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Journalism of Courage
Thank God for divine timing. The World Cup with its round-robin format provides for some delightful mathematical entertainment, which in turn provides the ideal launchpad for this mathematical blog.
But what exactly is a mathematical blog? It’s a forum where you will provide your brilliant answers for some simple puzzles you will find here week after week. Do send detailed replies to the email address at the bottom; we can debate and discuss them on this blog later. Not all the puzzles will be mathematical, for I also love logic puzzles and frequently create crosswords too, but maths will indeed be the pillar we will build our foundation on. Hence the World Cup kickoff.
Easy question: How many teams can win exactly three matches in a pool of seven?
Easy answer: all seven teams, theoretically.
This is possible if you “loop” the teams along the circumference of a circle — say A, B, C, D, E, F, G and back to A — and assume that any team defeats the three teams immediately clockwise but loses to the three teams immediately counterclockwise. Or vice versa.
A slightly tougher question is: if three wins are not necessarily enough, how many victories are a guarantee. Six means a no-contest, for only one team can win exactly six matches.
How many can win five matches each? Four different teams cannot do that, for that would give them 20 victories out of 18 matches (in any set of seven teams, a subset of four plays a total of 18 matches).
It is possible, however, for three teams to win five matches each if you “loop” them, i.e. A defeats B defeats C defeats A. Hence five victories are always enough to finish among the top four in a pool of seven teams.
That brings us to the moot question. If five victories are always enough but three are not necessarily so, what about four victories? Not enough, according to my “pool and loop” theory. It is possible for five teams to finish with exactly four victories each.
The following hypothetical points table shows how:
It’s immediately clear that the only team West Indies defeated was South Africa, who lost every match. But there 24 possible combinations for the other five teams to win four matches each. With a few riders, let’s narrow it down to a uniquely singular possibility.
(a) The two teams (other than WI and SA) that lost to Ireland both defeated India.
(b) The two teams that defeated Zimbabwe both lost to Pakistan.
(c) Zimbabwe lost to Ireland.
Puzzle#1A: Leaving out West Indies and South Africa, can you fill up the following results?
India defeated ____ and ____ but lost to ______ and ________.
Pakistan defeated ____ and ____ but lost to ______ and ________.
Ireland defeated ____ and ____ but lost to ______ and ________.
UAE defeated ____ and ____ but lost to ______ and ________.
Zimbabwe defeated ____ and ____ but lost to ______ and ________.
The missing money
Here’s something daft to end with. Three young working women rent a posh little flat at an upmarket location in a busy Indian metropolis. How much, they ask the landlord, who replies, somewhat absentmindedly, that it’s Rs 30,000 a month.
Turns out he’s made a mistake. After they have paid up for the first month and left, the owner of multiple flats realises he had meant to charge only Rs 25,000 for this one. Seeking to make amends at once, he summons his young son, hands him Rs 5,000 and sends him rushing after the three women with the refund.
Now the landlord, as you can see, belongs to that old-fashioned honest breed so rare to find today. But his son, alas, is the stuff a Jan Lokpal’s dreams are made of. Here’s how his devious mind works: When the three women agreed to pay Rs 30,000, they must have decided to split it @ Rs 10,000. Now there’s no way they can split Rs 5,000 three ways, so why not make it easy for them by refunding only Rs 3,000. He knows he will be found out by next month, of course, but the Rs 2,000 he will pocket now will be worth the inevitable thrashing from his father.
Everything goes to his plan. The women get back Rs 3,000, who share it @ Rs 1,000. That means they effectively paid Rs 10,000 – Rs 1,000 = Rs 9,000 apiece, which works out to Rs 27,000 for the three of them. The boy keeps Rs 2,000. That’s Rs 27,000 + Rs 2,000 = Rs 29,000. But we had begun with Rs 30,000, hadn’t we?
Puzzle#1B: What happened to the missing Rs 1,000? It’s a silly one, I know, but do write all the same; you have my email.
Please mail your replies to kabir.firaque@expressindia.com.
