A Checker Math Problem

[Dennis Cayton]

Greetings:

As some of you know, I am a retired public school math teacher, who now owns a private tutoring business. This is strictly a solo practice, which I have found to be the ideal retirement job.

Recently, I composed the following problem and assigned it to my Algebra students:
__________

During the 1st Round of a Checker tournament, three different styles of Checkers were played:

1) English Checkers
2) International Checkers
3) Canadian Checkers

English Checkers is a game played between two players on a 64-square checkerboard. Each player starts the game with 12 checker pieces, consisting of one of two contrasting colors, such as red and white.

International Checkers is a game played between two players on a 100-square checkerboard. Each player starts the game with 20 checker pieces, consisting of one of two contrasting colors, such as red and white.

Canadian Checkers is a game played between two players on a 144-square checkerboard. Each player starts the game with 30 checker pieces, consisting of one of two contrasting colors, such as red and white.

During the 1st Round of this Checker tournament, a total of 100 matches were played. Each match consisted of a total of 4 games, and each game was played to a final result.

Once again, each of these three different styles of Checkers were played in this tournament.

A total of 3,728 checker pieces, with no extra pieces remaining, were used to form the opening settings for all of the games played in the 1st Round of this tournament. Half of these checker pieces were red and the other half were white.

The total number of squares on all of the 100 checkerboards used in the 1st Round of this tournament was 9,364.

During the 1st Round of this tournament:

1) How many games of English Checkers were played?

2) How many games of International Checkers were played?

3) How many games of Canadian Checkers were played?

Show all of your work and justify each step.
__________

One of my students has already solved this problem correctly.

Please feel free to give it a try.

In addition, please feel free to critique this problem, as I am thinking about submitting it for publication. Therefore, any suggested revisions, including corrections of any inaccuracies, would be welcomed and appreciated.

If the wording of this problem strikes you as too verbose, I can understand.

All too often, however, mathematical "word problems" assume too much prior information by the students. As a result, a lot of vital information needed to solve such problems is left out, due to the fact that the student is assumed to know this information.

Sports problems are notorious for this sin of omission. I learned this firsthand, during my last retirement job, as a proofreader for mathematics textbooks.

Therefore, when composing a mathematical "word problem", it is often better to err on the side of being too wordy than not being wordy enough.

In closing.......

If you do attempt to solve this problem, have fun!

Good luck to all.

Best Wishes,

Dennis Cayton

[kiwinurse]

Everyone will be driven silly Dennis trying to figure it out.I think you should give the correct answer,after we all have tried,only ill have to wait until next week,as i have a new zealand national to attend in lower hutt,starting on friday.take care,jan

[Alex_Moiseyev]

English Checkers - 188
International Draughts - 116
Canadian Checkers - 96

1) Instead International Checkers you have to say "International Draughts"

2) In order to confuse solver more you can add to text extra unneccessary information about "black-white" squares and say that game is played only on black squares.

Overall - very nice math problem, congratulations sir !

In Russia students of 6th grade were able to solve this.

Regards,

Alex Moiseyev

[Alex_Moiseyev]

For those who were not graduated in Russian 6th grade I can help:

X - English Checkers number of boards, 4X - English Checkers number of games
Y - International Draughts number of boards, 4Y - International Draughts number of games
Z - Canadian Checkers numer of boards, 4Z - Canadian Checkers number of games

1) X+Y+Z = 100
2) 24X + 40Y + 60Z = 3,728
3) 64X + 100Y + 144Z = 9,364

X = 47, 4X = 188
Y = 29, 4Y = 116
Z = 24, 4Z = 96

[Danny_Alvarez]

i thought i was really smart and i had the answer ..... luckily i checked the posts and i see that Alex had the answer b4 me. he just expressed it in a different way. I have been looking for an alternate answer but im being bombarded by phonecalls right now.

i was working on number of matches trying to find a total that would fit 100 matches ....

so my answer is 47 matches for english checkers
29 matches for international
and 24 matches for canadian

i guess im not as fast as Alex ....
i wish i could tell you i did it quickly, it took me an hour and a lot of trial an error....
i started with what i thought was the obvious answer only to realize once i used a calculator that i was wrong by 4
41 english x 64
+ 40 international x 100
+ 19 canadian x 144 = 9360
the rest i worked from there trying to find a lowest common denominator and playing with the calculator

by the 3rd or fourth try i found it

Cheers Mr Cayton and Cheers Alex i would be interested in your method mate.

Danny Alvarez

[Alex_Moiseyev]

i would be interested in your method mate.

In Russia we called it "system of linear equations" I used an easy and classic Gauss method (optmization) to solve the system of equations because number of variables (3) was equal to number of equations (3) However for those who is familiar - matrix method to solve system also can be used.

Again, can you beleive or not, but we studied Gauss method in Russia in 6th grade.

I seriously suspect that one of 3 conditions in text is unneccessary, because there is one more "hidden condition" inside text - number of games could be integer only. This extra hidden condition will be an important component which could help us.

Just guessing ... Dennis can check.

Alex

[Danny_Alvarez]

my apologies Alex i hadnt seen ur second post where u explained in detail ur method....
i think my browser is playing up....
when i was about to post answer i couldnt see any posts and i thought i was first hahaha until i refreshed and saw ur first post

cheers and well done
Danny

[Alex_Moiseyev]

well done

Thanks. Math was my 2nd degree in college (first - computer science), my mom was a professional school math teacher for almost 40 years and my old son had bachelour degree (Carnegie Mellon University) in math. Math was always an important part of my life.

In college in Russia I was a volunteer for teaching disabled talented kids by advance math and ... this was a place (volunteer department) where I met my wife ! Ohhh ... 32 years ago.

[Danny_Alvarez]

I have always done the highest math level available due to my parents insistance that anything else wouldn't be acceptable.
in university i only got around to doing a bit of basic statistics .... but in general i have been averted to maths . I have always preferred chemistry and biology (but as usual maths is pretty much present in all the sciences in one way or another) .

32 years is a long time and is a test of true love, congrats to you both
I will personally be lucky if my wife puts up with me till tomorrow

cheers
Danny

[liam stephens]

Well done Dennis - very interesting. Here is another problem that is
Simple as A B C:

If a + b + c = 0 then can you prove that

(a x a x a) + (b x b x b) + (c x c x c) = (3 x a x b x c)

[or a(cubed) + b(cubed) + c(cubed) = 3abc]

[Dennis Cayton]

English Checkers - 188
International Draughts - 116
Canadian Checkers - 96

Greetings Alex:

Congratulations!

All of your answers above are correct.

I had a feeling you would be the first one to solve this problem if you saw it right away.

Instead International Checkers you have to say "International Draughts"

I debated with myself about this very issue.

Finally, for the sake of consistency, and because my target audience for this problem would be American math students, I decided to use the more familiar "Checkers" term for all three styles of play.

Overall - very nice math problem, congratulations sir!
Alex Moiseyev

Thank you, Alex!

Best Wishes,

Dennis Cayton

[Dennis Cayton]

i thought i was really smart and i had the answer ..... luckily i checked the posts and i see that Alex had the answer b4 me. he just expressed it in a different way. I have been looking for an alternate answer but im being bombarded by phonecalls right now.

i was working on number of matches trying to find a total that would fit 100 matches ....

so my answer is 47 matches for english checkers
29 matches for international
and 24 matches for canadian

i guess im not as fast as Alex ....
i wish i could tell you i did it quickly, it took me an hour and a lot of trial an error....
i started with what i thought was the obvious answer only to realize once i used a calculator that i was wrong by 4
41 english x 64
+ 40 international x 100
+ 19 canadian x 144 = 9360
the rest i worked from there trying to find a lowest common denominator and playing with the calculator

by the 3rd or fourth try i found it

Cheers Mr Cayton and Cheers Alex i would be interested in your method mate.

Danny Alvarez

Greetings Danny:

Congratulations!

You are the 2nd person to solve and find the correct number of matches played for each of the three styles of Checkers played in the 1st Round of this tournament.

The only thing you overlooked was the total number of "games" played for each style of Checkers, which is a simple matter of multiplying the number of each type of match played by 4.

You did very well to solve the number of matches played for each style of Checkers in an hour, especially if your primary methodology was one of "trial and error."

Once again, congratulations!



Best Wishes,

Dennis Cayton

[Dennis Cayton]

For those who were not graduated in Russian 6th grade I can help:

X - English Checkers number of boards, 4X - English Checkers number of games
Y - International Draughts number of boards, 4Y - International Draughts number of games
Z - Canadian Checkers numer of boards, 4Z - Canadian Checkers number of games

1) X+Y+Z = 100
2) 24X + 40Y + 60Z = 3,728
3) 64X + 100Y + 144Z = 9,364

X = 47, 4X = 188
Y = 29, 4Y = 116
Z = 24, 4Z = 96

Greetings Alex:

You have provided an excellent introduction to the solution to this problem above.

Please allow me to expand upon this solution as follows:
__________

Let X = The number of English Checker matches played.
Let 4X = The number of English Checker games played.
Let Y = The number of International Checker (or Draughts) matches played.
Let 4Y = The number of International Checker (or Draughts) games played.
Let Z = The number of Canadian Checker matches played.
Let 4Z = The number of Canadian Checker games played.

This brings us the system of three linear equations that Alex presented, and which I have repeated below:

1) X+Y+Z = 100
2) 24X + 40Y + 60Z = 3,728
3) 64X + 100Y + 144Z = 9,364

Note that the 2nd equation addresses the number of checker pieces for each style of Checkers, and that the 3rd equation addresses the number of squares on each of the three different sizes of the 100 checker boards.

There are many methods for solving this problem. What I present below is a purely algebraic "paper and pencil."

During my teaching career, I did not permit my students to use calculators to solve problems such as this one.

First, I will choose a variable to "attack" and "eliminate." I will choose the variable X for this purpose.

I will accomplish this by multiplying every term in the 1st equation by - 24:

- 24 ( X + Y + Z = 100 )

The result is:

- 24X - 24Y - 24Z = - 2400

I now combine this equation with the 2nd equation:

- 24X - 24Y - 24Z = - 2400
24X + 40Y + 60Z = 3728

This eliminates the variable X and results in the following equation:

16Y + 36Z = 1328

I will call this "New Equation Number One."

Next, I execute the same strategy with the 1st equation and the 3rd equation, by multiplying each term of the 1st equation by - 64 and combining it with the 3rd equation:

- 64 ( X + Y + Z = 100 )

This results in:

- 64X - 64Y - 64Z = - 6400

This is now combined with the 3rd equation:

- 64X - 64Y - 64Z = - 6400
64X + 100Y + 144Z = 9364

The result is:

36Y + 80Z = 2964

I will call this "New Equation Number Two."

Notice that the variable X has been eliminated for each of these two "New" equations. This gives us a reduced system of two equations in two variables.

I will now combine these two "New" equations together:

16Y + 36Z = 1328
36Y + 80Z = 2964

Each of the coefficients and constants above share a common factor of 4. Therefore, dividing each term by 4 results in the following equivalent system of equations:

4Y + 9Z = 332
9Y + 20Z = 741

My strategy is now to "attack" and "elminate" the variable Y. I can accomplish this by multiplying each term of the top "New" equation by 9 and each term of the bottom "New" equation by - 4:

9 ( 4Y + 9Z = 332 )
- 4 ( 9Y + 20Z = 741 )

This results in the following:

36Y + 81Z = 2988
- 36Y - 80Y = - 2964

This step eliminates the Y variable and solves for Z as follows:

Z (The total number of Canadian Checker matches ) = 24
4Z (The total number of Canadian Checker games ) = 96

I will now substitute Z = 24 for one of the two "New" equations:

4Y + 9Z = 332
4Y + 9 ( 24 ) = 332
4Y + 216 = 332
4Y = 116
Y (The total number of International Checker or Draughts matches) = 29
4Y (The total number of International Checker or Draughts games) = 116

We now know the following:

Y = 29
Z = 24

We can now substitute these two values into our 1st original equation:

X + Y + Z = 100
X + 29 + 24 = 100
X + 53 = 100
X (The total number of English Checker matches) = 47
4X (The total number of English Checker games) = 188
_________

There you have it.

Please forgive me for the long and tedious solution.

This is the "teacher" in me.

Please let me know if you spot any errors or typos.

Have to run.

I have a tutoring session to attend to in less than 30 minutes.

Best Wishes,

Dennis Cayton

[Alex_Moiseyev]

Nice explanation indeed ! it's defined here.

http://www.sosmath.com/matrix/system1/system1.html

I don't have anything to add to your excellent lesson, maybe just mention that this method is called "Gauss elimination" (see link above) and also give a simple verbal definition of Linear Equations and situation when the system of linear equations doesn't have solution or have multi-solutions

[Jolt]

Three variables usually means three equations. If you use the Gauss-Jordan Elimination with the help of a TI-89

1) X+Y+Z = 100
2) 24X + 40Y + 60Z = 3,728
3) 64X + 100Y + 144Z = 9,364


Matrix:
1 1 1 100
24 40 60 3,728
64 100 144 9,364

Reduces to:
1 0 0 47
0 1 0 29
0 0 1 96

This problem is very similar to solving for the unknowns in DC circuits using Kirschoff's circuit laws. Using matrices is the way to go when dealing with multiple variables. Definitely better than trying to figure out if one variable can be canceled or by substituting multiple times.