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Cheryl’s Birthday

“Hi! I’m Albert, and this is Bernard.”  Cheryl looked up at the cheerful-looking couple of boys who had approached her desk. “We didn’t recognize you from last year.”

“Hi! I’m Cheryl, and I just moved here.” she replied. She was slightly shy, and so appreciated the introduction.

“What are you doing?” inquired Bernard.  Cheryl held up the book of brain-teasers for Bernard to investigate.

“I got this book of puzzles for my birthday. They’re really fun!” piped Cheryl, cheerfully.

“When’s your birthday?” Albert asked.

Cheryl opened her mouth to answer, but stopped and thought for a second. “Why don’t you try to figure it out?”

Albert and Bernard looked at each other, bewildered. “How are we supposed to do that?” asked Albert.

Cheryl was already hard at work writing down a list of dates.  Then she looked up and handed the list to the confused boys.

May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17

“My birthday is one of these,” she said. “Now, I’m going to tell Albert the month of my birthday, and Bernard the day of my birthday.  The catch is that you can’t tell each other  what you know. Get it?”

The boys nodded uncertainly, still trying to figure out what Cheryl was getting to.  Cheryl stood up and whispered into Albert’s ear, and then walked around to Bernard and whispered in his ear.

“Can you figure it out now?” she asked the two.

“No!” exclaimed Albert. “How on Earth could I?”

“You can’t, yet.”  Cheryl could see that these two were going to need a little bit of help. She looked Albert in the eye and asked, “Does Bernard know the answer?”

Bernard started to say something, but Cheryl promptly hushed him. This question was intended for Albert.  Albert screwed up his face, thinking as hard as he could: The only way for Bernard to know the answer would be for him to have been told a day that only occurred in one month, so that the day implies the month and thus reveals the answer.  I know the month, and it doesn’t have any unique days, therefore Bernard can’t know the answer.

“No,” said Albert haltingly. “I don’t think so.”

“Very good!” Cheryl beamed. “Now, Bernard, what do you know?”

Bernard closed his eyes, took a deep breath, and thought. If Albert knows that I don’t know the date, that means that none of the months with unique dates are possible.  That’s helpful: June is the only month with 18, so it can’t be right. Also, May is the only month with a 19, so it can’t be right. That leaves July and August. Oh, since I know the day, there’s only one possibility left!

Bernard grinned. “I know the answer.”

Cheryl hopped forward excitedly. “Here, whisper it to me!” She leaned over as Bernard whispered into her ear.  She pulled back and held up her hand for a high five, which Bernard happily obliged.

Albert, feeling a bit left out, screwed up his face again in deep thought. If Bernard could figure out the month based on knowing the day and knowing that I knew he couldn’t know in the beginning, then Bernard must have figured out that May and June were wrong. Since he figured it out, that means that any shared dates in July and August must also be wrong, since he knows the date and wouldn’t be able to tell the difference.  That eliminates the 14s, leaving July 16, August 15, and August 17.

“Well, now I know, too!” said Albert, half-triumphant and half-whining. Clearly, he was jealous of having been beaten by Bernard, despite being aware of the fact that Bernard’s success enabled his own.  Cheryl leaned forward to Albert to hear his answer (pro forma keeping it a secret).  She straightened up, smiling again, and offered Albert the celebratory high-five.

With that, the two boys were greeted by some other friends, and excused themselves to wander off and meet with them. Meanwhile, Cheryl’s desk-neighbor Daisy was totally confused. As Cheryl sat down, Daisy asked “How did they figure that out?”

Cheryl looked up for just a second, and then replied “You can figure it out too!”

Daisy, not feeling at all confident, asked “Can I have a hint?”

Cheryl nodded obligingly, but said nothing. Instead, she took a piece of paper, drew a table on it, and handed it to Daisy.  Daisy took a long, considering look and the helpful-but-uninformative data structure.

May June July August
14 14
15 15
16 16
17 17

After a moment, Cheryl said “What do you know?”

What do I know? thought Daisy. I know that this puzzle is crazy. Never-the-less, Daisy was now resigned to working through the logic as best as she could. If Albert knew the Month, and he knew that Bernard couldn’t know the date, the only way that could be true is if all of the days in the correct month were in another month. That means any month with a day that isn’t in any other month can’t be the correct month.  So I can stop worrying about… May and June. She drew a vertical line through those columns. OK, now, that was enough information for Bernard to figure out the answer, which means that the answer couldn’t be a date that July and August shared. She crossed out the row with the 14s in it, leaving only July 16th and August 15 and 17. Now, knowing that Bernard could figure it out from that was enough information for Albert to figure it out.  If Albert had known that the Month was August, then he wouldn’t have been able to deduce the date. Since August has two possible dates left, that meant the answer must be…

“July 16th!” Daisy exclaimed happily. Cheryl grinned and offered the now-ceremonial high-five.

Just then the bell rang. “OK, everyone. Take your seats!” said the teacher in Singaporean Mandarin. “Welcome to 5th grade.”