Today’s three questions are variations of classic logic puzzles that used a cheap Take That reference as clickbait. Here’s my excuse.
1. The fork in the road
You are travelling to a Take That concert when you arrive at a fork in the road. Only one of the paths leads to the concert. Two people are standing there who know the way. One is a knight and the other a knave. As is well-known in logic puzzles, a knight will always tell the truth and a knave will always lie. What single question can you ask one of these people that guarantees you will know the correct path to the concert, A or B?
a) Which is the correct path?
b) Shall I choose path A?
c) What path would the other person tell me to choose?
d) Shall I choose path B?
Solution: c)
The point of this question is that you must ask a question that guarantees the correct answer whether you ask a knight or a knave. For questions a), b) and d), a knave and a knight will give opposite answers. Thus, if you ask these questions you cannot guarantee that you will know the correct path. But if you ask c) both a knight and a knave will recommend the same path. If they say B, the correct path is A, and vice versa.
2. The sharpest nibs in the box
Robbie, Gary,and Jason work at a shop that sells pens, erasers, and pencils.
Robbie says: “Seven pens and five erasers cost the same as six pencils.”
Gary says: “Four pens and nine pencils cost the same as five erasers.”
Jason says: “Six pencils and three erasers cost the same as four pens.”
Only one is lying. Can you tell us who?
a) Robbie
b) Gary
c) Jason
Solution Robbie.
Let A equal the price of pens, B equal the price of erasers, and C equal the price of pencils.
Robbie is saying that 7A+5B=6C.
Gary is saying that 4A+9C=5B.
Jason is saying that 6C+3B=4A.
Combining Robbie and Gary’s statements, we get 11A=−3C. This is impossible.
Combining Gary and Jason’s statements, we get 15C=2B. This is possible.
Combining Jason and Robbie’s statements, we get 8B=−3A. This is impossible.
Either Robbie is lying, which means that Gary and Jason are telling the truth. (This fits). Or Robbie is not lying, which means that Gary must be lying (since if Gary was also telling the truth then the combination of Robbie and Gary’s statements would not be impossible) and Jason must also be lying (for the same reasons.) But since there can be only one liar, we can eliminate this scenario, and thus Robbie is the liar.
3. The inevitable logic puzzle about hats
A group of people are in a room. Everyone is wearing either an orange or an indigo birthday hat. Each person can see the other people’s hats but not their own.
One of them shouts, “If you can see at least 6 orange hats and at least 6 indigo hats, raise your glass!”
Exactly 12 people raise their glasses.
How many people are in the room?
Solution 18
We know that there are at least 6 people with orange hats and 6 people with indigo hats. But since no one can see their own hats, there must be more people.
If there are 7+ people with an orange hat and 7+ people with an indigo hat, then there are 14+ people who would raise their glasses - more than the actual 12. Therefore there must be 6 people with a hat of one colour (say, orange), and more than 6 with a hat of the other colour (indigo).
The people with an orange hat can see only 5 other orange hats, so they don’t raise their glasses. But all people with an indigo hat will raise their glasses. Therefore there are 12 people with indigo hats. Add the 6 with an orange hat, and we get a total of 18.
I hope you enjoyed today’s puzzles. I’ll be back in two weeks.
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.
Thanks to Cuemath for today’s puzzles. Cuemath is a maths education platform that gamifies learning and claims to have taught more than 200,000 students around the world. The three puzzles (without the Take That references) are used by Cuemath to engage its pupils with logic.
I’m the author of several books of maths and puzzles, and also the co-author with Ben Lyttleton of the children’s book series Football School. The latest in the Football School series is The Greatest Ever Quiz Book, out now!
I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.
