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HR. CardColm

Colm Mulcahy, of Spelman College in Atlanta,  joins us to share his ice cream trick from his CardColm mathematical card trick column on the MAA website! You’re invited to explain how this works in the comments below.

Colm also shares a quick puzzle, tweeted on his What Would Martin Gardner Tweet feed @WWMGT. And finally we touch on the Gathering For Gardner and the Celebration of Mind, held all over the world around the time of Martin Gardner’s birthday, October 21.

And at last we get around to answering our quiz from a few weeks ago. There are indeed two solutions for correctly filling in the blanks in:

The number of 1’s in this paragraph is ___; the number of 2’s is ___; the number of 3’s is ____; and the number of 4’s is ___. 

[spoiler] namely (3,1,3,1) and (2,3,2,1) [/spoiler]

We can vary this puzzle at will, asking 

The number of 1’s in this paragraph is ___; the number of 2’s is ___; …..  and the number of N’s is ___. 

For N=2 or 3, there are no solutions (Asking that all the numbers we fill in are between 1 and N); for N=4 there are two. For N=5 there is just one, for N=6 there are none and beyond that there is just one. I think we’ll let the commenters explain that.

But here’s the cool thing. 

One way to approach the problem is to try filling in any answer at all, and then counting up what we have, filling that in, and repeating. Let’s illustrate, but first stipulate that we’ll stick with answers that are at least plausible– you can see that the total of all the numbers we fill in the blanks has to be 2N (since there are 2N total numbers in the paragraph). 

So here’s how this works. Suppose our puzzle is:

There are ___ 1’s;___ 2’s; ___ 3’s; ___ 4’s; ___ 5’s

Let’s pick a (bad) solution that totals 10, say, (2,4,1,2,1). So we fill in:

There are __2_ 1’s;   __4_ 2’s;    _1__ 3’s;      __2_ 4’s;     _1__ 5’s

That’s pretty wrong! There are actually three 1’s in that paragraph, three 2’s; at least there is just one 3, and two 4’s and one 5. In any case this gives us another purported solution to try: (3,3,1,2,1). Let’s fill that in:

There are __3_ 1’s;   __3_ 2’s;    _1__ 3’s;      __2_ 4’s;     _1__ 5’s

That attempt actually does have three 1’s; but has only two 2’s;  it does have three 3’s but only  one 4 and one 5. So let’s try (3,2,3,1,1):

There are __3_ 1’s;  __2_ 2’s;  _3__ 3’s;  __1_ 4’s;  _1__ 5’s

Lo and behold that works! We do in fact have three 1’s;  two 2’s; three 3’s and yes, one 4 and one 5.

So we can think of it this way: filling in a purported solution and reading off what we actually have gives another purported solution.

In this case (2,4,1,2,1) -> (3,3,1,2,1) -> (3,2,3,1,1) -> (3,2,3,1,1) etc,

We can keep following this process around, and if we ever reach a solution that gives back itself, we have a genuine answer, as we did here. 

So here’s an interesting thing to think about.

First, find, for N>=7, a correct solution; and a pair of purported solutions A,B  that cycle back and forth A->B->A->B etc.

Second, find a proof that this is all that can happen (unless I’m mistaken)–  any other purported solution eventually leads into  the correct one or that cycle.


Comments (8)

HQ. Newton v Leibnitz

A break from puzzling to discuss the history of the great Newton-Liebnitz dispute over the invention of Calculus, with the playwright Todd Taylor.


HE. On Cake and Coffee

Harry Kaplan joins us for discussion of cake and coffee– and leaves us with a counter-intuitive puzzle…

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HC. Strongly Connected Components

Samuel Hansen’s Strongly Connected Components podcast features interviews with all kinds of mathematical luminaries (that sounds familiar!) If you’ve been missing the Math Factor, be sure to check it out!

Here, we discuss, well, Chaim Goodman-Strauss—the tables are turned!


HA! Conway on Gardner

In this special segment, John H. Conway reminisces on his long friendship and collaboration with Martin Gardner. 

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GY. Chaitin on the Ubiquity of Undecidability

Greg Chaitin, author most recently of MetaMath!,  discusses the ubiquity of undecidability: incredibly all kinds of mathematical and physical systems exhibit utterly unpredictable, baffling behavior– and it’s possible to prove we can never fully understand why!

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G4G9: Report From the Festivities!

Quick interviews with folks here at the Gathering For Gardner, including Stephen Wolfram, Will Shortz,  Dale Seymour, John Conway and many others. 

Comments (1)

GU. Number Freak!


Puzzler Derrick Niederman tells us about his new book, Number Freak: From 1 to 200, the hidden language of numbers revealed, full of lore, mathematical amusements and numerical tidbits! 

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GT. The Largest Escher Exhibit Ever

The world’s largest ever exhibit of Escher’s works is on display, right now, at the Boca Raton Musuem of Art If you can, this is a must see event! We talk with the collector, Rock J. Walker about his fascination with this amazing work.


And of course we answer last week’s puzzle, and hear from listeners!



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GP, GQ, GR, GS: The Math Factor Catches Up (For Now)

A bit lazy, but we’re pretty far behind. Herewith, are

GP: Switcheroo!
GQ: Durned Ants
GR: VIth Anniversary Special
GS: I Met a Man

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