Earlier today, six puzzles were set involving people seated at round tables, each making statements about their ages relative to other diners. The solutions reveal the maximum and minimum possible number of truth tellers in each scenario.
Three people at a round table
Three people sit facing inwards at a round table, all of different ages. Each says: “I am younger than the person to my left.”
The greatest possible number of truth tellers is two. The oldest person must be lying, as no one is older than them. An example arrangement with two truth tellers has ages 30, 40, and 20 going clockwise around the table.
The smallest possible number of truth tellers is one. The youngest person must be telling the truth, since everyone else is older. An example with one truth teller has ages 40, 30, and 20 clockwise.
Seven people at a round table
Seven people of distinct ages sit at another round table. Each says: “I am younger than the person to my left, and I am younger than the person two places to my right.”
The greatest possible number of truth tellers is four. The two oldest people must be lying, ruling out six or seven. Five is also impossible: if five were telling the truth, the third oldest would be among them, forcing the three oldest into one of two arrangements, leaving no valid seat for the fourth oldest. An example with four truth tellers has ages 10, 20, 40, 70, 30, 50, and 60 clockwise.
The smallest possible number is one. The youngest must tell the truth. Arranging everyone in descending age order—70, 60, 50, 40, 30, 20, 10—ensures only the youngest tells the truth.
999 people at a round table
At a third table, 999 people of distinct ages each say: “I am younger than the person three places to my left, and older than the person three places to my right.”
The greatest possible number of truth tellers is 993. By colouring seats in a repeating gold, silver, and bronze pattern, each person’s statement concerns only those on same-coloured seats. Among the 333 gold-seated people, the youngest and oldest must lie, but all others can tell the truth if ages increase clockwise. Applying this to all three colours leaves six liars in total, so 999 minus 6 equals 993 truth tellers.
The smallest possible number is zero. Seating everyone with ages decreasing clockwise makes every statement false.
Daniel Griller contributed these puzzles. He authored the puzzle books Elastic Numbers and A Ring of Cats and Dogs, and Other Curious Puzzles. The next puzzle is due in two weeks.
