palette of
problems
David Rock and Joel Amidon
1. What is the greatest two-digit prime number whose two
digits are prime and that also sum to a prime number?
2. A bag contains 60 colored chips: 18 green, 15 red,
9 blue, 8 yellow, 6 black, and 4 white. If Louise is
picking chips randomly from a bag, what is the least
number of chips she must pick to guarantee that she
has at least 7 chips of the same color?
3. How many two-digit numbers exist such that both digits
in the number are square numbers? For this question, we
consider zero a square number because 0 × 0 = 0. Two
such possible numbers are 10 and 41 (both 4 and 1 are
squares).
4. What is the largest number that will always be a factor
of the sum of any four consecutive positive odd numbers?
5. Katarina missed an exam that she had to make up.
The class average without her score was 73 percent and
Katarina’s score was 84 percent. To determine the new
class average, what additional information is necessary?
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6. A fair six-sided die is tossed. After
it lands, the bottom face cannot be
seen. What is the probability that the
product of the visible numbers on the
five faces is divisible by 12?
7. The area of two different rectangles is each 72 square
centimeters. The length of the second rectangle is 5 cm
greater than the length of the first rectangle and its width
is 10 cm less than the width of the first rectangle. Find the
dimensions of the two rectangles.
8. Ming travels up a ski lift at 4 miles per hour (mph)
and skis down the hill at 28 mph. If the hill is the same
exact length as the distance traveled on the ski lift, and we
ignore any time spent at the top, what is Ming’s average
speed for the round trip in miles per hour?
9. A radio DJ announced that the tenth person to submit
an entry into the monthly drawing gets to put in an extra
entry for free. He also said that this extra entry will “double
your chances of winning the grand prize.” Explain whether
putting in two entries will double someone’s chances of
winning the prize compared with putting in one entry.
Vol. 18, No. 8, April 2013
Copyright © 2013 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Prepared by David Rock, [email protected], and Joel Amidon, [email protected], University of Mississippi, Oxford. Teachers,
student groups, or mathematics clubs are encouraged to submit single problems or groups of problems to the editor, David Rock, at
[email protected]. Published problems will be credited.
10. A set of five integers has a mean of 18, a mode of 15,
and a median of 19. What is the largest number in the set?
11. Alan was asked to multiply 5 by 2, add 2 to the product, divide the sum by 4, and add 8 to the quotient. Alan
used his calculator to perform all the calculations in one
step. His incorrect answer was 18.5. What was the right
answer? What did Alan do wrong?
12. Instead of buying packages of 20 batteries for $7.00
to power his video game controllers, Nathaniel decides to
buy 4 rechargeable batteries and a charger for $15.39.
How many times does he need to use the rechargeable
batteries to make them a better value?
13. Charlie has a plant that
doubles in height every
3 days and a wall that is
sinking to 1/3 its height
every 2 days. Currently,
the plant is 3 inches
tall and the wall is
15 feet tall. How
many days will it take
for the plant to surpass
the height of the wall?
14. Students have found that their ticket sales to a dance
can be accurately predicted by the rule
tickets sold = 180 – 10 × price.
Anders suggests that they charge $15 per ticket, so that
students collect $450 on the dance. Sunil thinks they can
make exactly the same amount of money by charging less
and having more people attend. What is Sunil’s ticket price?
15. Raquel has created a program for a robot. The robot
travels 10 meters and stops. The program is designed for
wheels that are 2.5 cm in diameter. When Raquel replaces
the wheels with ones having a 2.75 cm diameter, how
should she adjust the number of rotations so that the robot
travels the same distance?
16. Anna Grace’s old bedroom was a square with integer
dimensions. Her new bedroom is 3 feet wider and 2 feet
deeper. The increase in area is 46 ft.2. What were the
dimensions of Anna Grace’s old bedroom?
(Answers on page 518)
(Ed. note. Online solutions are available to NCTM members only.)
Download one of the free apps for your
smartphone. Then scan this tag to access the
Palette of Problems solutions that are online at
http://www.nctm.org/mtms042.
Vol. 18, No. 8, April 2013
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Mathematics Teaching in the Middle School
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