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The Problem:

Ed, the eldest child of the Family family, met a new girl named Candy at the beginning of his senior year in high school. He really liked here, so he wanted her phone number. He knew the first three digits of the number were $492$ because the town was so small that everyone has the same telephone prefix. She wouldn't give him the rest of the number at first, but he persisted. Finally, at the beginning of the lunch period, she handed him a piece of paper with several numbers on it. "The last four digits of my phone number are on this page," she explained. $$\begin{align} &3257 \,\,\,\,\,\,\,\,\,\,\, 4682 \,\,\,\,\,\,\,\,\,\,\, 8824 \,\,\,\,\,\,\,\,\,\,\, 0626 \,\,\,\,\,\,\,\,\,\,\, 4608\\ &\,\,\,\,\,\,\,\,\,\,\,8624 \,\,\,\,\,\,\,\,\,\,\, 4632 \,\,\,\,\,\,\,\,\,\,\, 6428 \,\,\,\,\,\,\,\,\,\,\, 8604 \,\,\,\,\,\,\,\,\,\,\,8428\\ &8064 \,\,\,\,\,\,\,\,\,\,\, 3195 \,\,\,\,\,\,\,\,\,\,\, 8420 \,\,\,\,\,\,\,\,\,\,\, 4218 \,\,\,\,\,\,\,\,\,\,\, 8240\\ &\,\,\,\,\,\,\,\,\,\,\,7915 \,\,\,\,\,\,\,\,\,\,\, 6420 \,\,\,\,\,\,\,\,\,\,\,4602 \,\,\,\,\,\,\,\,\,\,\, 2628 \,\,\,\,\,\,\,\,\,\,\, 4178\\ &3281 \,\,\,\,\,\,\,\,\,\,\, 2804 \,\,\,\,\,\,\,\,\,\,\, 4002 \,\,\,\,\,\,\,\,\,\,\, 4826 \,\,\,\,\,\,\,\,\,\,\, 0846\\ &\,\,\,\,\,\,\,\,\,\,\,4718 \,\,\,\,\,\,\,\,\,\,\, 4680 \,\,\,\,\,\,\,\,\,\,\, 6402 \,\,\,\,\,\,\,\,\,\,\, 0428 \,\,\,\,\,\,\,\,\,\,\, 2406\\ \end{align}$$ Ed protested, "But there must be thirty numbers here!"

Candy laughed. "That's right. But I'll give you some clues. If you really want my number, you'll figure it our."

"Okay," he said. "Shoot. I'm ready,"

Candy listed her clues:

  1. All the digits are even.

  2. All the digits are different.

  3. The digit in the tens place is less than the other digits.

  4. The sum of the two larger digits is $10$ more than the sum of the two smaller digits.

    During lunch, Ed frantically worked away and tried to figure out which number was Candy's. He looked up with only a few minutes left in the lunch period. "I don't have enough information," he protested. The bell rand, so they walked out of the lunchroom together.

The problem gets cut off. The rest of the problem is supposed to say

She said, "Okay, I'll tell you the sum of all the digits." She whispered the information in his ear. "Thanks!" Ed said, because this gave him enough information to figure out the number. "I'll call you tonight." What was Candy's phone number?

What I have tried: I eliminated most of the possibilities with the first three clues. So, before clue 4, I was left with:

4608, 8624, 6428, 8604, 4602, 2804, 4826, 6402, 2406

After using Clue 4, I was left with four possibilities:

4608, 8604, 2804, 6402

The problem implies that there is only ONE possible answer for Candy's phone number, but it appears I have found four. Did I go astray somewhere or use a clue wrong?

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(1), 6402 doesn't work because 6+4=10 is not 10 more than 2+0=2. (2), you copied the problem wrong. Candy says "Okay, I'll tell you the sum of all the digits", which should be sufficient information to solve the problem. – Yun William Yu Feb 20 '12 at 3:17
This is a variant of the census-taker problem, also known as the Ages of the three children puzzle, member of a family of problems with similar meta-information. I'm pretty sure Smullyan has discussed them in at least one of his books. – Arturo Magidin Feb 20 '12 at 3:24
@Yun William Yu Thanks for the catch. – Joe Feb 20 '12 at 3:38
u cant figure the answer out. the sum is not included in the problem – user109472 Nov 17 '13 at 16:07
He should just call every number and ask "is Candy there?" – DanielV Nov 17 '13 at 16:19
up vote 2 down vote accepted

Well I can narrow your options down to three. $6402$ is not an option because $6+4=10$ and $2+0=2$, so the sum of the two larger digits is not $10$ greater than the sum of the two smaller digits. If Yun William Yu is correct about the problem being copied wrong, the answer must be $2804$ because the sum of the digits in $4608$ and the sum of the digits $8604$ are equal. Since the sum of the digits of $2804$ is different, it must be the answer.

share|cite|improve this answer
Ah, silly mistake - I simply glossed over that fact. Thank you for your help Yun and Izzy. – Joe Feb 20 '12 at 3:39

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