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Earlier today I set you six ‘bat and ball’ puzzles, meaning puzzles that require you to overrule a wrong ‘gut’ answer. (Click here to read the original bat and ball puzzle.)
For each of the puzzles, I have included the gut answer, the correct answer and also the percentage of readers who got it right. (Readers were given four possible answers and invited to make a choice.)
In summary – a game of two halves. Most people aced questions 1,2 and 4. But most of you stumbled with 3, 5 and 6.
1. If it takes five elves five minutes to wrap five presents, how long would it take fifty elves to wrap fifty presents?
Gut answer: 50 minutes
Correct answer: 5 minutes
Percentage who got the right answer: 92.2 per cent.
2. A mother and daughter are fifty years old in total. The mother is 20 years older than the daughter. How old is the daughter?
Gut answer: 30
Correct answer: 15
Percentage who got the right answer: 95 per cent.
3. How many people is three trios of triplets thrice?
Gut answer: 3 x 3 x 3 x 3 = 81
Correct answer: 3 x 3 x 3 = 27
Percentage who got the right answer: 25 per cent (most popular answer was 81, with 72 per cent)
Comment: In my opinion, a ‘trio of triplets’ is 3 people, although it looks like many of you assumed it is 9 people. Here’s my reasoning: a triplet is one person. Thus a trio is three of them. If I had meant to refer to 9 people I would have set ‘a trio of sets of triplets.’ Likewise, a pair of twins is 2 people, not 4.
4. If you flip a coin 3 times, what is the probability of flipping at least 2 Heads?
Gut answer: something less than 50 per cent
Correct answer: 50 per cent
Percentage who got the right answer: 71 per cent.
Comment: There are 8 possibilities of three coin flips: HHH, HHT, HTH, THH, HTT, TTH, THT and TTT. In 4 of them there are at least two Heads.
5. This morning 100kg of marrows were put in the sun. They were 99% water, but through the day they lost moisture to evaporation and now they are only 98% water. How much do they weigh now?
Gut answer: something close to 100kg.
Correct answer: 50kg
Percentage who got the right answer: 38 per cent (most popular answer was 99kg, with 48 per cent)
Comment: This is a wonderful puzzle because it is so counter-intuitive.
When the marrows are 99 per cent water, the ratio of non-water to water is 1 to 99. When the marrows are 98 per cent water, the ratio of non-water to water is 2 to 98, or 1 in 49. Since, after the evaporation, the non-water is unchanged at 1kg, the total weight must be 50kg.
6. If I gave you £100 today and you paid me back £10 every month for a year, what is the annual interest rate on your loan?
Gut answer: 20 per cent.
Correct answer: a number between 21 and 119 per cent (it’s actually about 41.3 per cent, which is hard to calculate. Since the point of the puzzle was not to do a complicated calculation, but to realise that the question is more sophisticated than at first glance, I gave a wide range as one of the four options.)
Percentage who got the right answer: 18.2 per cent (most popular answer was 20 per cent, with 63 per cent)
Comment: This is the question that Shane Frederick asked Harvard students and they all got it wrong. (As I did too when Tim Harford first showed it to me .) The error comes in misunderstanding how interest works. When you borrow money, you only pay interest on money that you owe. If I borrowed £100 and, at the end of the year, paid £120 back in one go, then the interest rate would indeed be 20 per cent. Yet the question asks a more subtle question, since the amount owed changes through the year. You are effectively borrowing £100 for a month, then £90 for a month, and then £80 for a month, and so on, until after ten months you are borrowing nothing at all. A back of the envelope calculation makes this roughly equivalent to borrowing about £50 (the average of all the monthly amounts you owe) for the year. Since you are paying back £20 more than you borrowed, the interest is gong to be about 20/50 = 40 per cent. If anyone wants to calculate the precise sum below the line, please do!
You might complain that this question assumes a thorough grasp of how interest works. Fair comment. But the whole point of the exercise is to realise that it is important to stop and think. And when it comes to financial matters, this issue is particularly pertinent. Says Tim Harford: “Companies love expressing loans this way to disguise the true interest rate – which is why regulators have to force them to report interest payments in a consistent way.”
I hope you enjoyed today’s puzzles. I had a lot of fun! Please carry on the discussion below the line.
Thanks again to Tim Harford, whose excellent book How to Make The World Add Up is out now, and to Shane Frederick.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
My new book The Language Lover’s Puzzle Book celebrates the world of words and language, and features problems involving ancient, modern and invented languages, curious alphabets and writing systems, slang, A.I., and much more. David Crystal calls it: “The perfect companion for anyone who loves puzzles and languages.” Who am I to disagree! The book is out on November 5 but you can preorder at the Guardian Bookshop.
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