seen in the Bakhshali manuscript (which also featured the first recorded use of 0 for zero). 

But none of these symbols were universally adopted. The + and − signs that we use universally today were first introduced by French polymath Nicole d'Oresme in his Algorismus Propotionum (1356-61). However, as Edward Grant notes in his 1965 translation of d'Orseme's Algorismus, d'Oresme somewhat confusingly for us today called multiplication addition, and division subtraction. (For more about d'Oresme, the 'French Einstein of the 14th century', see this short biography.) 

Both the + and − signs were first used in print to indicate 'surplus' and 'deficit' ('what you add' and 'what you minus') by German mathematician Johannes Widman in his 1489 book 'Behende und hüpsche Rechenung auff allen Kauffmanschafft' (following the link will take you to the page where the symbols were first used; see the excerpted image above.) The subsequent universal adoption of + and − for addition and subtraction is evident from Heinrich Grammateus' Ayn new Kunstlich Buech (1521).

It is widely agreed that d'Oresme and subsequently Widman used + as an abbreviation for et, the Latin for 'and'. Eves, in describing Widman's use of +, and the subsequent universal adoption of such evident from Heinrich Grammateus' Ayn new Kunstlich Buech (1521), reinforces this belief 'as [et] was cursively written in manuscripts just before the time of the invention of printing' (p139). We remain unsure, however, of the reasons behind the adoption of the − symbol for subtraction.

The introduction of the equals sign

Welsh Doctor and mathematician Robert Recorde, born in 1510 in Tenby, Pembrokeshire, South Wales, was a popular author of a number of mathematical books — which he wrote, unusually for the time, in the English vernacular, thus making his writing more accessible than most scholarly books of the age, which were usually written in Latin. In his 1557 book, The Whetstone of Witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Cossike practise, with the rule of Equation: and the woorkes of Surde Nombers [1] (which you can peruse electronically here), Recorde was the first to use the plus and minus signs in English: 'There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made – and betokeneth lesse'. In the same book, Recorde also introduced the '=' symbol, 'to avoide the tediouse repetition of these woordes : is equalle to :', which he had already used some 200 times in the book [2].

During Recorde's time, much of the mathematical notation we take for granted today was not yet in use. In designing the symbol '=' — as 'a paire of paralleles, or Gemowe [twin] lines of one length, thus: =====, bicause noe 2 thynges can be moare equalle' — Recorde's initial motivation to abbreviate was quickly overtaken by something more profound, more enduring.  As Joseph Mazur (2014) eloquently puts it, 'the concise character of the symbol came with an unintended benefit: it enabled an unadorned picture in the brain that could facilitate comprehension'.

(As an aside, neither was there an easy way in the 16th century of denoting the powers of numbers, so Recorde coined the now unsurprisingly obsolete term ‘zenzizenzizenzic' [3] to ‘doeth represent the square of squares squaredly’, or in other words to denote the square of the square of a number's square: \({\left( {{{\left( {{n^2}} \right)}^2}} \right)^2} = {n^8}\).  Recorde also used another word (which didn't quite catch on), the 'sursolid', meaning to be raised to a prime number greater than three.  So a power of five would be the first sursolid, a power of seven the second sursolid, a power of eleven the third, and so on.)

The First Equation

In The Whetstone of Witte, using his newly invented '=' sign, Recorde published what is thus the first equation to appear in the symbolic form we use today. The equation was 14x + 15 = 71. Before Recorde, mathematicians would state an equality in words. But the introduction of the equals sign was more than just a function of efficiency (or laziness!). Recorde used the equals sign to express the fundamental role of equality in algebra, namely that if we perform the same manipulations on two equal expressions, i.e., to what we now call an equation, the results must still be equal. In Recorde’s words, 'if you abate even portions from thynges that bee equalle, the partes that remain shall be equall also.'

Notes, References & Links: