My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. Here’s my series on the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, 2b, and 2c: The 1089 trick.

Part 3a, 3b, and 3c: A geometric magic trick (see also here).

Part 4a, 4b, 4c, and 4d: A trick using binary numbers.

Part 5a, 5b, 5c, 5d: Predicting a digit that’s been erased from a number.

Part 6: Finale.

Part 7: The Fitch-Cheney 5-card trick.

Part 8a, 8b, 8c: A trick using Pascal’s triangle.

Fun With Permutations and Asimov’s Three Laws of Robotics

I’m not a big fan of science fiction, but I know enough to know that Isaac Asimov was one of the great science fiction novelists of the 20th century. The following was written by him in the October 1980 issue of The Magazine of Fantasy and Science Fiction and was reprinted in his book Counting the Eons, which was published in 1983. (I’m now holding the battered and torn pages of my copy of this book; I devoured Asimov’s musings on mathematics and science when I was young.)

Robotics has become a sufficiently well development technology to warrant articles and books on its history and I have watched this in amazement, and in some disbelief, because I invented it.

No, not the technology, the word.

In October 1941, I wrote a robot story entitled “Runaround,” first published in the March 1942 issue of Astounding Science Fiction, in which I recited, for the first time, my Three Laws of Robotics. Here they are:

A robot must not injure a human being or, through inaction, allow a human being to come to harm.

A robot must obey the orders give it by human beings except where those orders would conflict with the First Law.

A robot must protect its own existence, except where such protection would conflict with the First or Second Laws.

Clearly, the order in which the Three Laws of Robotics matters. Shuffling the order leads to $3! = 6$ possible permutations, and xkcd recently had some fun about what the consequences would be of those permutations.

Source: http://www.xkcd.com/1613/

Predicate Logic and Popular Culture (Part 61): Taylor Swift

Let $S(t)$ be the proposition “We are in style at time $t$ ,” let $C(t)$ be the proposition “We crash down at time $t$ ,” and let $B(t)$ be the proposition “We come back at time $t$ .” Translate the logical statement

$\forall t (\lnot S(t)) \Rightarrow (\forall t(C(t) \Rightarrow \exists u>t(B(u)))$ .

The straightforward way of translating this into English is, “If we never go out of style, then whenever we crash down we come back at a later time. This approximately matches the second half of the chorus of one of Taylor Swift’s hit songs.

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 60): Heartland

Let $L(x,t)$ be the proposition “ $x$ loves her at time $t$ .” Translate the logical statement

$\exists t<0(L(\hbox{I},t) \land \forall x \forall s < t (\lnot L(x,t)))$ ,

where $t = 0$ is now.

The clunky translation is “There was a time that I loved her, and nobody loved her before that time.” More succinctly, this is the title of the song that’s been played for countless father-daughter dances at wedding receptions since 2006. (I cannot tell a lie: I always turn into a sobbing and amorphous pile of mush whenever I hear this song.)

Predicate Logic and Popular Culture (Part 59): Taylor Swift

Let $T(x)$ be the proposition “You go talk to $x$ ,” and let $G(x)$ be the proposition “We are getting back together at time $t$ .” Translate the logical statement

$T(\hbox{your friends}) \land T(\hbox{my friends}) \land T(\hbox{me}) \land \forall t\ge 0 (\lnot G(t))$ ,

where time 0 is now.

Of course, this is the ending part of the chorus to “We Are Never Ever Getting Back Together.”

Predicate Logic and Popular Culture (Part 58): Taylor Swift

Let $Y(x)$ be the proposition “You are $x$ years old,” and let $L(x)$ be the proposition “ $x$ tell you that $x$ loves you,” and let $B(x)$ be the proposition “You believe $x$ .” Translate the logical statement

$(Y(15) \land \exists x(L(x))) \Rightarrow B(x)$ ,

where the domain is all people.

The straightforward way of translating this into English is, “If you are 15 years old and there exists someone who says that he/she loves you, then you believe him/her.” This approximately matches the chorus of one of Taylor Swift’s earliest hits.

Predicate Logic and Popular Culture (Part 57): Frozen

Let $C(t)$ be the proposition “The cold bothers me at time $t$ .” Translate the logical statement

$\lnot(\exists t\le 0 (C(t)))$ ,

where the domain is all times and $t=0$ is now.

The straightforward way of translating this into English is, “It is false that there exists a time in the past that the cold bothered me.” Also, DeMorgan’s Laws could be applied:

$\forall t\le 0(\lnot C(t))$ ,

which can be read “For all times in the past, the cold did not bother me.” Of course, this is the closing line of the chorus of the signature tune from Frozen.

Of course, I can’t mention Frozen without mentioning its parodies; this is the best one that I’ve seen.

Predicate Logic and Popular Culture (Part 56): The Byrds

Let $S(x,t)$ be the proposition “ $t$ is the season for $x$ .” Translate the logical statement

$\forall x \exists t (S(x,t))$ .

This pretty much matches the opening line of the 1960s hit song by The Byrds from Ecclesiastes 3.

Predicate Logic and Popular Culture (Part 55): The Quiet Man

Let $L(x)$ be the proposition “ $x$ is a lock,” let $B(x)$ be the proposition “ $x$ is a bolt,” and let $H(x)$ be the proposition “ $x$ is in your own mercenary little heart.” Translate the logical statement

$\forall x ( (L(x) \lor B(x)) \Rightarrow H(x))$ ,

where the domain is all people.

The straightforward way of translating this into English is, “If it’s a lock or a bolt, then it’s in your own mercenary little heart.” With a little more emphasis, this is one of the great lines uttered by John Wayne in the 1952 film The Quiet Man (a wonderful movie which really needs to be digitized and restored to its original brilliance).

Predicate Logic and Popular Culture (Part 54): Michael Jackson

Let $p$ be the proposition “Billie Jean is my lover,” let $q$ be the proposition “Billie Jean is a girl,” let $r$ be the proposition “Billie Jean claims I am the one,” and let $s$ be the proposition “The kid is my son.” Translate the logical statement $\lnot p \land q \land r \land \lnot s$ .