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Like everybody else, you too can be unique. Just keep shuffling

The first take-home lesson of this note is that you too can be unique. You’ll have to keep shuffling to get there, but it is an attainable goal.

Several years ago it dawned on me that the number of possible ways to order or permute the cards in a standard deck of size $52$ was inconceivably large. Of course it was — and still is — $52!$. That’s easy enough to scribble down (or even surpass spectacularly) without understanding just how far we are from familiar territory.

In what flipping dimension is a square peg in a round hole just as good as a round peg in a square hole?

In what flipping dimension is a square peg in a round hole just as good as a round peg in a square hole?

Let’s start at the beginning.

My Plus magazine puzzle from March asks “Which gives a tighter fit: a square peg in a round hole or a round peg in a square hole?” By “tighter” we mean that a higher proportion of the hole is occupied by the peg.

What’s the intersection of the set of mathematicians and the set of popes?

Hint: a man who started life with one name but later adopted the one he is today remembered as.

Some infinities (and egos) are bigger than others

Here’s a tale of a rational (or irrational?) legal battle from the 1990s re: Cantor’s diagonal argument.

Cantor’s diagonal argument from 1891 was truly revolutionary: an ingenious way to demonstrate that no matter what proposed list of all real numbers (or, say, just those between $0$ and $1$) is put forth, it’s easy to find a number which is definitely missing from the list.1

In a nutshell, Cantor was the first to show that some infinities are bigger than others.

Cantor’s diagonalisation argument for the reals is watertight, and has proved to be a model of elegance and simplicity in the century plus that has passed since it first appeared.

That didn’t stop engineer William Dilworth publishing A correction in set theory, in which he refutes Cantor’s argument, in the Transactions of the Wisconsin Academy of Sciences in 1974.

  1. One has to pay close attention to realise that the same proof doesn’t also establish that the rationals are uncountable, bearing in mind that the Cantor pairing function shows that the rationals most certainly are countable. See http://en.wikipedia.org/wiki/Countable_set []

A student is artistic and loves poetry. Is it more likely she’s studying Chinese or Business?

Let’s suppose that:

  1. $60\%$ of students who study Chinese are artistic and love poetry.
  2. $20\%$ of students who study Business are artistic and love poetry, and
  3. Only about $1\%$ of students study Chinese, whereas about $15\%$ of students study Business.

Thus, out of every $1000$ students, there are $10$ studying Chinese, of whom $6$ are artistic and love poetry, and also there are $150$ studying Business, of whom $30$ are artistic and love poetry.

So if a student is artistic and loves poetry, it’s $5$ times more likely she’s studying Business than Chinese.

So much for preconceptions (and “correlation”).

Knit your mother’s sweater

Here is a clever display of the prime factorisation of the numbers 1-200 on a sweater, from knitter Sondra Eklund.

Each prime is represented (as a square) by its own colour, and luckily there’s an infinite number of both. Composites are represented by squares composed of collections of smaller squares or rectangles of appropriate colours.

She has arranged the natural numbers in columns of width ten. Interesting geometric and visual patterns emerge, and on the other side she’s knitted a version with eight to a column, which makes it easier to work in Octal.

As Sondra says, “One of the cool things about this sweater is that it works in any language and on any planet!!!”

Thanks to Ivars Peterson (on Twitter at‏ @mathtourist) for the pointer.

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