Whoever you are, it’s important to keep your mind sharp. So when: your essays have just been handed in, you’ve come home for Christmas, and you’re suddenly surrounded by food and booze, nothing will help keep your brain whirring like a logic puzzle! Here’s three of my favourites from 2018.
The ‘tennis player with too many rackets’ puzzle
Mathematician Gihan Marasingha presented this puzzle on Radio 4 on September 14th 2018:
‘Each day for a week, a tennis player is to receive a number of tennis rackets as a gift. She knows that she’ll get a different number of rackets on each of the seven days and is told exactly how many she’ll receive in total. Using only this information, she deduces that on at least one day she’ll be presented with at least ten rackets. What is the minimum total number of rackets she could receive for her to know this?’
‘She’ll be given at least 43 rackets. Any fewer and she couldn’t guarantee she would receive ten or more on any one day. For example, if the total was 42, she could receive 9 + 8 + 7 + 6 + 5 + 4 + 3 (= 42) – and not necessarily get ten in one go.’
The ‘Brexit gets even more confusing’ puzzle
On September 28th 2018 this puzzle was featured on theConversation.com
‘At a party, the host arranges the guests in a circle and sticks a post-it note onto each of their foreheads. Each guest can see what’s written on everyone else’s post-it, but not their own.
“You are all Brexit negotiators,” says the host. “Some of you are representing the UK and some the EU, but all of you are negotiating for one or the other and there will be at least one negotiator for each side. The side you are representing is now written on your forehead. UK or EU – does anyone know which they are representing?”
Everyone shakes their head and says they don’t know.
“Does anyone know now?” asks the host.
Again, everyone shakes their head and says “I don’t know”.
But the host persists: “Now does anyone know?”
This cycle continues, to the perplexity of all and the annoyance of some. On the host’s sixth time of asking, the guests, by now wishing they’d gone elsewhere for the evening, again answer, for the sixth time, that they don’t know. But as soon as they’ve done so, some of the guests say, truthfully, “Now I know! I’m EU!”
Given the above facts, how many EU negotiators are there at the party? And what is the minimum number of guests? (Assume that all the guests have reasoned as well as they could have.)’
‘There are seven EU negotiators, and at least 14 guests.
So how can we know this? At the beginning of the game, the guests know only that there will be at least one EU and one UK negotiator. Suppose there was just one EU negotiator. She would see no other EUs, correctly infer that she was the sole EU, and answer “I know”.
All the guests can reason like this and know that all the others can do so, too. So when everyone says that they don’t know which side they are representing, they can infer that there isn’t exactly one EU. They then know that there are at least two, and that all the others know this as well.
The reasoning can then be repeated. If there were exactly two EUs, the EU negotiators would each see only one other, infer that they were the other EU, and answer that they know. So when all the guests again answer that they don’t know, all the guests can infer that there are at least three EUs, and that everyone else knows this as well.
Each time all the guests say “I don’t know”, they can rule out there being the next number of EUs. So, after the sixth round, everyone knows there are at least seven EUs (and that everyone knows this). Then the EUs, seeing only six other EUs, infer that they, too, are EUs, making seven in total. At that point, they can say, “Now I know!” (In this reasoning, it’s important that each guest knows that the others are reasoning in the same way as them. When a guest knows that there are at least three EUs, for example, she must know that all the other guests know it as well.) The reasoning also works for determining the number of UK negotiators, of course.’
Think those were easy? (No?) Well, try this next one on for size anyway!
‘The Hardest Logic Puzzle Ever’ – You better believe it.