BdMO National 2015 Secondary Problem 1. A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement: Witness One said exactly one of the four witnesses is a liar. Witness Two said exactly two of the four witnesses is a liar. Witness Three said exactly three of the four witnesses is a liar. Witness Four said exactly four of the four witnesses is a liar. Assume that each of the statements is either true or false.How many of the winesses are liars?
Problem
Source: Bangladesh National Mathematical Olyiad 2015
Tags: combinatorics unsolved, BdMO-2018, logic, logical thinking, contests, combinatorics
programjames1
26.02.2019 20:46
Note that only one of them can be correct, therefore there are $4-1=3$ liars, so Witness Three is correct.
ythomashu
26.02.2019 20:46
All of the witness' statements are contradictory, so only one is telling the truth. So the answer is 3
Olympus_mountaineer
26.02.2019 20:47
Isn't it a paradox?According the question all are liars or all are honest.
ythomashu
26.02.2019 20:47
Olympus_mountaineer wrote: Isn't it a paradox? no
Olympus_mountaineer
26.02.2019 20:49
But then $1$ is speaking truth and $3$ are telling lie.
ythomashu
26.02.2019 20:50
Olympus_mountaineer wrote: But then $1$ is speaking truth and $3$ are telling lie. which is what witness Three said
ubermensch
26.02.2019 20:58
Obviously exactly 1 of them can be saying the truth- 2 witnesses can't both be correct at the same time. Thus answer is 1.
ythomashu
26.02.2019 22:01
ubermensch wrote: Obviously exactly 1 of them can be saying the truth- 2 witnesses can't both be correct at the same time. Thus answer is 1. the question asks how many are lying not how many are telling the truth
ATGY
10.10.2020 13:26
These witnesses are contradicting themselves, which tells us that only one can be telling the truth, so 3 people are lying. This shows that only Witness 3 is telling the truth, so our answer is $\boxed{3}$.