De Econometrist neemt een statistische kijk op de wereld.
We are marooned on an island that has the following curious property: Everyone over a certain age lies all the time. More specifically, there is an age limit L — a positive integer — and all islanders who are younger than age L only tell the truth, while islanders who are at least L years old only tell lies.
We are greeted by five islanders who make the following statements:
A: “B is more than 20 years old.”
B: “C is more than 18 years old.”
C: “D is less than 22 years old.”
D: “E is not 17 years old.”
E: “A is more than 21 years old.”
A: “D is more than 16 years old.”
B: “E is less than 20 years old.”
C: “A is 19 years old.”
D: “B is 20 years old.”
E: “C is less than 18 years old.”
What is L? And what did we just learn about the ages of the islanders?
L was 19 years old. Islander A was 19 years old, B was 20, C was 18, D was at most 16 and E was at least 20.
The key is to find out who the liars are. Notice first that there are a few direct contradictions in the islanders’ statements: A and D contradict each other over B’s age, B and E contradict each other over C’s age, and E and C contradict each other over A’s age. So at least one member of the three pairs (A, D), (B, E) and (E, C) must be a liar. We also know that D and B aren’t both liars — that would make E both exactly 17 years old and more than 20 years old, which is impossible.
Given those facts, we have a few possibilities we can test. Let’s guess that A, B and E are liars and that C and D are truth-tellers, then check the consistency of their statements. There are two statements about each islander, which we can translate into true facts given our guess about the truthfulness of each of their speakers.
The statements about A check out: A is less than or equal to 21 and A is 19. That means A is 19.
The statements about B also check out: B is less than or equal to 20 and B is 20. That means B is 20.
The same goes for C: C is less than or equal to 18 and C is greater than or equal to 18. That means C is 18.
And D: D is less than 22 and D is less than or equal to 16. That means D is at most 16 years old.
And finally E: E is not 17 and E is greater than or equal to 20. That means E is at least 20.
Because all these statements check out, that means our guess worked! Since A, B and E are liars and C and D tell the truth, that means both of our truth-tellers are at most 18 and all of our liars are 19 or older. Therefore, L must equal 19.
And we’re done! Now get back to studying, you can do it!
Dit artikel is geschreven door Sjors Keet