Question

A says “ C is the knave,” B says, “A is the knight,” and C says “I am the spy.”

Short answer

The unique solution is, A is a knight, B is the spy and C is the knave.

Step-by-step solution

Introduction

In response to such directions,truth-tellersfrequently supply more comprehensive information in order to show their innocence. Liars, on the other hand, want to keep their mistakes hidden.

Explanation

From the given data,

A says, “C is the knave”, B says, “ A is the knight” and C says, “I am the spy”.

TheC statement is,

C cannot be a knight because a knight never lies about his identity.

Therefore,

C is either a knave or spy.

Simplification

$\neg \mathrm{Knight}\left(\mathrm{C}\right)\wedge \mathrm{Knave}\left(\mathrm{C}\right)\wedge \mathrm{Spy}\left(\mathrm{C}\right)\iff \mathrm{T}$

Then,

$\neg \mathrm{Knave}\left(\mathrm{C}\right)\iff \mathrm{T}$

$\mathrm{Knave}\left(\mathrm{C}\right)\iff \mathrm{F}$

Thus,

A is a knave.

Similarly,

$\neg \mathrm{Knight}\left(\mathrm{A}\right)\wedge \mathrm{Knave}\left(\mathrm{A}\right)\wedge \mathrm{Spy}\left(\mathrm{A}\right)\iff \mathrm{T}$

Then,

$\neg \mathrm{Knight}\left(\mathrm{A}\right)\iff \mathrm{T}$

Thus,

B must be a knight.

Therefore,

A is a knight, B is the spy and C is the knave.