# Category Archives: Problem Solving

Activities to help students learn how to solve math problems

# Sharin’ the Drivin’

Last summer my two older brothers and I took a long vacation trip, prostate driving through many states of our great country, sightseeing along the way. We took turns doing the driving for the 2,268 miles. The amount of time that each of us spent at the wheel came out to be in the ratio of our respective ages. Upon completing our grand tour, we calculated that we traveled at an average speed of 36 miles per hour.

If we brothers were born in consecutive years, what was the year of my birth?

# A “Mean” Product

My mother once taught in a small country three-room schoolhouse, ampoule where students of different age levels often had to study the same course together. For the course in American History, salve the kids in the 6th, 7th, and 8th grades were combined into one class. This means that their ages were between 10 and 14.

If the product of their ages was 41,760,576, what was the average, or arithmetic mean, of their ages?

# The Battle of the Books

Andrew the Assured, Brian the Big, Casey the Careful, Dootles the Dreamy and Edwina the Eager are all members of a reading club, ‘The Burping Bookworms’. They have a bizarre rule in the club: for each 5-person group, they must read 5 books in week, but in a special way–after they finish reading each book they must exchange them within the group, so that after several switches, each of them could finish reading the same 5 books at the same time. (Suppose they could all read at the same speed and swap books at the same time.)

After a pleasurable week of switching-and-reading, Casey the Careful was chatting with her friend, ‘3M’, Mazy the Mathematics Maniac.

Suddenly, Casey piped up, ‘You’re a math frenzy, Mazy, and you know about our ‘Burping Bookworms’ reading club, so I’ll give you a puzzle to solve.’

Casey took out a sheet of paper on which she jotted down facts:

1. The last book Andrew read was the second book Brian read;
2. The last book Casey read was the fourth book Brian read;
3. The second book Casey read was handed to her by Andrew;
4. The last book Dootles read was the third book Casey read;
5. The fourth book Brian read was the third book Edwina read;
6. The third book Dootles read was the book Casey had borrowed from the library (and read) in the first place.

‘Could you figure out the sequence in which we read these 5 books?’

Well, as a matter fact, Mazy was a real math maniac, and in a short time she was proudly handing her work over to Casey. Can you?

This problem was sent to WTM by Wanting Zhong, our friend from China.

# Murals by Max

Maximillian Smith is a modern painter of murals. His preferred size of designs are those painted on large rectangles whose length-to-width ratio is 2-to-1.

Two of his recent works of art have a unique mathematical connection with his daughter, Mindy, who became a for-real teenager just last year. You see, the numerical difference in the areas of these two paintings (as measured in square meters) is equal to Mindy’s age.

What is the sum of the two areas?

# A Little Bite of Pi

It is well known by all math students that  is an irrational number. This means in simple English that its decimal form, troche which begins 3.141592…, goes on forever without any repeating blocks of digits.

To celebrate Pi Day this year – March 14, or 3/14 – we propose the following problem.

Beginning with the “decimal” digits (i.e. those that come after the initial 3), write the first 100 digits on a chessboard, one digit at a time in the squares, returning to the first square when necessary. See the diagram for how to begin.

1. In which square of which row of the chessboard will the 100th digit be placed?
2. What is that digit?
3. What is the sum of the digits that are together in that square?

EXTRA: As this year is 2004, repeat items #1 and #2 for the case of the 2004th digit of .

# Sigma of P(n)

Let’s define a function over the non-negative integers in the following manner:

1. P(n) = n when n is a one-digit integer.
2. P(n) = the product of all the digits of n when n > 9.

Example: P(1729) = 126, because 1 × 7 × 2 × 9 = 126.

Evaluate the following:

Note: this problem was shared to WTM by Reza Kassai, of Shiraz, Iran.

# Arthur’s Arithmogons

Arthur has invented a new puzzle to show his friends. He calls it: Arithmogons.

He draws the following diagram first.

Then he puts any three integers he likes in the circles. In the squares he puts the sum of whatever two integers appear on the ends of the corresponding side.

Before showing this to a friend, he erases the integers in the circles. He tells his friends to find the circle numbers, knowing only those in the boxes.

Here is a sample of Arthur’s Arithmogons for you to solve.

EXTRA: while it’s possible to solve an Arithmogon puzzle by guess-&-check, it would be much more efficient to develop, and prove, a logical step-by-step approach. Such a method exists. Please find it and prove why it works.

# Time Is Power!

While taking a coffee break, my secretary, Sue, happened to glance at her digital watch. It showed the following time:

 1 : 4 4

“That’s rather curious,” she thought. “If I remove the two dots, I’ll have the number 144, which is the square of 12. I wonder how likely that sort of thing happens during my workday?”

Assuming that Sue works from 7 a.m. until 7 p.m., what is the probability that her watch will show a square number? Express your answer to the nearest hundredth of a percent.

EXTRA: what is the probability that a cube, 4th power, 5th power, etc. will occur in addition to squares?

SUPER-EXTRA: what is the probability for a power of any kind appearing on a watch set to a 24-hour day? (This means, times can range from 0:00 to 23:59. And “first” powers are not allowed throughout this problem.)

HYPER-EXTRA: if we now use the “seconds” digits that appear on many watches, what is the largest square number that can occur? Largest cube?

# Omega Numbers

A popular problem in mathematics classes about problem solving concerns finding the unit’s digit of a large power of a number. An example of this might be:

Find the unit’s digit of 24000.

Of course, the student solving this is not expected to compute 2 used as a factor 4000 times. The reasons should be obvious. Rather the solver begins by looking for patterns, and armed with that knowledge, deduce the answer in a simple, straight-forward manner.

We propose now the following variation on this theme:

State the 2-digit number formed
by the final pair of digits of 22004.

Please note: use of a simple calculator (with 8- or 10-digit displays) is permitted, however, such computing aid is not really even necessary. What is not permitted is the use of high-powered computing software, such as Mathematica.

# Portioning Out Peanuts

I have three younger brothers, buy cialis whose names are (in order by age) Gary, ailment Corky, shop and Steve. Though we have our natural differences, we do agree on one thing: we love to eat salted peanuts while watching football on TV!One Sunday I brought out a big bag of delicious salted, roasted peanuts to share with them. Being the math “guru” of the family, I decided on the following unique way to portion out the peanuts. I gave one-third of the quantity to Gary, one-fourth to Corky, and one-fifth to Steve, keeping the remaining portion for myself.

Though I received what was left after giving to my brothers, that does not necessarily imply that my portion was the smallest.

How did my portion rank in size, from most to least (1st, 2nd, 3rd or 4th)?

Extra: If I received between 50 and 60 peanuts as my share, how many peanuts were in the bag?