Several of my readers have asked me to explain how Doctor Cunningham arrives at the solution to the quiz posed to him by Inspector Jenkins upon their first meeting. The quiz appears in Chapter 4 at page 49.
To refresh your memory, here is the passage…
“Now a riddle. Listen carefully, as I won’t repeat myself. Jack and Jill must determine George’s birthday by month and day. They are both told it is one of the following: May 14, 15, 19; June 16,17,18; July 14, 16; August 14, 15, 17. One of the eleven days I just mentioned is George’s birthday. Jill is then separately told the specific number of the day that is George’s birthday, but nothing more. Jack is then separately told the specific month of George’s birthday, but nothing more. Jack next says, ‘I don’t know George’s birthday, but neither does Jill.’ Jill then says, “I didn’t know George’s birthday, but I know it now.’ Jack lastly says, ‘I know now his birthday too.’ What is George’s birthday?”
Doctor Cunningham sat back and smiled. “July 16.”
Now, you might note that Jenkins presents the quiz not to see if Cunningham can figure it out, but rather to see if the doctor will be truthful in admitting that he’d seen the quiz before. Earlier in their conversation Inspector Jenkins asks Doctor Cunningham if he received news of the man with the black box from the New York Times. No doubt the Inspector would have reviewed back issues of the Times upon arriving in New York in order to catch up on developments. He was familiar with the article Cunningham was referring to in their conversation. But Cunningham gives him more information to work with in his answer: “Yes, I read it every day, front to back…”
We later learn that Cunningham passes Inspector Jenkins’ test by admitting that he’d seen the quiz earlier in the Times and had already thought out the solution. “And how long do you think it took you to determine that answer?” asks the inspector. Cunningham responds truthfully, “The first time? The riddle was in the Times last weekend. It took me several minutes, I must confess. My Mary, Ms. Scott, solved it almost immediately…”
Based on that answer, Inspector Jenkins now knows three things: 1) Cunningham thinks laterally and can solve puzzles quickly, as this is indeed a hard puzzle, 2) he can be trusted to some extent, since he confessed he couldn’t solve the quiz instantly but needed several minutes to arrive at the solution, and 3) Ms. Scott must be incredibly sharp to solve such a difficult puzzle “almost immediately.”
But to the solution –
The quiz is best understood if the eleven possible dates given to Jack and Jill are written out so you can cross out dates and arrive at the solution by the process of elimination, like this:
May 14, 15, 19
June 16, 17, 18
July 14, 16
August 14, 15, 17
So, Jill has been told the number of the birthdate and Jack has been told the month. The first thing Jack says is that he doesn’t know the birthdate but neither does Jill. How can Jack say that with certainty? He can say that only if he knows already that the birth month is July or August, not May or June. That’s because May and June each have one number (19 for May and 18 for June) that appears only once in the entire set of dates, and only in those two months – if the number revealed to Jill were either the 18th or the 19th, then, she would know the birthdate without any further information. Another reason Jill can’t know the birthdate without more information is that every numbered date in July or August also occurs in one of the other named months, leaving her needing to know the month. Thus, Jack can say with certainty that Jill does not have enough information on her own to know the birthday only if May and June are off the table. So, cross out those six possible solutions.
May 14, 15, 19
June 16, 17, 18
July 14, 16
August 14, 15, 17
Jack’s saying that Jill cannot know the birthdate based on the number alone gives her a needed clue, though – if Jack can definitively say that Jill cannot know, she can deduce that the birthdate cannot be in May or June. (Again, if Jack’s separate revelation of the birth month had been either May or June, he would not be able to say that Jill couldn’t know the birthdate – she could know it because of the solitary occurrences of 18 and of 19. Either one of them, and thus the month, would be identifiable as the birthdate.) Jill knows, then, that in order for Jack to make such a statement, the birthday must fall in July or August. She then announces that she didn’t know the birthdate before, but with this clue Jack has provided, she does.
What clue does Jill’s pronouncement now provide to Jack, and thus to us? Clearly, Jill has been able to connect the number she was given with a month. Is her number 14? It couldn’t be, because she would still have to determine whether it was July 14, or August 14. So, 14 couldn’t have been Jill’s number, and now both you and Jack can cross off July 14 and August 14.
July 14, 16
August 14, 15, 17
That leaves July 16, August 15, and August 17 as the remaining possible answers, and Jill knows which of those single numbers is the correct one, allowing her to know the whole birthdate. Hm.
In saying that he now knows, too, Jack gives us the final clue to figure out what he and Jill know. How is that? A quick recap – 1) Jack knows the month, and Jill has deduced from Jack’s first statement that it cannot be either May or June; 2) Jill knows the number, and having deduced that the month is either July or August, Jill has figured out the birthdate; 3) Jack can rule out the 14th because Jill still would not have had enough information to choose between July and August. What further information has Jill given him in announcing that she now knows? Well, the number has to be 15, 16, or 17, obviously, and Jill, knowing what the number is, can now confidently know the whole birthdate. Remember – Jack already knows the month! If the birth month were August, he would still not know whole the birthdate because he would still have to determine whether it was on the 15th or the 17th. Jill took Jack’s clue (eliminating May and June as months), Jack took one clue from Jill (eliminating the number 14), and now you take the final clue from Jack’s pronouncement: Jack cannot know the whole birthdate if it’s in August, so you’re left deducing that the month must be July. Therefore, the only solution that allows both Jill and Jack to know the birthdate is July 16.