June 2024
Problem of the Month
Knights and Liars
Submitted by K. Sengupta, Calcutta, India
Six inhabitants A, B, C, D, E, and F of an island are discussing their respective ages.
Each one is either a Knight (Truth teller) or a Liar, over 18 but under 70 years of age, and the sum of their ages is 261.
A person 40 years old or older is a knight, unless his age is a multiple of 17, and then he is a liar.
A person under 40 is a liar, unless his age is a multiple of 13, and then he is a knight.
The six say:
A's Statement:
1. E is older than I am.
B's Statement:
1. A is 30 years younger than C.
C's Statement:
1. I am 51.
D's Statements:
1. C is 52.
2. I am not 29.
E's Statements:
1. A is a Liar.
2. F's age is not less than 40 years.
F's Statements:
1. D is a Liar.
2. B is 39.
Determine the ages of each of the six inhabitants.
Solution:
The ages are:
A: Liar - 38
B: Knight - 39
C: Liar - 68
D: Liar - 29
E: Knight - 26
F: Knight - 61
C must be a liar since he could not truthfully claim to be 51.
D must be a liar since he assigned the age of a Knight to Liar C.
This makes D 29 years of age.
F is a Knight since he correctly named D as a Liar.
This makes B 39 years of age and a Knight.
Since A is 30 years younger than Liar C, C cannot be under 57 years of age.
Therefore, C must be 68 (multiple of 17).
This makes A 38 years old and a Liar.
We now know that E is less than 38.
Since E correctly stated that A is a liar, he must be a Knight of age 26.
The second clue by E is overdefined since we now can subtract the sum of the five known ages from 261 to get 61 for F.