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Scenario [pdf]
Four Playful Mathematicians [Problem #2947]
Four playful mathematicians were eating lunch together at an outdoor cafe. They noticed a diamond-encrusted Rolex had been left behind at the
table next to them, so they quickly looked around for the person who had vacated the table. They saw the owner of the watch driving away. "Quick,
get the license plate!" urged one of the mathematicians, seeing that it was only a four-digit number.
When the car had disappeared around the corner, they put together these characteristics they'd noticed about the number:
1. The first said, "One of the digits is neither prime nor composite."
2. The second contributed, "The digit in the hundreds place is three times the digit in the thousands place."
3. The third added, "The digit in the ones place is four times greater than the digit in the tens place."
4. The fourth piped up with the last bit of information. "The smallest prime number is in the thousands place. We have enough information to help us locate
the owner of the watch."
Question: What number did the four playful mathematicians see on the license plate?
Comments and Sample Solutions
This puzzle was all about understanding numbers and logically working through the mathematicians's clues . The first step was really knowing place value, and
most submitters had no trouble demonstrating their mastery of this concept.
Another key to solving this puzzle was understanding prime and composite numbers, as well as knowing how to classify 0 and 1. This proved a bit more difficult
for some. If you struggled a bit here, you might find one of our Dr. Math pages, 1 and 0: Prime or Composite?, helpful.
The three solutions below give a good sampling of the submissions to this puzzle. In each case, the submitter has explained how the clues were used to
determine the digits of our mystery number. Notice how clues were combined to find some of the digits.
We encourage you to keep working on writing complete explanations of your work. When you use a math term, would it help to include a definition? If you didn't
know one, where did you find it? What else might help if you were teaching this puzzle to a friend?
Keep problem solving!
-Lisa & Kristina
Would you like to see your solution included here, or see your name on one of the lists below? Try this:
1. Write an explanation that would help another student understand how you solved it.
2. Use the "Check Answer" option.
3. If your answer is right,
write a comment about what you learned from this problem
or answer one of the other questions in our answer check.
If your answer isn't right,
write a revision and write a comment telling us what you fixed.
If your answer isn't right and you can't figure out how to revise,
write a comment telling us where you think you're stuck.
If you have done the things listed above, there's a much greater chance that your submission will be one of the ones that we read!
From: Kristina M , age 12 , West Rocks Middle School
My answer is 2614.
First, I figured out that the thousands place is a 2. I found this
out because the clue says "The smallest prime number is in the
thousands place." and two is the smallest prime number.
Second, I figured out that the hundreds place is a 6 because the clue
says "The digit in the hundreds place is three times the digit in the
thousands place." and 2x3=6
Next, I figured out that the tens place is a 1. I found this out
because the clue says "One of the digits is neither prime nor
composite." and a composite number is a number number bigger than one
that has more than two factors. A prime number is a numer that's only
factors are one and itself. This could also be 0, but if it was, then
there would be a 0 in the ones place also. Therefore, the answer is 1.
Finally, I figured out that the ones place is 4. I found this out
because the clue says "The digit in the ones place is four times the
digit in the tens place." The one has to be in the tens place to be
able to find out which number is in the ones place. Therefore, 1x4=4.
Therefore, the answer is 2614.
From: Ben L , age 9 , McDonogh School
The answer to the first question is 2614 and that is the license plate number.
First I figured out the fourth clue and the smallest
prime number is 2 so 2 is the answer to the fourth clue.
That means that 2 is in the thousands place.
Then I figured out the second clue and the answer to that
was 6 because 2 times 3 equals 6 so 6 is in the hundreds
place.
Then the answer to the first clue is 1 because 1 is neither
prime nor composite. The answer to the third clue is 4 because
the digit in the one's place is 4 times the digit in the
ten's place and 1 times 4 equals 4. I knew even if 0 is
not prime nor composite, it still doesn't fit the answer
because in the first clue it says that one of its digits
is neither prime nor composite and in the third clue it
says that the digit in the one's place is four times the
digit in the ten's place and if you take 0 times 4 you
get 0 and there can only be one number that is neither
prime nor composite.
Then if you take all the numbers that were the answers
to the clues in order I get 2614 and 2614 is your answer!!!
From: B , age 13 , Bowman Middle School
My final answer is two thousand six hundred fourteen.
First we realized that the license's plate number was a four digit
number. It was a thousand's place, hundred's place, ten's place, and
one's place. Then we learned that one of the digits is neither prime
nor composite. The digit in the hundred's place is three times the
digit in the thousand's place. Also the digit in the one's place is
four times the digit in the ten's place. The smallest prime number
is in the thousand's place.
Next we made a place value chart to figure out the answer to the
question. We made the thousand's place, hundred's place, ten's
place, and the one's place. After that we tried to put the clues
together, and get the right answer. We placed two in the thousand's
place. We got two because two is the smallest prime number. Next we
placed six in the hundred's place. We got six because it said
the digit in the hundred's place is three times the digit in the
thousand's place, so we multiplied two times three and got six. When
we got ready to find the digits for the tens and ones we knew it was
either two and eight or one and four because those were the only
digits that met the criteria for clue three. It wasn't two and eight
because it said that the digit in the thousand's place is the
smallest prime number. Therefore the two couldn't be in the ten's
place because the clue said the smallest prime number is in the
thousand's place. So we assume that no digit is used twice. For that
reason we could'nt use the two and eight. So thats why we placed the
one in the ten's place and the four in the one's place.
Finally we came to an answer of two thousand three hundred
fourteen. We got it because we used the method of the place value
chart. Thousands Hundreds
Tens
Ones
2
6
1
4
________ ________
____
____
We also used the prime and composite method,and also the
multiplication method. That's how we found the license's plate
number on the back of the car.
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