A Checker Math Problem

[Dennis Cayton]

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

If a + b + c = 0 then

(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]

Greetings Liam:

Many thanks for the above.

This is very interesting.

I plugged in three different sets of values for a, b, and c, in order to test and verify the above.

Sure enough, each set of values satisfied your equation above.

Maybe this idea could be used to compose an original problem on the Mathematics portion of the SAT Reasoning Test.

Composing original SAT Math problems has become a new and recent hobby of mine.

Thanks again!

Best Wishes,

Dennis Cayton

[Dennis Cayton]

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

Alex

Greetings Alex:

I believe it.

I have witnessed firsthand, far too many times, the lack of rigor that is found in the standards of mathematics education in the United States, when compared to those of other countries.

Several years ago, the following "problem" was found in a popular 4th grade mathematics textbook used in the United States:

"If math were a color, it would be the color _____________ because _____________________________."

Mathematics education in the United States has had an obsessive focus upon getting students to "feel good" about math, at the expense of the development of their mathematical knowledge and skills.

I have reviewed several elementary school level textbooks from the "Singapore Math" series, and the content of the math they perform is at least two grade levels ahead of the mathematics curriculums of our American students at comparable grade levels.

I know a high school student who recently came to the United States from Korea, and who told me they attend school until 6:00 p.m. every day.

One day, shortly before I retired from my teaching career, one of my 8th grade "Honors" students came up to my desk and asked me if she could borrow a calculator.

"What for?" I asked.

"I want to figure out 8 times 3", she replied.

I did not let her borrow a calculator. Instead, I just told her to go back to her desk and figure it out for herself.

She went home and told her father, who came up to see the principal of the school the next day. I was called into the office and given a reprimand for not letting her borrow a calculator.

Incidents such as these (and much worse), when endured year in and year out, can have the effect of wearing down a teacher's morale.

Having taught in the public schools for 24 years, I don't see how I lasted as long as I did.

Best Wishes,

Dennis Cayton

[B Salot]

Dennis,

I knew an instructor once who, after putting his thoughts in writing, always asked his students, as you did:

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

And, of course, he would always be sure to include an error or typo to test whether his students were paying attention.

Well, professor, I am not letting your intentional typo get by me. It is in this line copied from your extended explanation:

- 36Y - 80Y = - 2964

Bill Salot

[Alex_Moiseyev]

Mathematics education in the United States has had an obsessive focus upon getting students to "feel good" about math, at the expense of the development of their mathematical knowledge and skills.

WOW, you mirically read my thoughts !

I know a high school student who recently came to the United States from Korea, and who told me they attend school until 6:00 p.m. every day.

It is better to spend time in school rather then Internet or texting or idiotic TV "brainwash" shows !

One day, shortly before I retired from my teaching career, one of my 8th grade "Honors" students came up to my desk and asked me if she could borrow a calculator.

"What for?" I asked.

"I want to figure out 8 times 3", she replied.

I did not let her borrow a calculator. Instead, I just told her to go back to her desk and figure it out for herself.

She went home and told her father, who came up to see the principal of the school the next day. I was called into the office and given a reprimand for not letting her borrow a calculator.

Incidents such as these (and much worse), when endured year in and year out, can have the effect of wearing down a teacher's morale.

Having taught in the public schools for 24 years, I don't see how I lasted as long as I did.

You know what ... with many our disagreements on many other subjects, we are on the same side of barricade on this matter.

I came to USA in 1991, 20 years ago, and I can tell you - today it is not the same country where I came in ! American dream has been changed, attitude has been changed, service and respect dropped below ground zero.

I belong to the 1st American generation who can't tell that their kids will live better than parents.

[Dennis Cayton]

Dennis,

I knew an instructor once who, after putting his thoughts in writing, always asked his students, as you did:

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

And, of course, he would always be sure to include an error or typo to test whether his students were paying attention.

Well, professor, I am not letting your intentional typo get by me. It is in this line copied from your extended explanation:

- 36Y - 80Y = - 2964

Bill Salot

Greetings Bill Salot:

Congratulations!

You caught my intentional typo!



As they say in the textbook proofreading business:

"Nice catch!"

When I was in this line of work, they would usually assign two proofreaders to proofread each chapter independently. The rationale behind this practice was that one proofreader would be more likely to catch mistakes the other proofreader missed, and vice versa.

As a result, I always found mistakes the other proofreader had missed.

Conversely, he always found mistakes I had missed.

After each pair of proofreaders had finished each chapter, with all of their errors discovered and submitted by their final deadlines, then each chapter would be sent to another team of proofreaders for a final look.

In spite of this, the textbooks we proofread with such meticulous care would always still end up being published with errors remaining.

The "perfect" error free mathematics textbook has never crossed my eyes.

Thanks again!

Best Wishes,

Dennis Cayton

[B Salot]

I understand Tinsley also dabbled a little in mathematics.
Bill Salot

[Dennis Cayton]

Greetings:

I have just composed another Checker math problem.

This one is an SAT style problem written in the multiple-choice format.

Here goes:
__________

A square checkerboard is divided into y rows of y squares each. The value of y is an even integer, such that 7 < y < 13. If z is the total number of distinctly different squares which form the perimeter of the checkerboard, then which of the following could be a possible value of z?

I. 28

II. 32

III. 48

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III
__________

I know. This problem sounds convoluted and screwy to me too. But this is one of the types of problems that frequently appears on the SAT.

Let me know if you see any flaws in the wording and/or construction of this problem.

Best Wishes,

Dennis Cayton

[Jolt]

I'm going to take a guess and say that the answer is D, II & III

My reasoning:

1.) A square checkerboard is divided into y rows of y squares each => The number of squares is a perfect square, or y squared
2.) The value of y is an even integer, such that 7 < y < 13 => y can be 8, 10, 12 in which case there would be 64, 100, 144 squares

Finally, perimeter of a square is equal to 4s. The perimeter of an 8 sided square is 32, of a 10 sided square is 40, and of a 12 sided square is 48.

[jaguar72]

(A) I only

V/R,

Gary Jenkins/jaguar72

[Jason Solan]

A square checkerboard is divided into y rows of y squares each. The value of y is an even integer, such that 7 < y < 13. If z is the total number of distinctly different squares which form the perimeter of the checkerboard, then which of the following could be a possible value of z?

I feel like z is too ambiguously defined here. A 1x1 square is distinctly different from a 2x2 square and a 2x2 square that overlaps part of another 2x2 square is still distinctly different.

[jaguar72]

The value of y (as Jolt said) is 8, 10, or 12.

But z the number of "distinctly different perimeter squares" in an 8 x 8 board is 28, in a 10 x 10 board is 36, and in a 12 x 12 board is 44.

The only answer that fits is: (A) I only.

Then again, perhaps I am misinterpreting the entire problem...wouldn't be the first time... .

V/R,

Gary Jenkins/jaguar72

[Dennis Cayton]

A square checkerboard is divided into y rows of y squares each. The value of y is an even integer, such that 7 < y < 13. If z is the total number of distinctly different squares which form the perimeter of the checkerboard, then which of the following could be a possible value of z?

I feel like z is too ambiguously defined here. A 1x1 square is distinctly different from a 2x2 square and a 2x2 square that overlaps part of another 2x2 square is still distinctly different.

Greetings Jason:

Thank you very much for your feedback. You have made an excellent point.

My use of "distinctly different" was intended to serve as a hint NOT to count the corner squares twice and to count each of the perimeter or border squares once.

If a given checkerboard has y rows with y squares in each row, this tends to suggest that each of the squares are congruent. But maybe not.

Perhaps, some greater clarity could be added to this problem by specifying that each of the squares are congruent to each other.

If you have any suggestions as to how to add some greater clarity to this problem, especially with respect to the defnition of z, please feel free to do so.

Thanks again for your feedback.

Best Wishes,

Dennis Cayton

[Jason Solan]

The value of y (as Jolt said) is 8, 10, or 12.

But the number of "distinctly different perimeter squares" in an 8 x 8 board is 28, in a 10 x 10 board is 36, and in a 12 x 12 board is 44.

The only answer that fits is: (A) I only.

Then again, perhaps I am misinterpreting the entire problem...wouldn't be the first time... .

V/R,

Gary Jenkins/jaguar72

Thats how I interpreted the problem. My issue is that those are only 1x1 squares. There are also 4 (non overlapping) 4x4 squares that could make up the perimeter and (arguably) 1 8x8 square.



Dennis,
I'm not sure of better clarity, I just tend to overthink things when presented with problems. I saw what I thought you were going for and then overthought it and thought others might as well.

[Dennis Cayton]

(A) I only

V/R,

Gary Jenkins/jaguar72

The value of y (as Jolt said) is 8, 10, or 12.

But z the number of "distinctly different perimeter squares" in an 8 x 8 board is 28, in a 10 x 10 board is 36, and in a 12 x 12 board is 44.

The only answer that fits is: (A) I only.

Then again, perhaps I am misinterpreting the entire problem...wouldn't be the first time... .

V/R,

Gary Jenkins/jaguar72

Hello Gary Jenkins,

Congratulations!

You interpreted and answered the problem correctly.

Perhaps, I could have worded the latter half of the problem with greater clarity, by specifying that each of the squares within the checkerboard are congruent.

I might add that to my next revision.

The key to this problem was to recognize that the 4 corner squares are not to be counted twice.

For any given checkerboard, with y rows and y congruent squares in each row, then the number of these congruent squares which form the border or perimeter of the checkerboard can be solved by the the following:

4y - 4

For a standard 8 X 8 checkerboard, where y = 8, then the number of non-repeating border or perimeter squares is calculated as follows:

4 ( 8 ) - 4
32 - 4 = 28

ANSWER: CHOICE A (28)

None of the other choices work for the 10 X 10 or the 12 X 12 checkerboards.

Congratulations once again!

Best Wishes,

Dennis Cayton

[jaguar72]

Thats how I interpreted the problem. My issue is that those are only 1x1 squares. There are also 4 (non overlapping) 4x4 squares that could make up the perimeter and (arguably) 1 8x8 square.

Quite true, Jason. It's a good nuance but it did not occur to me at all. I just looked at the problem as being somewhat more straightforward than that.

V/R,

Gary Jenkins/jaguar72