Does Everyone Like Bill? (This Could Make You Smarter)

by Mary Martialay on October 26, 2010

You may have heard Rensselaer Professor Selmer Bringsjord’s recent WAMC Academic Minute on “very serious games.” (If not, listen to it here.) Bringsjord, chair of the cognitive science department and a professor of computer science, discussed whether logic and memory “serious games” – gaining popularity on the Internet as a way to boost cognitive ability – make you smarter. The run of the mill “serious games” don’t do much more than exercise a particular skill, like memorizing words in sequence, but Bringsjord said, “very serious games” can make you smarter.

He then offered an example of a “very smart game,” and, if you’re like me, you might have tried your hand at solving it (and in my case, failed). Here then, courtesy of Selmer Bringsjord, is the question with its answer:

The question: “Suppose that everyone likes anyone who likes someone, and suppose in addition that Alvin likes Bill. Now, does everyone like Bill? And why? Everyone who can play the game well, which means their answer is yes, will be guaranteed to have gotten smarter in the process.”

Here is an informal proof of the answer:

The premises are:
(1) Everyone likes anyone who likes someone.
(2) Alvin likes Bill.

Since (2) Alvin likes Bill, Alvin likes someone, and since (1) everyone likes anyone who likes someone, it follows that everyone likes Alvin, and therefore Bill likes Alvin.

(Most people, Bringsjord said, get this far.)

Since Bill likes Alvin, it follows that Bill likes someone, and since (1) everyone likes anyone who likes someone, therefore (presto!) everyone likes Bill.

Bringsjord said this problem was conceived by the famous cognitive scientist Johnson-Laird, who sent it as a challenge to Bringsjord’s colleague Yingrui Yang, who in turn passed it on to Bringsjord.

If you thought that was interesting, don’t miss the chance to listen to Bringsjord and other Rensselaer professors’ enlightening thoughts on WAMC’s Academic Minute.