Category: Engagement

My Mathematical Magic Show: Part 9

This mathematical trick was not part of my Pi Day magic show but probably should have been. I first read about this trick in one of Martin Gardner‘s books when I was a teenager, and it’s amazing how impressive this appears when performed. I particularly enjoy stumping my students with this trick, inviting them to figure out how on earth I pull it off.

Here’s a video of the trick, courtesy of Numberphile:

Summarizing, there’s a way of quickly determining x given the value of x^5 if x is a positive integer less than 100:

  • The ones digit of x will be the ones digit of x^5.
  • The tens digit of x can be obtained by listening to how big x^5 is. This requires a bit of memorization (and I agree with the above video that the hardest ones to quickly determine in a magic show are the ones less than 40^5 and the ones that are slightly larger than a billion):
    • 10: At least 10,000.
    • 20: At least 3 million.
    • 30: At least 24 million.
    • 40: At least 100 million.
    • 50: At least 300 million.
    • 60: At least 750 million.
    • 70: At least 1.6 billion.
    • 80: At least 3.2 billion.
    • 90: At least 5.9 billion.

 

 

 

 

 

 

 

Predicate Logic and Popular Culture (Part 92): Annie Get Your Gun

Let W(x,y) measure how well x can do y. Translate the logical statement

\forall y (W(\hbox{you},y) < W(\hbox{I},y)).

This needs no further introduction:

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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 91): The Scarlet Pimpernel

Let H(x,t) be the proposition “x happens at time t,” let V(x) be the proposition “x is a valley,” let M(x) be the proposition “x is a mountain,” let S(x,y,t) be the proposition “y must scale x at time t,” let W(x) be the proposition “x are perilous waters,” and let S(x,y,t) be the proposition “y must sail x at time t.” Translate the logical statement

\forall t \exists x (V(x) \land H(x,t)) \land \forall t \exists x(M(x) \land H(x,t) \land \exists y S(x,y,t))

\land \forall t \exists x(W(x) \land H(x,t) \land \exists y S(x,y,t)).

This matches the second half of the opening verse of the showstopper “Into The Fire” from the musical The Scarlet Pimpernel.

I also recommend Steve Amerson’s rendition:

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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 90): The Beatles

Let N(x) be the proposition “You need $latex x.” Translate the logical statement

N(\hbox{love}) \land \forall x (x \ne \hbox{love} \Rightarrow \lnot N(x)).

This matches the title of one of the Beatles’ greatest hits.

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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 89): Into the Woods

Let S(x) be the proposition “x is on your side” and let A(x) be the proposition “x is alone.” Translate the logical statement

(\exists x S(x)) \land (\forall x \lnot A(x)).

Of course, this matches the last two lines of “No One Is Alone,” which was released as a movie in 2014.

I’m personally partial to Steve Amerson’s rendition of this song:

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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 88): Bob Dylan

Let H(x) be the proposition “You have x,” let L(x) be the proposition “You have x to lose,” let $p$ be the proposition “You’re invisible now,” and let S(x) be the proposition “x is a secret to conceal.” Translate the logical statement

((\forall x \lnot H(x)) \Rightarrow (\forall x \lnot L(x))) \land p \land \forall x (S(x) \Rightarrow \lnot H(x)).

This matches the last two lines of the closing verse of this classic from Bob Dylan.

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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.