Hello guzzlers,
Last week some of the smartest sixth formers in the country sat Oxford University’s annual Mathematics Admissions Test. For today’s puzzle I’ve chosen one of its most fun questions from recent years.
It has five parts – successful applicants scored 4.73 on average out of the 5. So you need to get them all right!
(Logic puzzles like this one involving people wearing hats who can see only other people’s hats date from the 1930s. But logic puzzles as a genre date back at least to the time of Charlemagne. My new book Can You Solve My Problems? tells the history of puzzles in 125 brainteasers, and in it you’ll find lots more like this one to chew on.)
Now hold on to your hats!
Alice, Bob and Charlie are well-known expert logicians; they always tell the truth. They are sat in a row, as illustrated above. In each of the scenarios below, their father puts a red or blue hat on each of 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. All three of them are aware of this arrangement.
(i) 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?
(ii) 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?
(iii) 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?
(iv) 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?
(v) 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?
If you want to find the answers now - please submit your answers below. In each case write down only the colour of Charlie’s hat.
I’ll be back at 5pm GMT with a full explanation of the answers, and also the results of how well you all did. How many of you are Oxbridge material?! UPDATE: Results and explanations now up.
And, **PLEASE**, no spoilers until the answers are up! Don’t ruin everyone else’s fun. There are lots of other things to comment on, such as why the hats drawn in maths questions are always pointy ones, or what are the chances of having three expert logicians in the same family.
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I set a puzzle here every two weeks on a Monday. If you would like to suggest a puzzle email me.
My latest book is Can You Solve My Problems? A Casebook of Ingenious, Perplexing and Totally Satisfying Puzzles is a bumper buffet of delicious brain food from easily digestible morsels to fiendishly spicy dishes.
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Thanks to Dr Rebecca Cotton-Barratt at Oxford University for helping with this week’s puzzle.
