Learn / Library / Brainteasers

Parity of the others

BrainteasersMedium

A control room has $11$ monitoring units. Each unit is either calibrated (a truth-teller, always reporting truly) or faulty (a liar, always reporting falsely). Every unit reports: "The number of faulty units among the other $10$ units is even." In how many different ways can the states be assigned to the $11$ units so that all $11$ reports are consistent?

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Compare the number of liars a truth-teller sees with the number a liar sees, and see what it says about the total.

Solution

Let $L$ be the total number of liars. A truth-teller sees $L$ liars among the others, and the statement is true, so $L$ is even.
A liar sees $L - 1$ liars among the others, and the statement is false, so $L - 1$ is odd, which again means that $L$ is even.
So an assignment is consistent exactly when the total number of liars is even, whatever the types of individual people.
Exactly half of the $2^{11}$ assignments have an even number of liars, which is $2^{10} = 1024$ .

Answer: $1024$

Next question Back to the library