Puzzle – How many rectangles on a chess board? – Solutions
April 21, 2007 pm30 10:07 pm
This is the place for solutions to the rectangles on the chess board problem. The original problem is here.
Education, Math, Teaching, New York, Bronx, Union, Language, Travel
Categories
Recent Comments
- Anonymous on UFT Welfare Fund nest egg – bigger than most nests
- UFT Welfare Fund nest egg – bigger than most nests | JD2718 on Health Care Savings
- Anonymous on The Company You Keep
- Anonymous on The Company You Keep
- jd2718 on Chat GPT likes rhyming (and has trouble counting)
- 60 | JD2718 on Advice for myself
- 60 | JD2718 on Race
- 60 | JD2718 on Review and Resolutions
- ✡David✡ on Chat GPT likes rhyming (and has trouble counting)
- Anonymous on Do Not Apply: PS/MS4
- jd2718 on Feeling Congested? Pricing’s Not the Answer
- jd2718 on Chat GPT likes rhyming (and has trouble counting)
- New York City’s Hidden History – Freedom Day | JD2718 on The Day After NYC Freedom Day
- New York City’s Hidden History – Freedom Day | JD2718 on Freedom Day – NYC – February 3, 1964
- Does your dog bite? | JD2718 on Learning Loss
Archives
- #BlackLivesMatter 12-126 Administrative Code 12-126 AFT AFT Convention Algebra Regents American Federation of Teachers Andrew Cuomo ATRs Barack Obama Bill De Blasio blended learning blog carnival Carnival of Mathematics Carranza Common Core COVID Covid-19 Cuomo de Blasio elections Eric Adams health care health insurance Hillary Clinton Integrated Algebra Integrated Algebra 2 Regents Integrated Algebra Regents Integrated Geometry Integrated Trigonometry Lynne Winderbaum Math B Medicare Medicare Advantage Medicare Advantage Plus Michael Bloomberg Michael Mulgrew MLC MORE Caucus Mulgrew MulgrewCare municipal labor committee New Action New York State Education Department NYCCoronavirus NYC Mayoral Election NYSED NYSUT problem solving PSC Randi Weingarten Regents Regents exams remote learning remote teaching retirees Richard Carranza safety school closings School Reopening school safety school scheduling summer vacation teacher evaluation Teaching Fellows tenure UFT Delegate Assembly UFT elections UFT Executive Board United for Change United for Change / UFT Unity Unity Caucus US Presidential Election US Presidential Elections
Creative Commons License
Technorati
Meta
-
Recent Posts
- UFT Welfare Fund nest egg – bigger than most nests
- The Company You Keep
- 60
- New York City’s Hidden History – Freedom Day
- Does your dog bite?
- No Battle for Second Place
- Days are getting longer – are the weeks?
- Review and Resolutions
- Interview with PEP member Tom Sheppard, Part II
- Feeling Congested? Pricing’s Not the Answer (repost)
- 2023 – Looking Back
- Fighting Tuition Increases at CUNY, back in the day
- UFT Retired Teachers Chapter Election – coming this Spring
- A simple way to understand the UFT Healthcare Conflict
- Help me create a puzzle
A+ Favorite Posts
- A little math magic
- Bonus Question
- Content-free logic test
- Graduation Speech 2007
- I ban FOIL
- NYCDOE: lousy mini-schools, by the dozen
- Pedagogy: Outlook on teaching math
- Puzzle: proving a quadrilateral is a parallelogram
- Reviewing homework in math class
- The equine inequality
- Too much algorithmic honesty
- Whose schools? Our schools!
Education
Math
Math Education
Math Teachers
New York City
NY Teachers (except math)
Puzzles
Retired Blogs
Teachers Unions
The Wide World
JD2718 
A rectangle on a chess board is bounded by two vertical grid lines and two horizontal grid lines. Since a chess board has nine vertical and nine horizontal grid lines, the number of rectangles is C(9,2) * C(9,2) = 1296.
ACTUALLY THERE ARE 8 HORIZONTAL AND VERTICAL LINE
count it, there are 9, since there are 8 boxes
soumya, you are very stupid
allah has spoken!
u are very very stupid
soumya i dint know u r so dumb.
you lack basic knowledge in mathematics.
stupid soumya!!
a** h**** soumya
hehehe bas naam hi kaafi hai
be quite son of b****
Seriously? COUNT THEM!!!!
u r right
David has given the most easy and effective solution, what for all these stupid and confusing discussion……………….
hi dosto
Lovely answer David! The problem puzzled me for a long time.
thanx david!
Also, an alternative soln is summation of cubes from 1 to 8 ie (1+8+27+64+125+216+343+512 =1296)
-Jyoti
no there is not 1296 rectangle there is 1296-204=1092 rectangle
N3-N2 where n=1 2 3……8
1+8+27+64+125+216+343+512-(1+4+9+16+25+36+49+64)=1092
I didn’t understand your logic of summation. Why did you do this?
You’re right, adding the cube numbers. :)
thanx david guetta for the answer
great david !!!!
How do you work out c(9*2) ?
how do you do c(9^2) ?
c(9,2) would be combinations of nine things, taken two at a time. (9 x 8) / (1 x 2) = 36. How do you get it? There are 9 lines (dividing 8 rows). Take any two of them…
Very cool – a brand new approach for me. I have a few others, but this puzzle doesn’t seem to have caught my usual puzzlers’ imaginations. I was waiting for more response.
You can do it the same way as the square problem. The nth bottommost, mth rightmost square is the northwest corner of mn rectangles. The nth row has northwest corners of (1 + … + 8)n = 36n rectangles, so in total there are 36(1 + … + 8 ) = 36^2 = 1296 rectangles.
And in general, on an m*n board there are mn(m+1)(n+1)/4 squares, the same answer you’d get by generalizing David’s approach but by a different method.
My first solution was by dimensions. Consider a 3×3 chessboard.
There are 9 1×1’s, 6 2×1’s, 3 3×1’s
There are 6 1×2’s, 4 2×2’s, 2 3×2’s
There are 3 1×3’s, 2 2×3’s, 1 3×3
Trotting out the LaTeX, that’s
, which equals
From there we reach
, or,
as you’ve generalized for any m x n rectangle,
soumya…..u r really really stupid….
n jd2718 ur very very very genious………….
wt abt triangel gerneralised formula..?
Incidentally, this generalizes to n-dimensional boards. A board of size m1*m2*…*m(n) has m1(m1+1)…m(n)(m(n)+1)/2^n n-dimensional rectangular polytopes.
204 squares
Why are ther 204 squares in a chess board?
Because we count all the different sized squares, not just the little ones.
i make it 1365
there are 1296 rectangle in an chess board
i worked out a way an i got 1278
there are 1296 different rectangles in a chessboard..
how many squares and rectangles aee in a chess board…………………………………………………………………………………………………………………………………………………………………………
1296 rectangles + 204 squares
The easiest way to work this out, is to see a patern. So, see how many sqaures and rectangles there are in a 1 by 1 grid and add up total. Answer is 1. do that for 2,3,4,5,6,7 and 8. It may take a while but you will get the answer in the end.
too slow…for my homework, i get a third of a page to explain it
In a standard chessboard, you have 9 horizontal and 9 vertical lines. Choose any two horizontal and any two vertical lines to form a rectangle. Both of them can be chosen in 9c2 ways. So the total number of rectangles is 9c2 * 9c2, one for horiznotal, and one for vertical. Simple
can u explain the same for square…???
That’s gorgeous, thank you.
And, it generalizes eg how many boxes in a Rubik’s cube? (3 x 3 x 3)
dumm puzle
How many different rectangles which are not squares can be found on a chess board? ( a chess board has 8 rows and 8 columns of squares. a rectangle on the board is a collection of squares that form a rectangular piece of the whole board. Two rectangles are different if they have different sets of squares. A rectangle with an unequal number of rows and columns is not square.)
Hi hema, your question is part of the current CUNY math challenge – it wouldn’t be fair for me to answer.
I started from a 1*1 chess board and worked my way up to 1 4*4 chess board before I spotted a pattern.
1*1 = 1 square ‘a’
2*2 = 9 squares ‘b’
3*3 = 36 squares ‘c’
4*4 = 100 sqauers ‘d’
Let us call the number that is beign squared ‘n’.
I saw that the differences between n each time was going up by 1. For example:
number of squares on ‘a’ = 1
= 1*1
number of squares on ‘b’ = 9
= 3*3
number of squares on ‘c’ = 36
= 6*6
number of squares on ‘d’ = 100
= 10*10
The sequence is 1, 3, 6, 10 and therefore the differences are 2,3,4, and so I carried on with the sequence by adding 1 to the differences until I worked out that the total numbrer of rectangles on a chess board must be:
28*28 = 784
The number was lower than I expected, but I have yet to test out the hypothesis.
1296
there are 1296 rec. on chess board
there r 1296 rec. on chess board
and there are 204 squares
YAA Adding this to my bookmarks. Thank You
YAA Adding this to my bookmarks. Thank You
there are thousands of rectangles in a chess boards…
and 1296 is the number of squares..
but if u concentrate the number of rectangles…then it wud b lot more..
No, Ashish. There are 204 sqaures and 1092 rectangles and you add them together to get 1296.
there is 1…the board itself
Absolutely. But there are others as well.
What does 9c2 mean? Im really confused and trying figure this problem out but it is hard not understanding what everything means.. Can someone help, please!
9C2 is combinations of 9 things taken 2 at a time. For example, how many ways could you choose two books out of nine possible? We also write C(9,2) or
or 
there are 1296 – 204 = 1092 rectangles..
Finally the Answer I have been looking for ..
Out of those 9c2 * 9c2 quadrilaterals, some are squares…So no.of rectangles = no.of quadrilaterals – no.of Squares
1296 rectangles in a chess board
Hey pleas say hw its 1092 rectangles…explain
all square are rectangles but all rectangles are not square so the answer is not 1092 it is 1926
there is a nice solution to the how many squares or rectangles are on a chessboard problem here
http://puzzles.nigelcoldwell.co.uk/twentyseven.htm
with diagrams
hello champions
i think dat problem is related to combination. actually the wright answer is 1296.
thanks david u r so brilliant.
hi stop rude shutt your bitch fat gob
there are 1926
I figured it out in grade 3, honestly. I did it the long way but found a few patterns, and I got 1296 or 32 squared. I am very proud. :)
I found patterns when I figured it out in grade 3 so I didn’t take, like two years. I am NOT lying.
salo bakwas kr rhe ho ans kyo ni dete
288
64 squares
512 rectangles
1296
9C2 *9C2
1296
I’m stuck and this didn’t help at all
there is 1092 rectangles in chess board
and there is 204 square in chess board
respect sirs, i got there are 1296 rectangles on a chess board. but i think it is 1 less means 1295. can you please give me the reply? my email id= mail4sajee@gmail.com
248 squares
248 squares mail id :shivagupta1011@gmail.com
Stupid where is your solution