The logic you need to get into Oxford University
Find out if you have the sort of logic needed to get into Oxford University.
For his latest fortnightly puzzle in The Guardian, British writer Alex Bellos has posed a question from the 2012 Oxford mathematics admissions test.
It's all about the colour of hats being worn by three expert logicians sitting in a row.
In each of five scenarios, their father puts either red or blue hats on their heads. Alice can see Bob's and Charlie's hats, but not her own. Bob can see only Charlie's hat. Charlie can see none of the hats.
1. Their father puts a hat on each of their heads and says: "Each of your hats is either red or blue. At least one of you has a red hat." Alice then says "I know the colour of my hat." What colour is each person's hat?
2. Their father puts a new hat on each of their heads and again says: "Each of your hats is either red or blue. At least one of you has a red hat." Alice then says "I don't know the colour of my hat." Bob then says "I don't know the colour of my hat." What colour is Charlie's hat?
3. Their father puts a new hat on each of their heads and says: "Each of your hats is either red or blue. At least one of you has a red hat, and at least one of you has a blue hat." Alice says "I know the colour of my hat." Bob then says "Mine is red." What colour is each person's hat?
4. Their father puts a new hat on each of their heads and says: "Each of your hats is either red or blue. At least one of you has a red hat, and at least one of you has a blue hat." Alice then says "I don't know the colour of my hat." Bob then says "My hat is red". What colour is Charlie's hat?
5. Their father puts a new hat on each of their heads and says: "Each of your hats is either red or blue. Two of you who are seated adjacently both have red hats." Alice then says "I don't know the colour of my hat." What colour is Charlie's hat?
Bellos asked readers to submit only the colour of Charlie's hat.
He said the test was set by Oxford for applicants to computer science, mathematics & computer science, and computer science & philosophy.
Successful applicants scored 4.73 on average out of five, so it was a good idea to get all the answers right.
Of those who tried to answer the puzzle in The Guardian, about 80 per cent got each question right, Bellos said.
Logic puzzles about people who are wearing hats but can only see other people's hats date to the 1930s.
SPOILER ALERT:
The answers follow, so don't read any further unless you're sure you're ready.
1. Alice's hat is red, and the others are blue. It must be that Alice can see that neither of the others has a red hat, so can deduce the colour of her own. Charlie's hat is blue. 83.2 per cent of those who responded to The Guardian column got the colour of Charlie's hat correct.
2. Alice must be able to see a red hat, or would be able to deduce the colour of her own hat. Likewise, Bob must be able to see a red hat, or would be able to deduce the colour of his own hat (given Alice's answer). Hence Charlie's hat is red - 74.1 per cent got it right.
3. Alice must be able to see two hats of the same colour in order to deduce the colour of her hat. Bob knows this, and so deduces his hat is the same colour as Charlie's. Hence Alice's hat is blue, and Bob's and Charlie's are red - 78.8 per cent were correct.
4. Alice must be able to see two hats of opposite colours, or else she would be able to deduce her own hat colour. Bob knows this, so deduces his hat is a different colour from Charlie's. Hence Charlie's hat is blue - 89.2 per cent were right.
5. If Bob and Charlie had different colour hats, Alice would know that she and Bob both had red hats. Therefore Bob and Charlie both have red hats - 82.7 per cent right.
First appeared on Stuff.co.nz.
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