Did you solve it? The treacherous puzzle of the round table
The solutions to today’s puzzles
Earlier today I set you these six puzzles. Here they are again with solutions.
The round table
Three people sit at a round table, facing inwards, towards the centre of the table. No two of them are the same age.
Each of them says “I am younger than the person to my left.”
1. What is the greatest possible number of people telling the truth?
2. What is the smallest possible number of people telling the truth?
Solution
1. The oldest person must be lying, so the maximum is 2. Here is an example with 2 people telling the truth:
30-40-20 (Ages as you go clockwise around the table)
The answer is two.
2. The youngest person must be telling the truth, so the minimum is 1. Here is an example with 1 person telling the truth
40-30-20
The answer is one.
Seven people sit at a different round table, facing inwards, towards the centre of the table. No two of them are the same age.
Each of them says “I am younger than the person to my left, and I am younger than the person two places to my right”.
3. At this table, what is the greatest possible number of people telling the truth?
4. At this table, what is the smallest possible number of people telling the truth?
Solution
3. The two oldest people must be lying, so the answer cannot be six or seven.
It turns out that the answer cannot be five either. If five were telling the truth, the third oldest would be one of them, and so the three oldest people would have to be arranged in one of the two cases below (suppose the ages are 10, 20, 30, 40, 50, 60, 70).
* * * 60 * 50 70 (Ages as you go clockwise around the table)
* * * 70 * 50 60
But now there is nowhere for the next oldest (the 40-year-old) to sit and be telling the truth.
It is, however, possible for four people to be telling the truth:
10 20 40 70 30 50 60
The answer is four.
4. The youngest person must be telling the truth, so the answer cannot be zero. If you order everyone in descending order of ages, no one is telling the truth apart from the youngest:
70 60 50 40 30 20 10
The answer is one.
999 people sit at a third round table, facing inwards, towards the centre of the table. No two of them are the same age.
Each of them says “I am younger than the person three places to my left, and older than the person three places to my right”.
5. At this table, what is the greatest possible number of people telling the truth?
6. At this table, what is the smallest possible number of people telling the truth?
Solution
5. We might think about this by colouring the seats gold, silver and bronze in a repeating pattern, like so (in the illustration there are only 21 people, but the idea works for 999 people as well).
It can now be seen that each person makes a statement about people sitting on seats with the same colour as them; in particular, the closest people on either side with the same colour seat.
Consider those 333 people sitting in gold. The youngest and oldest of these must be lying, so there cannot be 333 or 332 gold truthtellers. But, as the example below helps to illustrate, it is possible for every other gold person to be telling the truth, by seating them with age increasing as we move clockwise around the table. Applying the same reasoning to those seated on silver or bronze, we can have all but six people telling the truth.
The same idea with 999 people has 999-6=993 telling the truth.
The answer is 993.
6. If the people are sat with age decreasing as we move clockwise around the table, then everybody is lying.
The answer is 0.
Thanks to Daniel Griller for today’s puzzles. He is the author of the excellent puzzle books Elastic Numbers and A Ring of Cats and Dogs, and Other Curious Puzzles.
I’ll be back in two weeks.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.