# You're reading: Posts By Card Colm

### The mathematics examinations faced by school leavers in the Republic of Ireland

This Friday, close to 13,000 students in the Republic of Ireland are set to take higher level maths in the Leaving Certificate, the state exams for 17-18 year old school leavers. That’s the highest number for two decades, and a 25% increase on last year’s all-time low of 10,400 who registered to sit the higher level exams. Typically, only about 80% of those show up for the higher level paper on the day–last year just 8,200 did–the rest playing safe and switching at the last minute to the ordinary level exams.

In 2011, a little over 55,000 Irish students overall, in a country with a population of 4.6 million, sat the Leaving Certificate in their final days of secondary education. This year, just under 54,000 school leavers are taking the Leaving, as it’s known. I hope they’ve studied hard, and wish them every success!

### 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”).