Neurobics

Caliban

You may have read the newspaper item where New Zealand was ranked ninth and last among the test nations by the renowned cricket authority Wisden. You may even have joined in discussions on the injustice of New Zealand being placed below Zimbabwe and Sri Lanka considering that we've beaten both. Why not relax with a purely recreational problem on the pastoral game.

Four men, Alan, Bernie, Colin and Darran made their debut in Shell Trophy cricket last season. All have other jobs; one is an accountant, another a farmer, one is an engineer and the last a shoe salesman. They play for Northern Districts, Canterbury, Auckland and Central Districts but not necessarily in that order. Alan and the engineer are both seam bowlers, Colin keeps wicket for Auckland, the shoe salesman is a wicket keeper too. Darran scored a century against Central Districts. Bernie did even better against Canterbury scoring 219 not out and against Central Districts he stumped four of the opposition. There is no farmer playing for Auckland. What is the name of the engineer and for which team does he play?

The renowned puzzlist Hubert Phillips, pen-name "Caliban", would have liked this problem. He published thousands like it for books, newspapers and magazines between the mid-thirties and the early sixties. Dover publications still has a number of his compilations on their list. All his conundrums require the potential solver to have a flair for seeing relationships, not necessarily numerical.

Here's one delightful problem which requires a clear head for figures but no great depth of mathematical knowledge:

Jane gave a tea party for her child relatives recently. They had great fun sorting themselves out at the table. There were nine children and their ages were consecutive. Jane wrote the numbers 1 to 9 on cards and gave one to each child (she also wrote one for herself with the number 9 on it). Then she told the children to sit at the table so their age was equal to the sum of the numbers on the cards of their immediate neighbours. The oldest child, Jake, sat at the end of the table opposite Jane and the youngest next but one on Jane's right, with Mary, who was 10 years old, sitting between them. How old was Jake? And where were all the guests sitting according to their ages and the numbers given to them?

Distracting Probabilities

1. On average, how many cards must be drawn from a well-shuffled standard pack of 52 cards to produce the first ace?

2. If a stick is broken in two at random, what is the average length of the smaller piece?

3. Samuel Pepys is said to have written to Isaac Newton to ask which of three events is more likely; that a person gets (a) at least one six when 6 dice are thrown, (b) at least 2 sixes when 12 dice are thrown, or (3) at least 3 sixes when 18 dice are thrown. What do you think?

4. I'm taking a gender-testing exam on day-old chicks for a new job. The examiner gives me four boxes each containing a chick which is either male or female. She asks me to test two, which I correctly prove to be male. She then says, "I meant to tell you, before you began, that there was at least one male chick. However, you know it now, without my telling you. Test again on a third chick." What she doesn't know is that I can't tell male and female chicks apart (but I need the job) and I've just guessed correctly so far. (a) What is the probability now that the next chick is male? (b) What would it have been if the examiner had not spoken?

Answers in next month's issue.

Russell Dear is a Mathematician living in Invercargill