Logic: Prisoners

May 3, 2001

Problem

Of three prisoners in a certain jail, one had normal vision, the second had only one eye, and the third was totally blind. All were of normal intelligence. The jailer told the prisoners that from three white hats and two red hats he would select three and put them on the prisoners’ heads. Each was prevented from seeing what colour was on his own head. They were brought together, and the jailer offered freedom to the prisoner with normal vision if he could tell what colour hat was on his own head. The prisoner said that he couldn’t tell. Next the jailer offered freedom to the prisoner with only one eye if he could tell what colour hat was on his own head. The second prisoner confessed that he couldn’t tell. The jailer did not bother making the offer to the blind prisoner, but agreed to extend the same terms to him when he made the request. The blind prisoner then smiled broadly and said:

“I do not need to have my sight; From what my friends have said, I clearly see my hat is !”

Solution. Firstly, call the prisoner with normal vision: P1; call the prisoner with one eye: P2; and call the prisoner who is blind: P3.

(i) There are 3 white hats and 2 red hats. (given)

(ii) If P2 and P3 both have red hats then P1 has a white hat. (follows from 1.) (iii) P1 does not know what colour his hat is. (given) (iv) P2 and P3 do not both have red hats. (follows from 2. and 3.)

(v) If P3 has a red hat then P2 has a white hat. (follows from 1. and 4.)

(vi) P2 does not know what colour his hat is. (given) (vii) P3 has a white hat. (follows from 1., 5., and 6.) Thus the missing word in the verse is: white.