Mth280 – Problem Solving Portfolio – Problem 5
From the Mth280 syllabus: A major method for assessing and facilitating your development of problem
solving skills will be a problem solving portfolio. The portfolio will contain a mix of assigned and student‐
selected problems. Assigned problems will be assigned sporadically throughout the semester. The final
entry will be a problem solving project of your own design.
Your portfolio is designed to demonstrate that you possess several key characteristics of effective
problem solvers:
1. Effective problem solvers have a variety of specific strategies available, and they can use
them appropriately to make progress on challenging problems.
2. Effective problem solvers are persistent and strategic when facing challenging problems.
3. Effective problem solvers find creative ways to represent the problem and to communicate
their results.
Your fifth (and final) assigned entry is described below. You may supplement with additional entries
from class if you feel it will show your growth and/or problem solving prowess.
You will also complete a problem solving project of your own design. Ideas for this project are on the
back of this sheet.
Further details on expectations for portfolio problems will be distributed next week.
Note: One way to address all three of the items above is to provide two or more different ways to solve a
problem (each using a distinct problem solving strategy). For instance, you might first put your ‘guess &
check’ hat on and show how that method leads to a solution. Then, put on your ‘solve a simpler related
problem’ hat and show how that method leads to the same solution but from a different angle. Be
thinking about this note as you consider the following task.
5. Precious gems. Shortly before the old king died, he willed that the kingdom’s 14 precious gems
would be divided among his four children, whom we shall call A, B, C, and D. The king did not specify
that this should be done fairly – indeed, he was very clear that the coins could reasonably all go to
one person, so long as all 14 precious gems were distributed. In how many ways can this be done?
(Please try to make some progress on this problem by Wednesday, 4/14, so that you can contribute
to a discussion about it at that time.)
Final Portfolio Entry – Your ‘problem solving project.’ The final entry in your portfolio will be a problem
solving project of your own design. A selection of project ideas appears below. Feel free to bring in
concepts, problems, or ideas from your other mathematics coursework! I can provide additional
potential topics, as needed.
Mth280 Project Ideas:
1. A certain prescription drug has an ‘elimination half life’ of n hours (you can find this number online
for many common drugs). This number represents the number of hours it takes for your body to
eliminate 50% of the drug from your system. If regular doses of p milligrams are taken every t hours,
calculate the level of the drug in the blood stream after d days. By how much does the drug level
fluctuate from dose to dose?
2. There are n points on a circle. A straight line segment is drawn between each pair of points. How
many intersections are there within the circle if no 3 lines are concurrent?
3. What is the largest power of 7 that will divide 1000! ?
4. What number, if divided by 10, leaves a remainder of 9; divided by 9 leaves a remainder of 8;
divided by 8 leaves a remainder of 7; … ; divided by 2 leaves a remainder of 1. One answer is
14,622,042,959. Find a smaller solution. Can you find the smallest solution?
5. In the history of college football’s Rose Bowl, no team’s final score has ever been 11 points. How
many different ways are there for a team to score 11 points?
6. Find a way to create a new square from the two squares shown.
For a hint, see http://www.figurethis.org/challenges/c45/challenge.htm#hint
7. Look in your cupboard. What combinations of cereal, milk, bread, and peanut butter
could you eat to get the minimum daily requirements of vitamins and minerals?
http://www.figurethis.org/challenges/c58/try_these4.htm
(Note: this is one of the follow‐up questions to the following problem at FigureThis.org:
“How much does each fish weigh?”
8. The Nation’s Debt. The United States government throws around numbers like 1 billion and 1
trillion as if we all know what they mean. For instance, this year the national debt surpassed $12
trillion, we allocated $700 billion to bail out Wall Street, and NASA’s budget was increased by $903.6
million to a total of about $18.7 billion. Few people have any idea how any of those numbers are.
Here are a few problems to help you get a handle on the size:
a. If dollars were seconds… Where were you 1000 seconds ago? 1 million seconds ago? 1
billion seconds ago? Were you even born? (It was 31 years ago.) How about 18.7 billion
seconds ago (that’s NASA’s budget, if dollars were seconds)? And what was going on in
the world 700 billion seconds ago? How about 12 trillion seconds ago?
b. If dollars were inches… Driving at 75 mph, how long would it take you to travel 1000
inches? 1 million inches? 1 billion inches? How about 12 trillion inches?